Random Motzkin paths near boundary

Włodzimierz Bryc Włodzimierz Bryc
Department of Mathematical Sciences
University of Cincinnati
2815 Commons Way
Cincinnati, OH, 45221-0025, USA. [email protected] , Alexey Kuznetsov Alexey Kuznetsov
Department of Mathematics and Statistics
York University, 4700 Keele Street
Toronto, Ontario, M3J 1P3, Canada [email protected] and Jacek Wesołowski Jacek Wesołowski, Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warszawa, Poland [email protected]

Limits of Random Motzkin paths with KPZ related asymptotics

Abstract.

We study Motzkin paths of length $L$ with general weights on the edges and end points. We investigate the limit behavior of the initial and final segments of the random Motzkin path viewed as a pair of processes starting from each of the two end points as $L$ becomes large. We then study macroscopic limits of the resulting processes, where in two different regimes we obtain Markov processes that appeared in the description of the stationary measure for the KPZ equation on the half line and of conjectural stationary measure of the hypothetical KPZ fixed point on the half line. Our results rely on the behavior of the Al-Salam–Chihara polynomials in the neighbourhood of the upper end of their orthogonality interval and on the limiting properties of the $q$ -Pochhammer and $q$ -Gamma functions as $q\nearrow 1$ .

1. Introduction and main results

In this paper we study Markov processes that arise as limits of random Motzkin paths with random endpoints. A Motzkin path of length $L$ is a sequence of steps on the integer lattice that starts at point $(0,n_{0})$ with the initial altitude $n_{0}$ and ends at point $(L,n_{L})$ at the final altitude $n_{L}$ for some non-negative integers $n_{0},n_{L},L$ . Steps can be upward, downward, or horizontal, along the vectors $(1,1)$ , $(1,-1)$ and $(1,0)$ , respectively, and the path cannot fall below the horizontal axis. The path is uniquely determined by the sequence of altitudes ${\boldsymbol{\gamma}}=(n_{0},n_{1},\dots,n_{L})$ with $n_{j}-n_{j-1}\in\{-1,0,1\}$ , $j=1,\dots,L$ .

A random Motzkin path can be generated by assigning a discrete probability measure on the set of all Motzkin paths and choosing ${\boldsymbol{\gamma}}$ at random according to this probability measure. We will write ${\boldsymbol{\gamma}}=(\gamma_{0},\dots,\gamma_{L})$ and will use the notation ${\boldsymbol{\gamma}}^{(L)}$ if we need to explicitly indicate its dependence on the parameter $L$ . In our main results we are interested in the assignment of probability $\mathds{P}_{L}({\boldsymbol{\gamma}})$ which depends on two boundary parameters $\rho_{0},\rho_{1}\in[0,1)$ and two parameters $q\in[0,1)$ and $\sigma\in[0,1]$ that determine the edge weights. As the probability $\mathds{P}_{L}({\boldsymbol{\gamma}})$ of selecting a Motzkin path ${\boldsymbol{\gamma}}$ we take the following expression:

(1.1)

\mathds{P}_{L}({\boldsymbol{\gamma}})=\frac{1}{\mathfrak{C}_{L}}\rho_{0}^{\gamma_{0}}\rho_{1}^{\gamma_{L}}(2\sigma)^{H({\boldsymbol{\gamma}})}\prod_{k=0}^{L}[\gamma_{k}+1]_{q}

where $[n]_{q}=1+\dots+q^{n-1}$ denotes the usual $q$ -number, and $H({\boldsymbol{\gamma}})=\#\{j\in[1,L]:\gamma_{j}=\gamma_{j-1}\}$ is the number of horizontal steps, and $\mathfrak{C}_{L}$ is the normalizing constant. This weighting of the Motzkin paths was inspired by a formula in (Barraquand and Le Doussal,, 2023, Section 2.3), where $\sigma=1$ . A more general setup is discussed in Section 2.

The above setup differs from the most commonly studied random Motzkin paths chosen uniformly from all Motzkin paths which start and end at altitude 0. Such random Motzkin paths correspond to random walks conditioned on staying non-negative and returning to $0$ at time $L$ ; it is well known that their asymptotic fluctuations are described by the Brownian excursion Kaigh, (1976) and their behavior near boundaries is described by an explicit Markov chain, see Keener, (1992). It turns out that new phenomena and new asymptotic fluctuations arise in the presence of boundary parameters $\rho_{0},\rho_{1}$ , see Bryc and Wang, (2023), Bryc and Wang, (2024). Our goal is to extend Bryc and Wang, (2023) to allow more general weights that depend on parameter $q$ , and to study the boundary limit Markov chain in two different asymptotic regimes. As limits in the two asymptotic regimes we recover Markov processes that appeared in the description given in Bryc and Kuznetsov, (2022) of the non-Gaussian term of the stationary measure of the KPZ equation on the half-line, see Barraquand and Le Doussal, (2022), Barraquand and Corwin, (2023), and the non-Gaussian term in the conjectural stationary measure of the hypothetical KPZ fixed point on the half line in Barraquand and Le Doussal, (2022).

We use the following standard notation for the $q$ -Pochhammer symbols:

	$\displaystyle(a;q)_{n}$	$\displaystyle=\prod_{k=0}^{n-1}(1-aq^{k}),$	$\displaystyle(a_{1},\dots,a_{k};q)_{n}$	$\displaystyle=(a_{1};q)_{n}(a_{2};q)_{n}\dots(a_{k};q)_{n},$
	$\displaystyle(a;q)_{\infty}$	$\displaystyle=\prod_{k=0}^{\infty}(1-aq^{k}),$	$\displaystyle(a_{1},\dots,a_{k};q)_{\infty}$	$\displaystyle=(a_{1};q)_{\infty}(a_{2};q)_{\infty}\dots(a_{k};q)_{\infty}.$

Our first main result is the following limit theorem for the boundaries of the random Motzkin path. Let ${\boldsymbol{\gamma}}^{{}^{(L)}}=\{\gamma^{(L)}_{k}\}_{k\geq 0}$ be a sequence of the initial altitudes of a random Motzkin path of length $L$ , appended with $0$ for $k>L$ , and let $\widetilde{{\boldsymbol{\gamma}}}^{{}^{(L)}}=\{\widetilde{\gamma}^{(L)}_{k}\}_{k\in\mathds{Z}_{\geq 0}}$ be a sequence of the final altitudes, $\widetilde{\gamma}_{k}^{(L)}=\gamma_{L-k}$ , $k=0,1,\dots,L$ , appended with $0$ for $k>L$ .

Theorem 1.1.

Suppose that $0\leq\rho_{0},\rho_{1},q<1$ and $0<\sigma\leq 1$ . Then

\left({\boldsymbol{\gamma}}^{{}^{(L)}},\,\widetilde{{\boldsymbol{\gamma}}}^{{}^{(L)}}\right)\Rightarrow\left({\boldsymbol{X}},\,{\boldsymbol{Y}}\right),

as $L\to\infty$ , where on the left hand side we have random Motzkin paths with respect to measure (1.1) and on the right-hand side we have two independent Markov chains ${\boldsymbol{X}}=\left\{X_{k}\right\}_{k\geq 0}$ , ${\boldsymbol{Y}}=\left\{Y_{k}\right\}_{k\geq 0}$ with the same transition probabilities on $\mathds{Z}_{\geq 0}$ given by

\mathds{P}(X_{k}=n+\delta|X_{k-1}=n)=\frac{1-q^{n+1}}{1+\sigma}\cdot\begin{cases}\frac{s_{n+1}}{2s_{n}},&\delta=1,\\ \sigma,&\delta=0,\\ \frac{s_{n-1}}{2s_{n}},&\delta=-1,\\ 0,&\mbox{\em otherwise},\end{cases}

for $n=0,1,\ldots$ and with the initial laws

(1.2)

\mathds{P}(X_{0}=n)=\frac{1}{C_{0}}\rho_{0}^{n}s_{n},\;\mathds{P}(Y_{0}=n)=\frac{1}{C_{1}}\rho_{1}^{n}s_{n},\quad n=0,1,\ldots,

where $s_{-1}=0$ and

s_{n}=\sum_{k=0}^{n}\frac{(a;q)_{k}}{(q;q)_{k}}\;\frac{(b;q)_{n-k}}{(q;q)_{n-k}},\quad n=0,1,\ldots,

with $a=-q(\sigma+i\sqrt{1-\sigma^{2}})$ , $b=-q(\sigma-i\sqrt{1-\sigma^{2}})$ ; the normalizing constants are

C_{j}=\frac{(a\rho_{j},b\rho_{j};q)_{\infty}}{(\rho_{j};q)_{\infty}^{2}},\quad j=0,1.

We note that if $q=0$ then $s_{n}=n+1$ , recovering transition probabilities in (Bryc and Wang,, 2023, Theorem 1.1), who use the parameter $\sigma$ that is twice our $\sigma$ . It is natural to expect that as in (Bryc and Wang,, 2023, Theorem 1.6), there is a version of this result that holds also for $\rho_{1}\geq 1$ . However, this is beyond the scope of this paper.

Theorem 1.1 will be deduced from a more general Theorem 2.4. The proof appears in Section 4 and relies on properties of the Al-Salam–Chihara polynomials, which we discuss in Section 3.

The next two theorems give macroscopic continuous-time limits of the family of Markov processes $\{X_{k}\}$ . In the statements of Theorems 1.2 and 1.3 below, $\xrightarrow{\textit{f.d.d.}}$ denotes convergence of finite dimensional distributions.

In our first result, we take $\rho_{0}\nearrow 1$ at an appropriate rate but keep $q$ fixed. Then the normalized Markov process $\left\{X_{k}\right\}$ converges to the Bessel process, which for $\sigma=1$ appeared in the description of the non-Gaussian term in the conjectural stationary measure for the hypothetical KPZ fixed point on the half-line in (Bryc and Kuznetsov,, 2022, Theorem 2.6 ).

Theorem 1.2.

Fix $0\leq q<1$ , $0<\sigma\leq 1$ and ${\mathsf{c}}>0$ . Let $\{X_{k}^{(N)}\}_{k\in\mathds{Z}_{\geq 0}}$ be a Markov process from Theorem 1.1 with the initial law that depends on $N$ through $\rho_{0}=e^{-{\mathsf{c}}/\sqrt{N}}$ . Then

(1.3)

\frac{1}{\sqrt{N}}\left\{X_{\left\lfloor Nt\right\rfloor}^{(N)}\right\}_{t\geq 0}\xrightarrow{\textit{f.d.d.}}\left\{\xi_{t}^{({\mathsf{c}})}\right\}_{t\geq 0}\;\mbox{ as }N\to\infty,

where $\{\xi_{t}^{({\mathsf{c}})}\}_{t\geq 0}$ is the 3-dimensional Bessel process with transition probabilities

(1.4)

\mathds{P}(\xi_{t}^{({\mathsf{c}})}=dy|\xi_{0}^{({\mathsf{c}})}=x)=\frac{y}{x}\,{\mathsf{q}}_{\frac{t}{1+\sigma}}(x,y)\,dy,

where ${\mathsf{q}}_{t}$ , $t>0$ , is the transition kernel of the Brownian motion killed at hitting zero, i.e.

(1.5)

{\mathsf{q}}_{t}(x,y)=\frac{1}{\sqrt{2\pi t}}\left(\exp(-\tfrac{(y-x)^{2}}{2t})-\exp(-\tfrac{(y+x)^{2}}{2t})\right),\quad x,y>0,

and with the initial distribution

(1.6)

\mathds{P}(\xi_{0}^{({\mathsf{c}})}=dx)={\mathsf{c}}^{2}xe^{-{\mathsf{c}}x}{\bf 1}_{\{x>0\}}dx.

In our second result, we take both $\rho_{0}\nearrow 1$ and $q\nearrow 1$ at appropriate rates. Then, under appropriate centering and normalization, the Markov process $\left\{X_{k}\right\}$ converges to the Markov process on $\mathds{R}$ , which for $\sigma=1$ appeared in the description of the non-Gaussian term in the representation of the stationary measure for the KPZ equation on the half line given in (Bryc and Kuznetsov,, 2022, Theorem 2.3). (This is only a one-parameter subset of the two-parameter family of such measures conjectured in Barraquand and Le Doussal, (2022) and proved rigorously in Barraquand and Corwin, (2023).) In the statement, $K_{\nu}(z)$ is a modified Bessel K function (Macdonald function) of imaginary index $\nu$ and positive argument $z$ , see e.g. (Olver et al.,, 2023, 10.32.E9).

Theorem 1.3.

Fix $0<\sigma\leq 1$ and ${\mathsf{c}}>0$ . Let $\{X_{k}^{(N)}\}_{k\in\mathds{Z}_{\geq 0}}$ be a Markov process from Theorem 1.1 with parameters $q=e^{-2/\sqrt{N}}$ and $\rho_{0}=e^{-{\mathsf{c}}/\sqrt{N}}$ . Then

(1.7)

\frac{1}{\sqrt{N}}\left\{X_{\left\lfloor Nt\right\rfloor}^{(N)}-\sqrt{N}\log\sqrt{2N(1+\sigma)}\right\}_{t\geq 0}\xrightarrow{\textit{f.d.d.}}\left\{\zeta_{t}^{({\mathsf{c}})}\right\}_{t\geq 0}\;\mbox{ as }N\to\infty,

where $\zeta^{({\mathsf{c}})}$ is a Markov process with transition probabilities

(1.8)

\mathds{P}(\zeta_{t}^{{}^{({\mathsf{c}})}}=dy|\zeta_{0}^{{}^{({\mathsf{c}})}}=x)=\frac{K_{0}(e^{-y})}{K_{0}(e^{-x})}\;{\mathsf{p}}_{\frac{t}{1+\sigma}}(x,y)\,dy,\quad x,y\in\mathds{R},

where ${\mathsf{p}}_{t}$ , $t>0$ , is the Yakubovich transition kernel, Yakubovich, (2011), defined by

(1.9)

{\mathsf{p}}_{t}(x,y)=\frac{2}{\pi}\int_{0}^{\infty}e^{-tu^{2}/2}K_{{\textnormal{i}}u}(e^{-x})K_{{\textnormal{i}}u}(e^{-y})\frac{du}{|\Gamma(iu)|^{2}},

and with the initial distribution

(1.10)

\mathds{P}(\zeta_{0}^{({\mathsf{c}})}=dx)=\frac{4}{2^{\mathsf{c}}\Gamma({\mathsf{c}}/2)^{2}}e^{-{\mathsf{c}}x}K_{0}(e^{-x})\,dx.

Theorems 1.2 and 1.3 will follow from local limits in Theorem 4.2 and Theorem 4.3.

The paper is organized as follows. In Section 2 we introduce random Motzkin paths with general weights, we prove matrix representation, integral representation, and a general boundary limit theorem. In Section 3 we recall some properties of the Al–Salam–Chihara polynomials, and we prove new asymptotic results for their behavior near the boundary of the interval of orthogonality. In Section 4 we use these results to prove Theorem 1.1 and two local limit theorems, which we use to complete proofs of Theorems 1.2 and 1.3. In Section 5 we discuss properties of the $q$ -Pochhammer and $q$ -Gamma functions and derive the limits and bounds needed in our proofs.

2. General limit theorem for random Motzkin paths at the boundary

In this section we introduce a more general setting and prove a version of Theorem 1.1 for more general random Motzkin paths.

Recall that a Motzkin path of length $L=1,2,\dots$ is a sequence of lattice points $({\boldsymbol{x}}_{0},\dots,{\boldsymbol{x}}_{L})$ such that ${\boldsymbol{x}}_{j}=(j,n_{j})\in\mathds{Z}_{\geq 0}\times\mathds{Z}_{\geq 0},\;j=0,1,\dots,L$ . An edge $({\boldsymbol{x}}_{j-1},{\boldsymbol{x}}_{j})$ is called a ( $j$ -th) step, and only three types of steps are allowed: upward steps, downward steps, and horizontal steps. The edge $({\boldsymbol{x}}_{j-1},{\boldsymbol{x}}_{j})$ is an upward step if $n_{j}-n_{j-1}=1$ , a downward step if $n_{j}-n_{j-1}=-1$ , and a horizontal step if $n_{j}-n_{j-1}=0$ , see, e.g., Flajolet and Sedgewick, (2009, Definition V.4, p. 319) or Viennot, (1985). Each such path can be identified with a sequence of non-negative integers that specify the starting point $(0,n_{0})$ and consecutive values $n_{j}$ along the vertical axis at step $j\geq 1$ . We shall write ${\boldsymbol{\gamma}}=(\gamma_{0},\gamma_{1},\dots,\gamma_{L})$ with $\gamma_{j}=n_{j}$ for such a sequence and refer to ${\boldsymbol{\gamma}}$ as a Motzkin path. By $\mathcal{M}_{m,n}^{(L)}$ we denote the family of all Motzkin paths ${\boldsymbol{\gamma}}$ of length $L$ with the initial altitude $\gamma_{0}=m$ and the final altitude $\gamma_{L}=n$ . We also refer to $\gamma_{0}$ and $\gamma_{L}$ as the boundary/end points of the path.

We are interested in probabilistic properties of random Motzkin paths. To introduce a probability measure on the set of the Motzkin paths, we assign weights to the edges and to the end-points of a Motzkin path. The weights for the edges arise multiplicatively from three sequences ${\boldsymbol{a}}=(a_{j})_{j\geq 0},{\boldsymbol{b}}=(b_{j})_{j\geq 0},{\boldsymbol{c}}=(c_{j})_{j\geq 0}$ of positive real numbers. For a path ${\boldsymbol{\gamma}}=(\gamma_{0}=m,\gamma_{1},\dots,\gamma_{L-1},\gamma_{L}=n)\in\mathcal{M}_{m,n}^{(L)}$ we define its (edge) weight

(2.1)

w({\boldsymbol{\gamma}})=\prod_{k=1}^{L}a_{\gamma_{k-1}}^{\varepsilon_{k}^{+}}b_{\gamma_{k-1}}^{\varepsilon_{k}^{0}}c_{\gamma_{k-1}}^{\varepsilon_{k}^{-}},

where

\varepsilon_{k}^{+}({\boldsymbol{\gamma}}):={\bf 1}_{\gamma_{k}>\gamma_{k-1}},\;\varepsilon_{k}^{-}({\boldsymbol{\gamma}}):={\bf 1}_{\gamma_{k}<\gamma_{k-1}},\;\varepsilon_{k}^{0}({\boldsymbol{\gamma}}):={\bf 1}_{\gamma_{k}=\gamma_{k-1}},k=1,\dots,L.

That is, the edge weight is multiplicative in the edges, we take ${\boldsymbol{a}}$ , ${\boldsymbol{b}}$ and ${\boldsymbol{c}}$ as the weights of the upward steps, horizontal steps and downward steps, and the weight of a step depends on the altitude of the left-end of an edge. Note that the value of $c_{0}$ does not contribute to $w({\boldsymbol{\gamma}})$ . Since $\mathcal{M}_{i,j}^{(L)}$ is a finite set, the normalization constants

\mathfrak{W}^{(L)}_{m,n}=\sum_{{\boldsymbol{\gamma}}\in\mathcal{M}_{m,n}^{(L)}}w({\boldsymbol{\gamma}})

are well defined for all $m,n\geq 0$ .

Figure 1. Motzkin path

{\boldsymbol{\gamma}}=(2,1,1,0,1,1,2,1,0,1)\in\mathcal{M}^{(9)}

with weight contributions marked at the edges. The probability of selecting the path shown above from

\mathcal{M}^{(9)}

\mathds{P}_{9}({\boldsymbol{\gamma}})=\frac{\alpha_{2}\beta_{1}}{\mathfrak{C}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},9}}b_{1}^{2}a_{0}^{2}a_{1}c_{1}^{2}c_{2}^{2}

In addition to the weights of the edges, we also assign weights to the two boundary points. To this end we choose two additional non-negative sequences ${\boldsymbol{\alpha}}=(\alpha_{m})_{m\geq 0}$ and ${\boldsymbol{\beta}}=(\beta_{n})_{n\geq 0}$ such that

(2.2)

\mathfrak{C}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}},L}:=\sum_{m,n\geq 0}\alpha_{m}\mathfrak{W}^{(L)}_{m,n}\beta_{n}<\infty.

With finite normalizing constant (2.2), the countable set $\mathcal{M}^{(L)}=\bigcup_{m,n\geq 0}\mathcal{M}_{m,n}^{(L)}$ becomes a probability space with the discrete probability measure

(2.3)

\mathds{P}_{L}({\boldsymbol{\gamma}})\equiv\mathds{P}_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}},L}(\{{\boldsymbol{\gamma}}\})=\frac{\alpha_{\gamma_{0}}\beta_{\gamma_{L}}}{\mathfrak{C}_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}},L}}\,w({\boldsymbol{\gamma}}),\quad{\boldsymbol{\gamma}}\in\mathcal{M}^{(L)}.

For an illustration, see Figure 1. We now consider a random vector $(\gamma_{0}^{(L)},\dots,\gamma_{L}^{(L)})\in\mathcal{M}^{(L)}$ sampled from $\mathds{P}_{L}$ , and extend it to an infinite process denoted by

(2.4)

{\boldsymbol{\gamma}}^{(L)}:=\left\{\gamma_{k}^{(L)}\right\}_{k\geq 0},\quad\mbox{ with }\quad\gamma_{k}^{(L)}:=0\mbox{ if }k>L.

Similarly, we also introduce the infinite reversed process by

(2.5)

\widetilde{{\boldsymbol{\gamma}}}^{(L)}:=\left\{\widetilde{\gamma}_{k}^{(L)}\right\}_{k\geq 0}\quad\mbox{ with }\quad\widetilde{\gamma}_{k}^{(L)}:=\begin{cases}\gamma_{L-k}^{(L)},&\mbox{ if }k=0,\dots,L,\\ 0,&\mbox{ if }k>L.\end{cases}

We are interested in the limit processes for both ${\boldsymbol{\gamma}}^{(L)}$ and $\widetilde{{\boldsymbol{\gamma}}}^{(L)}$ as $L\to\infty$ . In these limits, we do not scale the sequences nor the locations.

Recall that weak convergence of discrete-time processes means convergence of finite-dimensional distributions (Billingsley,, 1999). For integer-valued random variables, the latter follows from convergence of probability generating functions. We will therefore fix $K$ , $z_{0},z_{1}\in(0,1]$ and $t_{1},\dots,t_{K},s_{1},\dots,s_{K}>0$ and the goal is to determine two discrete time processes ${\boldsymbol{X}}=\{X_{k}\}_{k\geq 0}$ and ${\boldsymbol{Y}}=\{Y_{k}\}_{k\geq 0}$ such that

(2.6)

\lim_{L\to\infty}\mathds{E}\left[z_{0}^{\gamma_{0}^{(L)}}\prod_{j=1}^{K}t_{j}^{\gamma_{j}^{(L)}-\gamma_{j-1}^{(L)}}\prod_{j=1}^{K}s_{j}^{\gamma_{L-j}^{(L)}-\gamma_{L+1-j}^{(L)}}z_{1}^{\gamma_{L}^{(L)}}\right]\\ =\mathds{E}\left[z_{0}^{X_{0}}\prod_{j=1}^{K}t_{j}^{X_{j}-X_{j-1}}\right]\mathds{E}\left[z_{1}^{Y_{0}}\prod_{j=1}^{K}s_{j}^{Y_{j}-Y_{j-1}}\right].

Indeed, the above expressions uniquely determine the corresponding probability generating functions for small enough arguments. For example,

\mathds{E}\left[\prod_{j=0}^{K}v_{j}^{X_{j}}\right]=\mathds{E}\left[z_{0}^{X_{0}}\prod_{j=1}^{K}t_{j}^{X_{j}-X_{j-1}}\right]

with $z_{0}=v_{0}\dots v_{K}$ and $t_{j}=v_{j}v_{j+1}\dots v_{K}$ .

2.1. Matrix representation and integral representation

We will need a convenient representation for the left hand side of (2.6). We introduce a tri-diagonal matrix

{{\boldsymbol{M}}}_{t}:=\left[\begin{matrix}b_{0}&a_{0}t&0&0&\cdots\\ c_{1}/t&b_{1}&a_{1}t&0&\\ 0&c_{2}/t&b_{2}&a_{2}t&\\ 0&0&c_{3}/t&\ddots&\ddots\\ \vdots&&&\ddots&\ddots\end{matrix}\right],

and infinite column vectors

\vec{V}_{{\boldsymbol{\alpha}}}(z):=[\alpha_{n}\,z^{n},\,n=0,1,\ldots]^{T}\quad\mbox{and}\quad\vec{W}_{{\boldsymbol{\alpha}}}(z):=[\beta_{n}\,z^{n},\,n=0,1,\ldots]^{T}.

We use the following matrix representation for the left hand side of (2.6).

Lemma 2.1 (matrix ansatz).

For every $K=0,1,\dots$ such that $2K\leq L$ , we have

(2.7)

\mathds{E}\left[z_{0}^{\gamma_{0}^{(L)}}\prod_{j=1}^{K}t_{j}^{\gamma_{j}^{(L)}-\gamma_{j-1}^{(L)}}\prod_{j=1}^{K}s_{j}^{\gamma_{L-j}^{(L)}-\gamma_{L+1-j}^{(L)}}z_{1}^{\gamma_{L}^{(L)}}\right]\\ =\frac{1}{\mathfrak{C}_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}},L}}\vec{V}_{{\boldsymbol{\alpha}}}(z_{0})^{T}{{\boldsymbol{M}}}_{t_{1}}{{\boldsymbol{M}}}_{t_{2}}\cdots{{\boldsymbol{M}}}_{t_{K}}{{\boldsymbol{M}}}_{1}^{L-2K}{{\boldsymbol{M}}}_{1/s_{K}}\cdots{{\boldsymbol{M}}}_{1/s_{2}}{{\boldsymbol{M}}}_{1/s_{1}}\vec{W}_{{\boldsymbol{\beta}}}(z_{1}),

where

(2.8)

\mathfrak{C}_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}},L}=\vec{V}_{{\boldsymbol{\alpha}}}(1)^{T}M_{1}^{L}\vec{W}_{{\boldsymbol{\beta}}}(1)

is the normalization constant.

Proof.

Formula (2.7) follows from a more general formula

(2.9)

\sum_{m,n=0}^{\infty}\alpha_{m}z_{0}^{m}\beta_{n}z_{1}^{n}\sum_{\gamma\in\mathcal{M}_{m,n}^{{}^{(L)}}}\prod_{j=1}^{L}t_{j}^{\gamma_{j}-\gamma_{j-1}}w({\boldsymbol{\gamma}})=\vec{V}_{{\boldsymbol{\alpha}}}(z_{0})^{T}{{\boldsymbol{M}}}_{t_{1}}{{\boldsymbol{M}}}_{t_{2}}\cdots{{\boldsymbol{M}}}_{t_{L}}\vec{W}_{{\boldsymbol{\beta}}}(z_{1}).

applied to the sequence $(t_{1},\dots,t_{L})$ of the form $(t_{1},\dots,t_{K},1,\dots,1,1/s_{K},\dots,1/s_{1})$ .

To prove (2.9), consider an infinite matrix ${{\boldsymbol{W}}}(L)$ with entries

(2.10)

[{{\boldsymbol{W}}}(L)]_{m,n}:=\sum_{\gamma\in\mathcal{M}_{m,n}^{{}^{(L)}}}\prod_{j=1}^{L}t_{j}^{\gamma_{j}-\gamma_{j-1}}w({\boldsymbol{\gamma}}).

It is clear that the left hand side of (2.9) is

\vec{V}_{{\boldsymbol{\alpha}}}(z_{0})^{T}{{\boldsymbol{W}}}(L)\vec{W}_{{\boldsymbol{\beta}}}(z_{1}).

It remains to verify that

{{\boldsymbol{W}}}(L)={{\boldsymbol{M}}}_{t_{1}}{{\boldsymbol{M}}}_{t_{2}}\cdots{{\boldsymbol{M}}}_{t_{L}}.

With the case $L=0$ being trivial, we proceed by induction. In view of (2.10) we have

[{{\boldsymbol{W}}}(L+1)]_{m,n}=t_{L+1}[{{\boldsymbol{W}}}(L)]_{m,n-1}a_{n-1}+[{{\boldsymbol{W}}}(L)]_{m,n}b_{n}+t_{L+1}^{-1}[{{\boldsymbol{W}}}(L)]_{m,n+1}c_{n+1}

Thus ${{\boldsymbol{W}}}(L+1)={{\boldsymbol{W}}}(L){{\boldsymbol{M}}}_{t_{L+1}}$ . ∎

The key step in our proof of boundary limit theorem is an integral representation for the left hand side of (2.6) based on the orthogonality measure of the orthogonal polynomials determined by the edge weights of Motzkin paths.

Following Viennot, (1985) and Flajolet and Sedgewick, (2009) with the sequences ${\boldsymbol{a}},{\boldsymbol{b}},{\boldsymbol{c}}$ of edge weights for the Motzkin paths that appeared in (2.1) we now associate real polynomials $p_{-1}(x)=0$ , $p_{0}(x)=1,p_{1}(x),\dots$ defined by the three step recurrence

(2.11)

xp_{n}(x)=a_{n}p_{n+1}(x)+b_{n}p_{n}(x)+c_{n}p_{n-1}(x),\quad n=0,1,2,\dots

(With the usual conventions that $p_{-1}(x)=0$ , $p_{0}(x):=1$ and $a_{n}>0$ , the recursion determines polynomials $\{p_{n}(x)\}$ uniquely, with $p_{1}(x)=(x-b_{0})/a_{0}$ .)

By Favard’s theorem (Ismail,, 2009), polynomials $\{p_{n}(x)\}$ are orthogonal with respect to a probability measure $\nu$ on the real line, which we assume to be compactly supported, and thus unique. It is well known that the $L_{2}$ norm $\|p_{n}\|_{2}^{2}:=\int_{\mathds{R}}(p_{n}(x))^{2}\nu(dx)$ is given by the formula

(2.12)

\|p_{n}\|_{2}^{2}=\prod_{k=1}^{n}\frac{c_{k}}{a_{k-1}}.

We need some conditions on the weights of the end points ${{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}$ in (2.3).

Assumption 2.1.

We assume that

(

A_{1}

)

The series

(2.13)

\phi_{{\boldsymbol{\alpha}}}(x,z)=\sum_{n=0}^{\infty}\alpha_{n}z^{n}p_{n}(x)\quad\mbox{ and }\quad\psi_{{\boldsymbol{\beta}}}(x,z)=\sum_{n=0}^{\infty}\beta_{n}z^{n}\frac{p_{n}(x)}{\|p_{n}\|_{2}^{2}}

converge absolutely on the support of $\nu$ for all $z\in\mathds{C},|z|\leq 1$ .

(

A_{2}

)

The function $x^{L}\phi_{\alpha}(x,1)\psi_{\beta}(x,1)$ is integrable with respect to the measure $\nu$ , and

(2.14)

\int_{\mathds{R}}\sum_{m,n=0}^{\infty}\alpha_{n}\frac{\beta_{m}}{\|p_{m}\|_{2}^{2}}\left|x^{L}p_{n}(x)p_{m}(x)\right|\nu(dx)<\infty.

Consider the following two infinite column vectors

\vec{P}(x):=\left[p_{i}(x),\,i=0,1,\ldots\right]^{T}\quad\mbox{and}\quad\vec{Q}(x):=[\widetilde{p}_{i}(x),\,i=0,1,\ldots]^{T}

where

(2.15)

\widetilde{p}_{n}(x)=p_{n}(x)/\|p_{n}\|_{2}^{2}=p_{n}(x)\prod_{k=1}^{n}\frac{a_{k-1}}{c_{k}}.

Note that with $a_{-1}:=0$ , polynomials $\widetilde{p}_{n}(x)$ satisfy the three step recurrence

x\widetilde{p}_{n}(x)=c_{n+1}\widetilde{p}_{n+1}(x)+b_{n}\widetilde{p}_{n}(x)+a_{n-1}\widetilde{p}_{n-1}(x).

In particular,

(2.16)

{{\boldsymbol{M}}}_{1}^{L}\vec{P}(x)=x^{L}\vec{P}(x)\quad\mbox{ and }\quad\vec{Q}^{T}(x){{\boldsymbol{M}}}_{1}^{L}=x^{L}\vec{Q}^{T}.

The left hand side of the expression (2.7) has the following integral representation.

Lemma 2.2.

If (2.14) holds and $2K\leq L$ , then

(2.17)

\mathds{E}\left[z_{0}^{\gamma_{0}^{(L)}}\prod_{j=1}^{K}t_{j}^{\gamma_{j}^{(L)}-\gamma_{j-1}^{(L)}}\prod_{j=1}^{K}s_{j}^{\gamma_{L-j}^{(L)}-\gamma_{L+1-j}^{(L)}}z_{1}^{\gamma_{L}^{(L)}}\right]=\frac{1}{\mathfrak{C}_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}},L}}\int x^{L-2K}\Psi_{0}(x)\Psi_{1}(x)\nu(dx),

where

\Psi_{0}(x)={\vec{V}_{{\boldsymbol{\alpha}}}(z_{0})^{T}{{\boldsymbol{M}}}_{t_{1}}{{\boldsymbol{M}}}_{t_{2}}\cdots{{\boldsymbol{M}}}_{t_{K}}\vec{P}(x)}\quad\mbox{and}\quad\Psi_{1}(x)={\vec{W}_{{\boldsymbol{\beta}}}(z_{1})^{T}\widetilde{{{\boldsymbol{M}}}}_{s_{1}}\widetilde{{{\boldsymbol{M}}}}_{s_{2}}\cdots\widetilde{{{\boldsymbol{M}}}}_{s_{K}}\vec{Q}(x)},

with $\widetilde{{{\boldsymbol{M}}}}_{s}={{\boldsymbol{M}}}_{1/s}^{T}$ . Moreover, $\mathfrak{C}_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}},L}$ , introduced in (2.8), can be written as

\mathfrak{C}_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}},L}=\int x^{L}\left(\vec{V}_{{\boldsymbol{\alpha}}}(1)^{T}\vec{P}(x)\right)\left(\vec{W}_{{\boldsymbol{\beta}}}(1)^{T}\vec{Q}(x)\right)\nu(dx).

Proof.

We use expression (2.9) with $(t_{1},\dots,t_{L})$ of the form $(t_{1},\dots,t_{K},1,\dots,1,1/s_{K},\dots,1/s_{1})$ . Proceeding somewhat formally, we write the identity matrix as the integral, ${{\boldsymbol{I}}}=\int\,\vec{P}(x)\vec{Q}(x)^{T}\,\nu(dx)$ and rewrite the product of matrices in (2.7) as

		$\displaystyle{{\boldsymbol{M}}}_{t_{1}}{{\boldsymbol{M}}}_{t_{2}}\cdots{{\boldsymbol{M}}}_{t_{K}}{{\boldsymbol{M}}}_{1}^{L-2K}{{\boldsymbol{M}}}_{1/s_{K}}\cdots{{\boldsymbol{M}}}_{1/s_{2}}{{\boldsymbol{M}}}_{1/s_{1}}$
	$\displaystyle=$	$\displaystyle{{\boldsymbol{M}}}_{t_{1}}{{\boldsymbol{M}}}_{t_{2}}\cdots{{\boldsymbol{M}}}_{t_{K}}{{\boldsymbol{M}}}_{1}^{L-2K}\left(\int\,\vec{P}(x)\vec{Q}(x)^{T}\,\nu(dx)\right)\,{{\boldsymbol{M}}}_{1/s_{K}}\cdots{{\boldsymbol{M}}}_{1/s_{2}}{{\boldsymbol{M}}}_{1/s_{1}}$
	$\displaystyle=$	$\displaystyle\int{{\boldsymbol{M}}}_{t_{1}}{{\boldsymbol{M}}}_{t_{2}}\cdots{{\boldsymbol{M}}}_{t_{K}}\left({{\boldsymbol{M}}}_{1}^{L-2K}\vec{P}(x)\right)\,\vec{Q}(x)^{T}\,{{\boldsymbol{M}}}_{1/s_{K}}\cdots{{\boldsymbol{M}}}_{1/s_{2}}{{\boldsymbol{M}}}_{1/s_{1}}\nu(dx)$
	$\displaystyle\stackrel{{\scriptstyle\eqref{mpq}}}{{=}}$	$\displaystyle\int x^{L-2K}{{\boldsymbol{M}}}_{t_{1}}{{\boldsymbol{M}}}_{t_{2}}\cdots{{\boldsymbol{M}}}_{t_{K}}\vec{P}(x)\,\vec{Q}(x)^{T}{{\boldsymbol{M}}}_{1/s_{K}}\cdots{{\boldsymbol{M}}}_{1/s_{2}}{{\boldsymbol{M}}}_{1/s_{1}}\nu(dx).$

Therefore, the right hand side of (2.7) becomes

\tfrac{1}{\mathfrak{C}_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}},L}}\,\int x^{L-2K}\left(\vec{V}_{{\boldsymbol{\alpha}}}(z_{0})^{T}{{\boldsymbol{M}}}_{t_{1}}{{\boldsymbol{M}}}_{t_{2}}\cdots{{\boldsymbol{M}}}_{t_{K}}\vec{P}(x)\right)\,\left(\vec{Q}(x)^{T}{{\boldsymbol{M}}}_{\frac{1}{s_{K}}}\cdots{{\boldsymbol{M}}}_{\frac{1}{s_{2}}}{{\boldsymbol{M}}}_{\frac{1}{s_{1}}}\vec{W}_{{\boldsymbol{\beta}}}(z_{1})\right)\nu(dx)

and the result follows from the symmetry of the dot product, $\vec{u}^{T}\vec{v}=\vec{v}^{T}\vec{u}$ . (Assumption (2.14) is responsible for Fubini’s theorem, which is used to switch the integral with the infinite sums in the last expression. This avoids formal integration of the product of infinite matrices.) ∎

We will need the following elementary lemma. (See also (Bryc and Wang,, 2023, Lemma A.1).)

Lemma 2.3.

Let $\mu$ be a probability measure with $B\in{\textnormal{supp}}(\mu)\subseteq[A,B]$ with $|A|<B<\infty$ . Let $F$ be a real function which is bounded on $\mathrm{supp}(\mu)$ and left-continuous at $B$ .

Then

(2.18)

\lim_{L\to\infty}\frac{\int F(x)x^{L}\mu(dx)}{\int x^{L}\mu(dx)}=F(B).

Proof.

Let $X$ be a random variable with the law $\mu$ . First we will derive some estimates on the moments of $X$ and $|X|$ . Fix $\varepsilon>0$ such that $B-\varepsilon>|A|$ . Since $B\in{\textnormal{supp}}(\mu)$ , we have $C:=\mathds{P}(X>B-\varepsilon/2)>0$ . And since ${\textnormal{supp}}(\mu)\subseteq[A,B]$ we have

\mathds{E}[X^{L}]=\mathds{E}[X^{L}{\bf 1}_{\{X>B-\varepsilon/2\}}]+\mathds{E}[X^{L}{\bf 1}_{\{X\leq B-\varepsilon/2\}}]\geq C(B-\varepsilon/2)^{L}-|A|^{L}\\ =(B-\varepsilon/2)^{L}\left(C-\frac{|A|^{L}}{(B-\varepsilon/2)^{L}}\right).

The inequalities $B-\epsilon/2>B-\epsilon>|A|$ imply that the term $|A|^{L}/(B-\epsilon/2)^{L}$ converges to zero as $L\to+\infty$ , therefore $\mathds{E}[X^{L}]>\frac{1}{2}C(B-\varepsilon/2)^{L}$ for large enough $L\in{\mathbb{N}}$ . Next, we use the inequality $|x|^{L}\leq x^{L}+2|A|^{L}$ , which is valid for all $x\in[A,B]$ and $L\in{\mathbb{N}}$ , and obtain

\frac{\mathds{E}[|X|^{L}]}{\mathds{E}[X^{L}]}\leq 1+2\frac{|A|^{L}}{\mathds{E}[X^{L}]}<1+\frac{4}{C}\frac{|A|^{L}}{(B-\epsilon/2)^{L}},

which gives us an upper bound $\mathds{E}[|X|^{L}]<2\mathds{E}[X^{L}]$ , valid for $L$ large enough.

Now we are ready to establish (2.18). By taking $F-F(B)$ instead of $F$ in (2.18), without loss of generality, we can assume that $F(B)=0$ (clearly, $F-F(B)$ remains left-continuous at $B$ ). Now we estimate, for $L\in{\mathbb{N}}$ large enough

	$\displaystyle\left\|\frac{\mathds{E}[F(X)X^{L}]}{\mathds{E}[X^{L}]}\right\|$	$\displaystyle\leq\frac{\mathds{E}[\|F(X)\|\|X\|^{L}{\bf 1}_{\{X\leq B-\varepsilon\}}]}{\mathds{E}[X^{L}]}+\frac{\mathds{E}[\|F(X)\|\|X\|^{L}{\bf 1}_{\{X>B-\varepsilon\}}]}{\mathds{E}[X^{L}]}$
		$\displaystyle\leq\frac{2}{C}\\|F\\|_{\infty}\frac{(B-\varepsilon)^{L}}{(B-\varepsilon/2)^{L}}+2\sup_{x\in[B-\varepsilon,B]}\|F(x)\|,$

where we used the estimates:

\mathds{E}[|F(X)||X|^{L}{\bf 1}_{\{X\leq B-\varepsilon\}}]\leq\|F\|_{\infty}(B-\varepsilon)^{L}\quad\mbox{and}\quad\mathds{E}[X^{L}]\geq\tfrac{1}{2}C(B-\tfrac{\varepsilon}{2})^{L}

for the first term and

\mathds{E}[|F(X)||X|^{L}{\bf 1}_{\{X>B-\varepsilon\}}]\leq\sup_{x\in[B-\varepsilon,B]}|F(x)|\times\mathds{E}[|X|^{L}]\quad\mbox{and}\quad\mathds{E}[|X|^{L}]<2\mathds{E}[X^{L}]

for the second term. Since $(B-\epsilon)^{L}/(B-\epsilon/2)^{L}$ converges to zero when $L\to+\infty$ , we obtain

\limsup\limits_{L\to+\infty}\left|\frac{\mathds{E}[F(X)X^{L}]}{\mathds{E}[X^{L}]}\right|\leq 2\sup_{x\in[B-\varepsilon,B]}|F(x)|.

Taking the limit $\epsilon\to 0^{+}$ , gives us the desired result (2.18) in the case $F(B)=0$ . ∎

2.2. Limit theorem

The following result is a version of (Bryc and Wang,, 2023, Theorem 1.1) for non-constant weights of edges. It describes the joint limiting behavior of processes ${\boldsymbol{\gamma}}^{(L)}$ and $\widetilde{{\boldsymbol{\gamma}}}^{(L)}$ introduced in (2.4) and (2.5).

Theorem 2.4.

Suppose that $\textnormal{supp}(\nu)$ , the support of the orthogonality measure $\nu$ of the polynomials (2.11), satisfies $B\in{\textnormal{supp}}(\nu)\subseteq[A,B]$ with $B>|A|$ and that $(A_{1})$ and $(A_{2})$ from Assumption 2.1 hold. Then $\pi_{n}:=p_{n}(B)>0$ , $n=0,1,\ldots$ and

\left({\boldsymbol{\gamma}}^{(L)},\,\widetilde{{\boldsymbol{\gamma}}}^{(L)}\right)\Rightarrow\left({\boldsymbol{X}},{\boldsymbol{Y}}\right)

as $L\to\infty$ , where ${\boldsymbol{X}}=\left\{X_{k}\right\}_{k\geq 0}$ and ${\boldsymbol{Y}}=\left\{Y_{k}\right\}_{k\geq 0}$ are independent Markov chains with the same transition probabilities

(2.19)

\mathsf{Q}_{n,m}=\frac{1}{B}\begin{cases}a_{n}\frac{\pi_{n+1}}{\pi_{n}},&m=n+1,\\ b_{n},&m=n,\\ c_{n}\frac{\pi_{n-1}}{\pi_{n}},&m=n-1,\\ 0,&\mbox{otherwise},\end{cases}

with $m,n=0,1,\ldots$ , and with the initial laws given by

(2.20)

\mathds{P}(X_{0}=n)=\frac{1}{C_{{\boldsymbol{\alpha}}}}\alpha_{n}\pi_{n},\quad\mathds{P}(Y_{0}=n)=\frac{1}{C_{{\boldsymbol{\beta}}}}\beta_{n}\widetilde{\pi}_{n},\;n=0,1,\ldots,

where $C_{{\boldsymbol{\alpha}}}$ , $C_{{\boldsymbol{\beta}}}$ are the normalizing constants and $\widetilde{\pi}_{n}=\widetilde{p}_{n}(B)$ with $\widetilde{p}_{n}$ given in (2.15).

Proof.

We first verify that transition probabilities (2.19) and (2.24) are well defined, i.e. $\pi_{n}>0$ , $n=0,1,\dots$ . Clearly, $\pi_{0}=1$ . For $n\geq 1$ , the polynomial $p_{n}(x)$ has leading term $a_{0}^{-1}a_{1}^{-1}\dots a_{n-1}^{-1}x^{n}$ , thus $p_{n}(x)\to+\infty$ as $x\to+\infty$ . If $\pi_{n}=p_{n}(B)$ were negative, this would imply that $p_{n}$ has a zero outside of the support of the measure of orthogonality, which is impossible, see e.g. Ismail, (2009). The case $p_{n}(B)=0$ is also impossible, since the interlacing property of the zeroes of orthogonal polynomials would imply that $p_{n+1}$ has a zero in the interval $(B,\infty)$ .

To determine the limit as $L\to\infty$ of the left hand side of (2.6), we re-write the right hand side of (2.17) as

\frac{\int x^{L}\nu(dx)}{\int x^{L}\left(\vec{V}_{{\boldsymbol{\alpha}}}(1)^{T}\vec{P}(x)\right)\left(\vec{W}_{{\boldsymbol{\beta}}}(1)^{T}\vec{Q}(x)\right)\nu(dx)}\cdot\frac{\int x^{L-2K}\nu(dx)}{\int x^{L}\nu(dx)}\cdot\frac{\int x^{L-2K}\Psi_{0}(x)\Psi_{1}(x)\nu(dx)}{\int x^{L-2K}\nu(dx)}

Using Lemma 2.3, each of the factors above converges as $L\to\infty$ and we get

(2.21)

\lim_{L\to\infty}\mathds{E}\left[z_{0}^{\gamma_{0}^{(L)}}\prod_{j=1}^{K}t_{j}^{\gamma_{j}^{(L)}-\gamma_{j-1}^{(L)}}\prod_{j=1}^{K}s_{j}^{\gamma_{L-j}^{(L)}-\gamma_{L+1-j}^{(L)}}z_{1}^{\gamma_{L}^{(L)}}\right]\\ =\frac{1}{\left(\vec{V}_{{\boldsymbol{\alpha}}}(1)^{T}\vec{P}(1)\right)\left(\vec{Q}(1)^{T}\vec{W}_{{\boldsymbol{\beta}}}(1)\right)}\cdot\frac{1}{B^{2K}}\cdot\Psi_{0}(B)\Psi_{1}(B)=\frac{\Psi_{0}(B)}{B^{K}C_{{\boldsymbol{\alpha}}}}\cdot\frac{\Psi_{1}(B)}{B^{K}C_{{\boldsymbol{\beta}}}}.

We will show that the latter expressions matches the product on the right hand side of (2.6).

To this end it suffices to prove

(2.22)

\frac{1}{B^{K}}\,\vec{V}_{{\boldsymbol{\alpha}}}(z)^{T}{{\boldsymbol{M}}}_{t_{1}}{{\boldsymbol{M}}}_{t_{2}}\cdots{{\boldsymbol{M}}}_{t_{K}}\vec{P}(B)=\sum_{n\geq 0}\alpha_{n}\pi_{n}z^{n}\,\mathds{E}(T_{1:K}|X_{0}=n),

where $T_{i:K}=\prod_{j=i}^{K}t_{j}^{X_{j}-X_{j-1}}$ , as well as

(2.23)

\frac{1}{B^{K}}\,\vec{W}_{{\boldsymbol{\beta}}}(z)^{T}\widetilde{{{\boldsymbol{M}}}}_{s_{1}}\widetilde{{{\boldsymbol{M}}}}_{s_{2}}\cdots\widetilde{{{\boldsymbol{M}}}}_{s_{K}}\vec{Q}(B)=\sum_{n\geq 0}\beta_{n}\widetilde{\pi}_{n}z^{n}\,\mathds{E}(S_{1:K}|Y_{0}=n),

where $S_{i:K}=\prod_{j=i}^{K}s_{j}^{Y_{j}-Y_{j-1}}$ , and

(2.24)

\mathds{P}(Y_{k+1}=n+\delta|Y_{k}=n)=\frac{1}{B}\begin{cases}c_{n+1}\frac{\widetilde{\pi}_{n+1}}{\widetilde{\pi}_{n}},&\delta=1,\\ b_{n},&\delta=0\\ a_{n-1}\frac{\widetilde{\pi}_{n-1}}{\widetilde{\pi}_{n}},&\delta=-1.\end{cases}

Note that the processes ${\boldsymbol{X}}$ and ${\boldsymbol{Y}}$ have the same transition probabilities, since

\frac{\widetilde{\pi}_{n+1}}{\widetilde{\pi}_{n}}=\frac{\pi_{n+1}}{\pi_{n}}\frac{a_{n}}{c_{n+1}}.

We prove only (2.22) since the proof of (2.23) follows along the same lines. We use induction with respect to $K$ .

The case of $K=0$ is immediate since the left hand side of (2.22) is $\vec{V}_{{\boldsymbol{\alpha}}}(z)^{T}\vec{P}(B)=\sum_{n\geq 0}\,\alpha_{n}\pi_{n}\,z^{n}$ . For $K>0$ we first note that the left hand side of (2.22) can be written as

R_{K}:=\frac{1}{B^{K-1}}\,\vec{V}_{\widetilde{\alpha}(z,t_{1})}(z)^{T}{{\boldsymbol{M}}}_{t_{2}}\cdots{{\boldsymbol{M}}}_{t_{K}}\vec{P}(B),

where $\widetilde{\alpha}_{n}(z,t)=(\alpha_{n-1}a_{n-1}\frac{t}{z}+\alpha_{n}b_{n}+\alpha_{n+1}c_{n+1}\frac{z}{t})/B$ , $n\geq 0$ . Consequently, by induction assumption for $K-1$ we get

R_{K}=\sum_{n\geq 0}\,\widetilde{\alpha}_{n}(z,t_{1})\pi_{n}\,\mathds{E}(T_{2:K}|X_{1}=n)\\ =\sum_{n\geq 0}\left(\alpha_{n-1}a_{n-1}\frac{t_{1}}{z}+\alpha_{n}b_{n}+\alpha_{n+1}c_{n+1}\frac{z}{t_{1}}\right)\frac{\pi_{n}}{B}\,\mathds{E}(T_{2:K}|X_{1}=n)\\ =\sum_{n\geq 0}\,\alpha_{n}\pi_{n}\Big{(}\tfrac{t_{1}}{z}\,\mathds{E}(T_{2:K}|X_{1}=n+1)\,\tfrac{a_{n}\pi_{n+1}}{B\pi_{n}}+\mathds{E}(T_{2:K}|X_{1}=n)\,\tfrac{b_{n}}{B}+\tfrac{z}{t_{1}}\,\mathds{E}(T_{2:K}|X_{1}=n-1)\,\tfrac{c_{n}\pi_{n-1}}{B\pi_{n}}\Big{)}\\ =\sum_{n\geq 0}\alpha_{n}\pi_{n}\,\mathds{E}\left(\left(\frac{t_{1}}{z}\right)^{X_{1}-n}T_{2:K}|X_{0}=n\right)=\sum_{n\geq 0}\alpha_{n}\pi_{n}z^{n}\,\mathds{E}(T_{1:K}|X_{0}=n).

Formula (2.21) shows that the limit processes ${\boldsymbol{X}}$ and ${\boldsymbol{Y}}$ are independent. This ends the proof. ∎

For proofs of local limit theorems we need to determine the limit of $\sqrt{N}\mathds{P}\left(X_{\left\lfloor Nt\right\rfloor}=y_{N}\middle|X_{0}=x_{N}\right)$ as $N\to\infty$ for suitably chosen sequences $(y_{N})$ and $(x_{N})$ . The following formula is useful in computing such limits.

Proposition 2.5.

For a Markov process $\{X_{k}\}$ , we have

(2.25)

\mathds{P}(X_{k}=n|X_{0}=m)=\frac{\pi_{n}}{\pi_{m}}\frac{1}{B^{k}}\int_{[A,B]}x^{k}p_{m}(x)\widetilde{p}_{n}(x)\,\nu(dx).

(This resembles (Karlin and McGregor,, 1959, p. 67).)

Proof.

Recall notation (2.1). Fix ${\boldsymbol{\gamma}}\in\mathcal{M}_{m,n}(k)$ . Due to cancellations,

\mathds{P}(X_{1}=\gamma_{1},X_{2}=\gamma_{2},\dots,X_{k}=\gamma_{k}|X_{0}=\gamma_{0})=\frac{w({\boldsymbol{\gamma}})}{B^{k}}\frac{\pi_{\gamma_{k}}}{\pi_{\gamma_{0}}},

\mathds{P}(X_{k}=n|X_{0}=m)=\frac{\pi_{n}}{B^{k}\pi_{m}}\sum_{{\boldsymbol{\gamma}}\in\mathcal{M}_{m,n}^{(k)}}w({\boldsymbol{\gamma}}).

By Viennot, (1985, (5)) (see also (Bryc and Wang,, 2023, Proposition 2.1)) we have

\int_{\mathds{R}}p_{m}(x)p_{n}(x)x^{L}\,\nu(dx)=\|p_{n}\|_{2}^{2}\sum_{\gamma\in\mathcal{M}_{m,n}^{(L)}}w({\boldsymbol{\gamma}}).

We get

\mathds{P}(X_{k}=n|X_{0}=m)=\frac{\pi_{n}}{B^{k}\pi_{m}\|p_{n}\|_{2}^{2}}\int_{\mathds{R}}\,x^{k}p_{m}(x)p_{n}(x)\,\nu(dx).

∎

3. Properties of the Al-Salam–Chihara polynomials

The general theory developed in Section 2 can be advanced further when it is specialized to the setting considered in Section 1. To proceed, we need to recall the definition and properties of the Al-Salam–Chihara polynomials.

We will be working with Al-Salam-Chihara polynomials $\{Q_{n}\}$ in real variable $x$ , defined by the three-step recursion

(3.1)

2xQ_{n}(x;a,b|q)\\ =Q_{n+1}(x;a,b|q)+(a+b)q^{n}Q_{n}(x;a,b|q)+(1-q^{n})(1-abq^{n-1})Q_{n-1}(x;a,b|q),\;n=0,1,\dots,

where parameters $a,b$ are either real or complex conjugate, $|ab|<1$ , and $q\in[0,1)$ . As usual, we initialize the recursion with $Q_{-1}\equiv 0$ and $Q_{0}\equiv 1$ .

It is known, see Ismail, (2009); Koekoek et al., (2010) or Koekoek and Swarttouw, (1994), that if $|a|<1,|b|<1$ , then the polynomials $\{Q_{n}\}$ are orthogonal with respect to the unique probability density supported on $[-1,1]$ and given by

(3.2)

g(x)=\frac{(q,ab;q)_{\infty}}{2\pi\sqrt{1-x^{2}}}\frac{\left|(e^{2i\theta};q)_{\infty}\right|^{2}}{\left|(ae^{i\theta},be^{i\theta};q)_{\infty}\right|^{2}},\quad x=\cos\theta.

It is also known that the generating function of $\{Q_{n}\}$ is well defined for complex $|t|<1$ , and is given by the infinite product:

(3.3)

\sum_{n=0}^{\infty}\frac{Q_{n}(\cos\theta;a,b|q)}{(q;q)_{n}}t^{n}=\frac{(at,bt;q)_{\infty}}{(e^{i\theta}t,e^{-i\theta}t;q)_{\infty}}.

Invoking the $q$ -binomial theorem, see e.g. (Gasper and Rahman,, 2004, (1.3.2)), the latter implies the following formula, which can also be recalculated from (Berg and Ismail,, 1996, p. 50) or (Ismail,, 2009, (15.1.12)):

(3.4)

\frac{Q_{n}(\cos\theta;a,b|q)}{(q;q)_{n}}=\sum_{k=0}^{n}\frac{(be^{i\theta};q)_{k}}{(q;q)_{k}}e^{-ik\theta}\frac{(ae^{-i\theta};q)_{n-k}}{(q;q)_{n-k}}e^{i(n-k)\theta}.

In particular, at the boundary of the interval of orthogonality, we have

(3.5)

\frac{Q_{n}(1;a,b|q)}{(q;q)_{n}}=\sum_{k=0}^{n}\frac{(a;q)_{k}(b;q)_{n-k}}{(q;q)_{k}(q;q)_{n-k}}.

3.1. Maximum of the Al-Salam–Chihara polynomials

Chebyshev polynomials $U_{n}$ and $T_{n}$ on the interval $[-1,1]$ are bounded in absolute value by their value at $x=1$ . Similar result holds for the $q$ -Hermite polynomials, see (Ismail,, 2009, Theorem 13.1.2). For our proofs we need to extend this property to the Al-Salam–Chihara polynomials.

Proposition 3.1.

Let $q\in[0,1)$ , and $a,b$ be real or a complex conjugate pair such that $0\leq ab\leq 1$ and $a+b\leq 0$ . Then for all $x\in[-1,1]$ we have

(3.6)

|Q_{n}(x;a,b|q)|\leq Q_{n}(1;a,b|q).

Remark 3.1.

A similar bound seems to be implied by (Ismail,, 2009, (15.1.4)) without explicit assumption that $a+b\leq 0$ . We note that (3.6) does not hold for arbitrary $a,b$ :

Q_{2}(\cos\theta;-qe^{i\alpha},-qe^{-i\alpha};q)=2q^{3}\cos(2\alpha)+4(q+1)q\cos(\alpha)\cos(u)+2\cos(2u)+[4]_{q}.

In particular, $Q_{2}(\cos\theta;q,q;q)=3q^{3}+q^{2}-4(q+1)q\cos(\theta)+q+2\cos(2\theta)+1$ has a strict maximum at $\theta=\pi$ which exceeds the value at $\theta=0$ by $8q(1+q)$ .

Corollary 3.1.

Let $q\in[0,1)$ , $r\in[0,1]$ , $\alpha\in[-\pi/2,\pi/2]$ . Denote $a=-re^{i\alpha}$ and $b=-re^{-i\alpha}$ . Then for all $x\in[-1,1]$ bound (3.6) holds.

Proof.

Indeed, $a+b=-r\cos\alpha\leq 0$ and $ab=r^{2}\in[0,1]$ ∎

Proof of Proposition 3.1.

We will write $Q_{n}$ as a trigonometric polynomial

Q_{n}(x;a,b|q)=\sum\limits_{m=0}^{n}a(n,m)T_{m}(x),

where $T_{m}(\cos\theta)=\cos(m\theta)$ is a Chebyshev polynomial of the first kind, $m=0,1,\dots$ . We are going to show that $a(n,m)\geq 0$ for all $0\leq m\leq n$ . Since for $x=\cos\theta\in[-1,1]$ we have $|T_{m}(x)|\leq 1=T_{m}(1)$ , this will imply the bound (3.6).

To use (Szwarc,, 1992, Theorem 1), we rewrite the recursion for the Al-Salam–Chihara polynomials into his notation as

xQ_{n}(x)=\gamma_{n}Q_{n+1}(x)+\beta_{n}Q_{n}(x)+\alpha_{n}Q_{n-1}(x),

where

\gamma_{n}=1/2,\quad\beta_{n}=\frac{a+b}{2}q^{n},\quad\alpha_{n}=\frac{(1-q^{n})(1-abq^{n-1})}{2}.

The Chebyshev polynomials satisfy recursion

xT_{n}(x)=\gamma_{n}^{\prime}T_{n+1}(x)+\beta_{n}^{\prime}T_{n}(x)+\alpha_{n}^{\prime}T_{n-1}(x),

where

\gamma_{n}^{\prime}=\frac{1}{2}(1+\delta_{n=0}),\quad\beta_{n}^{\prime}=0,\quad\alpha_{n}^{\prime}=\frac{1}{2}\delta_{n>0}.

It is then clear that the assumptions of (Szwarc,, 1992, Theorem 1) are satisfied. Thus $a(m,n)\geq 0$ for all $0\leq m\leq n$ , $n=0,1,\dots$ , ending the proof. ∎

3.2. Pointwise asymptotics near the end of the interval of orthogonality

We will need pointwise asymptotics for the Al-Salam–Chihara polynomials at the upper endpoint of the orthogonality interval. Such pointwise limits have been studied for orthogonal polynomials both on finite intervals and on unbounded domains Aptekarev, (1993); Baik et al., (2003); Deift et al., (2001, 1999); Ismail and Li, (2013); Ismail, (1986, 2005); Ismail and Wilson, (1982); Ismail et al., (2022); Kuijlaars et al., (2004); Lubinsky, (2020); Nevai, (1984). In particular, (Aptekarev,, 1993, Theorem 1) gives general conditions on the orthogonality measure and on the three-step recursion of the orthonormal polynomials $\{q_{n}(x)\}$ which imply

(3.7)

\lim_{n\to\infty}\frac{1}{n^{\alpha+1/2}}\,q_{n}\left(1-\tfrac{u^{2}}{2n^{2}}\right)=\frac{J_{\alpha}(u)}{u^{\alpha}}

uniformly on compact subsets of complex plane, where $J_{\alpha}$ is the Bessel function.

Lubinsky (Lubinsky,, 2020, Theorem 1.1) proves a version of (3.7) which can be re-written as

(3.8)

\lim_{n\to\infty}\frac{q_{n}\left(1-\frac{u^{2}}{2n^{2}}\right)}{q_{n}(1)}=2^{\alpha}\Gamma(\alpha+1)\frac{J_{\alpha}(u)}{u^{\alpha}}

uniformly on compact subsets of complex plane. For our density function (3.2) the assumptions in (Lubinsky,, 2020, Theorem 1.1) are satisfied with $\alpha=1/2$ . Indeed, one can re-write (3.2) as

(3.9)

g(x)=\frac{2\sqrt{1-x^{2}}(q,ab;q)_{\infty}\left|(qe^{2i\theta};q)_{\infty}\right|^{2}}{\pi\left|(ae^{i\theta},be^{i\theta};q)_{\infty}\right|^{2}},\;x=\cos\theta.

Note that for $\alpha=1/2$ , the right hand side of (3.8) simplifies to $u^{-1}\sin u$ and that the limit under normalization (3.7) is discussed in (Lubinsky,, 2020, Theorem 1.3).

Pointwise asymptotics for the Al-Salam–Chihara polynomials at the endpoints of the interval of orthogonality follows from the above results. However, our proof is quite direct, and the approach extends to the case of varying orthogonality measures, so we include it for completeness. Our asymptotic result for the Al-Salam-Chihara polynomials with varying orthogonality measure as $q\nearrow 1$ seem to be new and does not follow from Levin and Lubinsky, (2020).

The following asymptotics holds for fixed $q<1$ .

Theorem 3.2.

Fix $a,b$ real or complex conjugate with $|a|,|b|<1$ , $q\in[0,1)$ and $u>0$ . Let $\{Q_{n}\}$ be the Al-Salam–Chihara polynomials (3.1). If $u_{M}\to u$ , then

(3.10)

\lim_{M\to\infty}\tfrac{1}{M}\,Q_{M}\left(1-\tfrac{u_{M}^{2}}{2M^{2}};a,b|q\right)=\frac{\sin u}{u}\;\frac{(a,b;q)_{\infty}}{(q;q)_{\infty}}.

Remark 3.2.

We will also need the case $u=0$ , with

(3.11)

\lim_{n\to\infty}\frac{1}{n}Q_{n}(1;a,b|q)=\frac{(a,b;q)_{\infty}}{(q;q)_{\infty}}.

We also note the following extension of bound (3.6): there is a constant $C=C(a,b,q)$ such that for all $n=0,1\dots$ and $x\in[-1,1]$ we have

(3.12)

|Q_{n}(x;a,b|q)|\leq C(n+1).

Proof of Theorem 3.2.

The use of $M$ in (3.10) is solely to avoid notation clash when we use it in the proof of Theorem 4.2, so for the proof, we replace $M$ by $n$ .

The series (3.3) converges for all $|t|<1$ , thus we can write

(3.13)

\frac{Q_{n}(x;a,b|q)}{(q;q)_{n}}=\frac{1}{2\pi{\textnormal{i}}}\oint_{|t|=r}t^{-n-1}\frac{(at,bt;q)_{\infty}}{(e^{{\textnormal{i}}\theta}t,e^{-{\textnormal{i}}\theta}t;q)_{\infty}}dt,

where $r\in(0,1)$ and the integration is done in the counter-clockwise direction. We fix now $x=\cos\theta\in(-1,1)$ , so that $\theta\in(0,\pi)$ and we note that the integrand

F(t):=t^{-n-1}\frac{(at,bt;q)_{\infty}}{(e^{{\textnormal{i}}\theta}t,e^{-{\textnormal{i}}\theta}t;q)_{\infty}}=t^{-n-1}\frac{(at,bt;q)_{\infty}}{(e^{{\textnormal{i}}\theta}tq,e^{-{\textnormal{i}}\theta}tq;q)_{\infty}}\times\frac{1}{(1-e^{{\textnormal{i}}\theta}t)(1-e^{-{\textnormal{i}}\theta}t)}

has two simple poles at $t=e^{\pm{\textnormal{i}}\theta}$ inside the disk $|t|<q^{-1}$ . By the Cauchy Residue Theorem we can write

(3.14)

\frac{Q_{n}(x;a,b|q)}{(q;q)_{n}}=-{\textnormal{Res}}(F;e^{{\textnormal{i}}\theta})-{\textnormal{Res}}(F;e^{-{\textnormal{i}}\theta})+\frac{1}{2\pi i}\oint_{|t|=R}t^{-n-1}\frac{(at,bt;q)_{\infty}}{(e^{{\textnormal{i}}\theta}t,e^{-{\textnormal{i}}\theta}t;q)_{\infty}}dt,

where $R$ is in the interval $(1,q^{-1})$ .

Recalling that $\left|1-|z|\right|\leq|1-z|\leq 1+|z|$ and $|t|=R>1$ we get

\left|\frac{(at,bt;q)_{\infty}}{(e^{{\textnormal{i}}\theta}t,e^{-{\textnormal{i}}\theta}t;q)_{\infty}}\right|\leq\frac{(-|a|R,-|b|R;q)_{\infty}}{(R;\,q)^{2}_{\infty}}.

Therefore, the third term in (3.14) is bounded by

\frac{(-|a|R,-|b|R;q)_{\infty}}{(R;\,q)^{2}_{\infty}\,R^{n}}\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}0.

Since under our assumptions $\theta=\theta_{n}$ with $\cos\theta_{n}=1-u_{n}^{2}/(2n^{2})$ is such that $n\theta_{n}\to u$ , we see that

\frac{1}{n}{\textnormal{Res}}(F;e^{\pm{\textnormal{i}}\theta_{n}})=\frac{e^{\mp{\textnormal{i}}(n+1)\theta_{n}}}{n(e^{\pm{\textnormal{i}}\theta_{n}}-e^{\mp{\textnormal{i}}\theta_{n}})}\frac{(ae^{\pm{\textnormal{i}}\theta_{n}},be^{\pm{\textnormal{i}}\theta_{n}};q)_{\infty}}{(e^{\pm 2{\textnormal{i}}\theta_{n}}q;q)_{\infty}(q;q)_{\infty}}\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}\frac{e^{\mp iu}(a,b;q)_{\infty}}{\pm 2{\textnormal{i}}u\,(q;\,q)_{\infty}^{2}}.

Consequently, returning to (3.14) we get

\tfrac{1}{n}\,Q_{n}\left(1-\frac{u_{n}^{2}}{2n^{2}};a,b\middle|q\right)\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}\frac{(e^{iu}-e^{-iu})(a,b;q)_{\infty}}{2{\textnormal{i}}u(q;q)_{\infty}}.

∎

Proof of Remark 3.2.

The limit (3.11) follows from (3.5) and an elementary lemma that if $\alpha_{n}\to\alpha$ , $\beta_{n}\to\beta$ then $n^{-1}\sum_{k=1}^{n}\alpha_{n}\beta_{n-k}\to\alpha\beta$ . (To verify the latter, write $\tfrac{1}{n}\sum_{k=1}^{n}\alpha_{k}\beta_{n-k}=\tfrac{1}{n}\sum_{k=1}^{n}(\alpha_{k}-\alpha)\beta_{n-k}+\frac{\alpha}{n}\sum_{k=1}^{n}\beta_{k}\to 0+\alpha\beta$ .) To prove (3.12) we use (3.4), which by triangle inequality gives a bound with explicit constant:

\sup_{x\in[-1,1]}|Q_{n}(x;a,b|q)|\leq\frac{(-|a|,-|b|;q)_{n}}{(q;q)_{n}}(n+1).

∎

The following asymptotics holds for $q\nearrow 1$ . The statement is somewhat cumbersome as we will need to apply this result to $a,b$ that vary with $q$ .

Theorem 3.3.

If $\widetilde{a},\widetilde{b}$ are real in $[0,\infty)$ , or a complex conjugate pair with $\textnormal{Re}(\widetilde{a})\geq 0$ , then with $m=\left\lfloor Mx\right\rfloor+\left\lfloor M\log\left(M\sqrt{(1+\widetilde{a})(1+\widetilde{b})}\right)\right\rfloor$ , $q_{M}=e^{-2/M}$ and $a_{M}=-q_{M}\widetilde{a}$ , $b_{M}:=-q_{M}\widetilde{b}$ we have

(3.15)

\lim_{M\to\infty}\frac{(q_{M};q_{M})_{\infty}^{2}}{M(a_{M},b_{M};q_{M})_{\infty}}\;\frac{Q_{m}(\cos(u/M);a_{M},b_{M}|q_{M})}{(q_{M};q_{M})_{m}}=K_{{\textnormal{i}}|u|}(e^{-x}),\;u\in\mathds{R}.

Proof.

The proof relies on two technical estimates that we will prove in Section 5. In the proof, for simplicity of notation we suppress the dependence of $q$ on $M$ . Fix $u\in{\mathbb{R}}$ and let $a=-q\widetilde{a}$ , $b=-q\widetilde{b}$ , where $q=e^{-2/M}$ .

In view of (3.13) with $r=q$ , we can write

(3.16)

\frac{Q_{m}(\cos(u/M);a,b|q)}{(q;q)_{m}}=\frac{1}{2\pi{\textnormal{i}}}\oint_{|t|=q}\frac{(at,bt;q)_{\infty}}{(q^{{\textnormal{i}}u/2}t,q^{-{\textnormal{i}}u/2}t;q)_{\infty}}t^{-m-1}dt,

where we are integrating counterclockwise along a circle of radius $q<1$ . Next we change the variable of integration $t=q^{z}$ , so that $dt=\ln(q)q^{z}dz=-(2/M)tdz$ , and obtain

(3.17)

\frac{Q_{m}(\cos(u/M);a,b|q)}{(q;q)_{m}}=\frac{1}{M\pi{\textnormal{i}}}\int_{1-M\pi{\textnormal{i}}/2}^{1+M\pi{\textnormal{i}}/2}\frac{(aq^{z},bq^{z};q)_{\infty}}{(q^{{\textnormal{i}}u/2+z},q^{-{\textnormal{i}}u/2+z};q)_{\infty}}e^{2mz/M}dz.

From (3.17), in view of (5.2), we get

(3.18)

\frac{(q;q)_{\infty}^{2}}{M(a,b;q)_{\infty}}\;\frac{Q_{m}(\cos(u/M);a,b|q)}{(q;q)_{m}}\\ =\frac{1}{\pi{\textnormal{i}}}\int_{1-M\pi{\textnormal{i}}/2}^{1+M\pi{\textnormal{i}}/2}\tfrac{\Gamma_{q}({\textnormal{i}}u/2+z)\Gamma_{q}(-{\textnormal{i}}u/2+z)}{M^{2}(1-q)^{2-2z}}\;\tfrac{(-\widetilde{a}q^{1+z},-\widetilde{b}q^{1+z};q)_{\infty}}{(-\widetilde{a}q,-\widetilde{b}q;q)_{\infty}}\;e^{2mz/M}dz=\frac{1}{\pi{\textnormal{i}}}\int_{1-{\textnormal{i}}\infty}^{1+{\textnormal{i}}\infty}f_{M}(z)dz,

where

(3.19)

f_{M}(z)={\mathbf{1}}_{\{|\textnormal{Im}(z)|\leq M\pi/2\}}\frac{\Gamma_{q}({\textnormal{i}}u/2+z)\Gamma_{q}(-{\textnormal{i}}u/2+z)(-\widetilde{a}q^{1+z},-\widetilde{b}q^{1+z};q)_{\infty}}{M^{2-2z}(1-q)^{2-2z}(-\widetilde{a}q,-\widetilde{b}q;q)_{\infty}}\;e^{2mz/M-2\log M}.

We now write $z=1+{\textnormal{i}}s$ . In order to use the dominated convergence theorem, we verify that for every $x,u\in\mathds{R}$ (recall that $m$ depends on $x$ ) there are constants $A,B,M_{*}$ such that for all $M>M_{*}$ and all real $s$ we have

(3.20)

|f_{M}(1+{\textnormal{i}}s)|\leq Ae^{-B|s|}.

To verify (3.20) we write $|f_{M}(1+{\textnormal{i}}s)|=f^{(1)}\cdot f^{(2)}\cdot f^{(3)}$ , where

	$\displaystyle f^{(1)}$	$\displaystyle={\mathbf{1}}_{\{\|s\|\leq M\pi/2\}}\|\Gamma_{q}(1+{\textnormal{i}}s+{\textnormal{i}}u/2)\Gamma_{q}(1+{\textnormal{i}}s-{\textnormal{i}}u/2)\|,$
	$\displaystyle f^{(2)}$	$\displaystyle=\frac{\|(-\widetilde{a}q^{2+{\textnormal{i}}s},-\widetilde{b}q^{2+{\textnormal{i}}s};q)_{\infty}\|}{\|(-\widetilde{a}q,-\widetilde{b}q;q)_{\infty}\|},$
	$\displaystyle f^{(3)}$	$\displaystyle=\left\|\frac{e^{2(1+{\textnormal{i}}s)m/M}}{M^{2}(1-q)^{2-2(1+{\textnormal{i}}s)}}\right\|=e^{2(m/M-\log M)}.$

From (5.24) we see that there exist constants $A,B,M_{*}>0$ such that $f^{(1)}\leq Ae^{-B|s|}$ for all real $s$ . From (5.26) we see that $f^{(2)}\leq 1$ . With

\frac{m}{M}=\frac{\left\lfloor Mx\right\rfloor+\left\lfloor M\log(M\sqrt{(1+\widetilde{a})(1+\widetilde{b})})\right\rfloor}{M}\leq x+\log\,\sqrt{(1+\widetilde{a})(1+\widetilde{b})}+\log\,M,

we see that

f^{(3)}\leq e^{2\left(x+\log\sqrt{(1+\widetilde{a})(1+\widetilde{b})}\right)}.

Thus (3.20) holds.

We can now apply the dominated convergence theorem and pass to the limit inside the integral (3.18). We fix $z=1+{\textnormal{i}}s$ and compute $\lim_{M\to\infty}f_{M}(z)$ by computing the limit for the factors in (3.19).

Clearly, ${\mathbf{1}}_{\{|\textnormal{Im}(z)|<M\pi/2\}}\to 1$ as $M\to\infty$ and by (5.3) we get

\lim_{q\to 1^{-}}\Gamma_{q}({\textnormal{i}}u/2+z)\Gamma_{q}(-{\textnormal{i}}u/2+z)=\Gamma({\textnormal{i}}u/2+z)\Gamma(-{\textnormal{i}}u/2+z).

Since $\lim_{M\to\infty}\,M(1-q_{M})=2$ , from (5.1) we get

\lim_{M\to\infty}\frac{(-\widetilde{a}q_{M}^{1+z},-\widetilde{b}q_{M}^{1+z};q_{M})_{\infty}}{M^{2-2z}(1-q_{M})^{2-2z}(-\widetilde{a}q_{M},-\widetilde{b}q_{M};q_{M})_{\infty}}=\tfrac{1}{4}\frac{2^{2z}}{\left((1+\widetilde{a})(1+\widetilde{b})\right)^{z}}.

Finally, since

\frac{m}{M}=x+\log M+\log\sqrt{(1+\widetilde{a})(1+\widetilde{b})}+O(\tfrac{1}{M}),

we get

e^{2mz/M-2z\log M}\sim e^{2zx+2z\log M+2z\log\sqrt{(1+\widetilde{a})(1+\widetilde{b})}-2z\log M}=\left((1+\widetilde{a})(1+\widetilde{b})\right)^{z}e^{2zx}.

Putting all the factors together,

\lim_{M\to\infty}f_{M}(z)=\frac{1}{4}\Gamma({\textnormal{i}}u/2+z)\Gamma(-{\textnormal{i}}u/2+z)2^{2z}e^{2zx}.

Thus by the dominated convergence theorem,

\lim_{M\to\infty}\frac{1}{\pi{\textnormal{i}}}\int_{1-{\textnormal{i}}\infty}^{1+{\textnormal{i}}\infty}f_{M}(z)dz=\frac{1}{4\pi{\textnormal{i}}}\int_{1-{\textnormal{i}}\infty}^{1+{\textnormal{i}}\infty}\Gamma({\textnormal{i}}u/2+z)\Gamma(-{\textnormal{i}}u/2+z)2^{2z}e^{2zx}dz.

This completes the proof by the Mellin-Barnes type formula for the Bessel K function:

K_{{\textnormal{i}}u}(e^{-x})=\frac{1}{4\pi{\textnormal{i}}}\int_{c-{\textnormal{i}}\infty}^{c+{\textnormal{i}}\infty}\Gamma(s+{\textnormal{i}}{u}/{2})\Gamma(s-{\textnormal{i}}{u}/{2})2^{2s}e^{2xs}ds,

where $c>0$ , $u,x\in\mathds{R}$ . (This is (Olver et al.,, 2023, 10.32.13) [10.32.13] https://dlmf.nist.gov/10.32.E13 with $\nu={\textnormal{i}}u$ , $z=e^{-x}>0$ , and shifted variable of integration $t=s+{\textnormal{i}}u/2$ .) ∎

4. Motzkin paths with $q$ -number weights

We now return to the setting from Section 1, providing additional details and further results for the case of edge weights $a_{n}=[n+2]_{q}$ , $b_{n}=2\sigma[n+1]_{q}$ , $c_{n}=[n]_{q}$ .

We note that with the above choice of the edge weights, probability measure (1.1) is the same as (2.3) with the boundary weights $\alpha_{n}=\rho_{0}^{n}[n+1]_{q}$ and $\beta_{n}=\rho_{1}^{n}$ . Indeed, we note that for an upward step we have $a_{\gamma_{k-1}}=[\gamma_{k-1}+2]_{q}=[\gamma_{k}+1]_{q}$ , for a horizontal step we have $b_{\gamma_{k-1}}=2\sigma[\gamma_{k-1}+1]_{q}=2\sigma[\gamma_{k}+1]_{q}$ and for a downward step we have $c_{\gamma_{k-1}}=[\gamma_{k-1}]_{q}=[\gamma_{k}+1]_{q}$ . So formula (2.1) gives $w({\boldsymbol{\gamma}})=(2\sigma)^{H({\boldsymbol{\gamma}})}\prod_{k=1}^{L}[\gamma_{k}+1]_{q}$ and thus (2.3) gives (1.1).

Recursion (2.11) becomes

(4.1)

xp_{n}(x)=[n+2]_{q}p_{n+1}(x)+2\sigma[n+1]_{q}p_{n}(x)+[n]_{q}p_{n-1}(x).

Then (2.12) gives

(4.2)

\|p_{n}\|_{2}^{2}=\prod_{k=1}^{n}\frac{[k]_{q}}{[k+1]_{q}}=\frac{1}{[n+1]_{q}},\mbox{ hence $\widetilde{p}_{n}(x)=[n+1]_{q}p_{n}(x)$.}

Lemma 4.1.

Suppose $0\leq\sigma\leq 1$ . Polynomials $\{p_{n}\}$ are orthogonal with respect to the density supported on the interval $[A,B]$ with

(4.3)

A=-2\frac{1-\sigma}{1-q},\quad B=2\frac{1+\sigma}{1-q}.

Moreover,

(4.4)

\pi_{n}=p_{n}(B)=\frac{1}{[n+1]_{q}}\sum_{k=0}^{n}\frac{(a;q)_{k}}{(q;q)_{k}}\;\frac{(b;q)_{n-k}}{(q;q)_{n-k}},

where

(4.5)

a=-{q}\left(\sigma+i\sqrt{1-\sigma^{2}}\right),\;b=-{q}\left(\sigma-i\sqrt{1-\sigma^{2}}\right).

We note that $a,b$ are the solutions of the system of equations

(4.6)

a+b=-2\sigma q,\quad ab=q^{2}.

Proof.

To determine the orthogonality measure as well as the sequence $\pi_{n}=p_{n}(B)$ , we establish a connection between polynomials $\{p_{n}\}$ and monic Al-Salam-Chihara polynomials $\{Q_{n}\}$ , defined by the three step recursion (3.1) with complex conjugate parameters (4.5).

It is straightforward to verify that if $\{p_{n}\}$ satisfy recursion (4.1) then polynomials

Q_{n}(x;a,b|q):=\left(1-q\right)^{n}[n+1]_{q}!\;p_{n}\left(2\;\frac{x+\sigma}{1-q}\right),\;n=0,1,\dots

satisfy recursion (3.1) with parameters (4.5). Conversely, with $y=2\frac{x+\sigma}{1-q}$ , we have

(4.7)

p_{n}(y)=\frac{1}{[n+1]_{q}(q,q)_{n}}Q_{n}(x;a,b|q).

Therefore, with $g$ defined by (3.2) (recall, that $g$ is supported on $[-1,1]$ ), polynomials $\{p_{n}\}$ are orthogonal with respect to the weight function (probability density function)

(4.8)

\frac{1-q}{2}\;g\left(\tfrac{(1-q)x}{2}-\sigma\right)

which is supported on the interval $[A,B]$ given by (4.3). Combining (3.5) and (4.7) we see that

(4.9)

\pi_{n}=p_{n}(B)=\frac{Q_{n}(1;a,b|q)}{[n+1]_{q}(q;q)_{n}}

is given by (4.4), see (3.5). ∎

With the above preparations, we can now prove Theorem 1.1.

Proof of Theorem 1.1.

We apply Theorem 2.4. With $\alpha_{n}=\rho_{0}^{n}[n+1]_{q}$ and $\beta_{n}=\rho_{1}^{n}$ , conditions $(A_{1})$ and $(A_{2})$ from Assumption 2.1 hold by (4.7), (4.2) and (3.12). The assumptions on the support of the orthogonality measure follow from Lemma 4.1. The probabilities $\pi_{n}=\frac{s_{n}}{[n+1]_{q}}$ are given by (4.4), so the transition matrix is just a recalculation of (2.19). Recalling that $\widetilde{\pi}_{n}=[n+1]_{q}\pi_{n}$ , see (4.2), the initial laws (1.2) arise from (2.20). Finally, the formulas for the normalization constants are calculated from the relation (4.7) by applying (3.3) with $t=\rho_{j}$ and $\theta=0$ . ∎

4.1. Local limit theorems

We will deduce Theorem 1.2 and Theorem 1.3 from two local limit theorems. These theorems use (2.25) to study convergence of the transition probabilities for the Markov process $\{X_{k}\}$ from Theorem 1.1 under the 1:2 scaling of space and time.

We have the following local limit theorem for fixed $q$ .

Theorem 4.2.

Let $\{X_{k}\}$ be the Markov process introduced in Theorem 1.1. If $0\leq q<1$ , and $0<\sigma\leq 1$ then for $x,y,t>0$ we have

(4.10)

\lim_{N\to\infty}\sqrt{N}\,\mathds{P}\left(X_{\left\lfloor Nt\right\rfloor}=\left\lfloor y\sqrt{N}\right\rfloor\middle|X_{0}=\left\lfloor x\sqrt{N}\right\rfloor\right)=\frac{y}{x}\,{\mathsf{q}}_{\frac{t}{1+\sigma}}(x,y),

where ${\mathsf{q}}_{t}$ , $t>0$ , is given in (1.5).

Furthermore, if $\rho_{0}=1-{\mathsf{c}}/\sqrt{N}$ varies with $N$ for some fixed constant ${\mathsf{c}}>0$ , then denoting by $X_{0}^{(N)}$ a random variable with the initial law (1.2), for $x>0$ we have

(4.11)

\lim_{N\to\infty}\sqrt{N}\,\mathds{P}\left(X_{0}^{(N)}=\left\lfloor x\sqrt{N}\right\rfloor\right)={\mathsf{c}}^{2}xe^{-{\mathsf{c}}x}.

Proof.

We use (2.25) with

(4.12)

k=\left\lfloor Nt\right\rfloor,\quad m=\left\lfloor x\sqrt{N}\right\rfloor,\quad n=\left\lfloor y\sqrt{N}\right\rfloor.

(To avoid notation clash, we shall use $w$ and $\widetilde{w}$ instead of $x,y$ for the variables of integration, and in formulas such as (4.7).)

In view of (3.11) and (4.7), it is clear that for $x,y>0$ we have

(4.13)

\lim_{N\to\infty}\frac{\pi_{\left\lfloor y\sqrt{N}\right\rfloor}}{\pi_{\left\lfloor x\sqrt{N}\right\rfloor}}=\frac{y}{x}.

Indeed, the constant on the right hand side of (3.11) is non-zero, as our $a,b$ have modulus $q<1$ , see (4.5). So, see (4.8) and (4.3), we need to find the limit of the expression

\tfrac{1-q}{2}\tfrac{\sqrt{N}}{B^{k}}\int_{A}^{B}\widetilde{w}^{k}p_{m}(\widetilde{w})\widetilde{p}_{n}(\widetilde{w})g\left(\tfrac{1-q}{2}\widetilde{w}-\sigma\right)d\widetilde{w}=\tfrac{\sqrt{N}}{(1+\sigma)^{k}[m+1]_{q}}\int_{-1}^{1}(w+\sigma)^{k}\frac{Q_{m}(w)}{(q;q)_{m}}\frac{Q_{n}(w)}{(q;q)_{n}}g(w)dw.

(Here, we used (4.7) and identity/substitution $\widetilde{w}/B=(w+\sigma)/(1+\sigma)$ .)

Next we observe that the integral over the interval $[-1,0]$ is negligible. Indeed, on this interval $|w+\sigma|\leq\sigma\vee(1-\sigma)=r(1+\sigma)$ , which together with (3.12) shows that

\tfrac{\sqrt{N}}{[m+1]_{q}}\left|\int_{-1}^{0}\left(\tfrac{w+\sigma}{1+\sigma}\right)^{k}\tfrac{Q_{m}(w)}{(q;q)_{m}}\tfrac{Q_{n}(w)}{(q;q)_{n}}g(w)dw\right|\leq C\sqrt{N}(m+1)(n+1)r^{k}\int_{-1}^{0}g\left(w\right)dw\to 0

as $N\to\infty$ , since $k,m,n$ are given by (4.12) and $r\in[0,1)$ .

It remains to analyze the case of the integral over $[0,1]$ . Changing the variable of integration by setting $u=\sqrt{2N(1-w)}$ , i.e. $w=1-\frac{u^{2}}{2N}$ , we get

(4.14)

\frac{\sqrt{N}}{[m+1]_{q}}\int_{0}^{1}\left(\frac{w+\sigma}{1+\sigma}\right)^{k}\,\frac{Q_{m}(w)}{(q;q)_{m}}\frac{Q_{n}(w)}{(q;q)_{n}}g(w)dw\\ \frac{1}{[m+1]_{q}}\int_{0}^{\sqrt{2N}}\left(1-\tfrac{u^{2}}{2(1+\sigma)N}\right)^{k}\frac{Q_{m}\left(1-\tfrac{u^{2}}{2N}\right)}{\sqrt{N}(q;q)_{m}}\frac{Q_{n}\left(1-\tfrac{u^{2}}{2N}\right)}{\sqrt{N}(q;q)_{n}}\sqrt{N}g\left(1-\tfrac{u^{2}}{2N}\right)udu.

Clearly, $\frac{1}{[m+1]_{q}}\to 1-q$ . The next observation is that for $u\in[0,\sqrt{2N}]$ and $k=\left\lfloor Nt\right\rfloor$ , we have

(4.15)

\left(1-\frac{u^{2}}{2(1+\sigma)N}\right)^{k}\sim e^{-\frac{u^{2}t}{2(1+\sigma)}}\mbox{ as $N\to\infty$}.

Now we refer to (3.9) for $g\left(1-\tfrac{u^{2}}{2N}\right)$ . Noting that $\theta\sim\frac{u}{\sqrt{N}}\sim\sqrt{1-(1-\frac{u^{2}}{2N})^{2}}$ , we get

(4.16)

\sqrt{N}g\left(1-\frac{u^{2}}{2N}\right)\sim\frac{2\,u}{\pi}(q;q)_{\infty}^{3}\frac{(ab;q)_{\infty}}{(a,b;q)^{2}_{\infty}}=\frac{2u}{\pi(1-q)}\;\frac{(q;q)_{\infty}^{4}}{(a,b;q)^{2}_{\infty}}

for $u\in[0,\sqrt{2N}]$ , where the last equality is by (4.6).

We now determine the asymptotics for the remaining factors of the integrand on the right hand side of (4.14). We apply Theorem 3.2 with $M=\left\lfloor\sqrt{N}x\right\rfloor\to\infty$ and with $u_{M}/M:=u/\sqrt{N}$ so that $u_{M}=uM/\sqrt{N}=ux+O(1/M)$ . Recalling (4.12), we see that $m=M$ and thus

(4.17)

\frac{Q_{m}\left(1-\frac{u^{2}}{2N}\right)}{\sqrt{N}}=\frac{\left\lfloor\sqrt{N}x\right\rfloor}{\sqrt{N}}\;\frac{Q_{M}\left(1-\frac{u_{M}^{2}}{2M^{2}}\right)}{M}\to x\,\frac{\sin(ux)}{ux}\;\frac{(a,b;q)_{\infty}}{(q;q)_{\infty}}=\frac{\sin(ux)}{u}\;\frac{(a,b;q)_{\infty}}{(q;q)_{\infty}}

as $N\to\infty$ . The same argument with $n$ from (4.12) gives

(4.18)

\frac{Q_{n}\left(1-\frac{u^{2}}{2N}\right)}{\sqrt{N}}\to\frac{\sin(uy)}{u}\;\frac{(a,b;q)_{\infty}}{(q;q)_{\infty}}\mbox{ as $N\to\infty$}.

To verify uniform integrability, we apply elementary bounds

(q;q)_{\infty}\leq|(z;q)_{\infty}|\leq(-q;q)_{\infty},\quad|z|\leq q

within (3.9). We get

\sqrt{N}\,g\left(1-\tfrac{u^{2}}{2N}\right)\leq\sqrt{N}\sqrt{1-\left(1-\tfrac{u^{2}}{2N}\right)^{2}}\;\frac{2(-q;q)_{\infty}^{2}}{\pi(1-q)(q;q)_{\infty}}\;\leq C(q)|u|.

Together with an elementary bound $(1-x/N)^{k}\leq e^{-xt}$ for $0\leq x\leq N$ and the fact that $[m+1]_{q}\to 1/(1-q)$ this show that the integrand on the right hand side of (4.14) is bounded by an integrable function $Cu^{2}e^{-cu^{2}}$ , so we can use dominated convergence theorem to pass to the limit under the integral. Combining (4.13), (4.14), (4.15), (4.16), (4.17) and (4.18) we conclude that the left hand side of (4.10) becomes

\frac{y}{x}\,\frac{(q;q)_{\infty}^{4}}{(a,b;q)_{\infty}^{2}}\;\frac{2}{\pi}\int_{0}^{\infty}\lim_{N\to\infty}\left({\bf 1}_{u<\sqrt{2N}}\left(1-\frac{u^{2}}{2(1+\sigma)N}\right)^{k}\frac{Q_{m}\left(1-\frac{u^{2}}{2N}\right)}{\sqrt{N}(q;q)_{m}}\frac{Q_{n}\left(1-\frac{u^{2}}{2N}\right)}{\sqrt{N}(q;q)_{n}}\right)\,u^{2}\,du\\ =\,\frac{y}{x}\,\frac{2}{\pi}\int_{0}^{\infty}e^{-\frac{u^{2}t}{2(1+\sigma)}}\sin(ux)\sin(uy)\,du=\frac{y}{x}\,\sqrt{\frac{1+\sigma}{2\pi t}}\,\left(e^{-\frac{(y-x)^{2}(1+\sigma)}{2t}}-e^{-\frac{(y+x)^{2}(1+\sigma)}{2t}}\right)

as required.

To prove (4.11) we use (3.11). Using notation (4.12), from (1.2) and (3.5) we get

(4.19)		$\displaystyle\sqrt{N}\mathds{P}(X_{0}^{(N)}=m)=\sqrt{N}\frac{(\rho_{0};q)_{\infty}^{2}}{(a\rho_{0},b\rho_{0};q)_{\infty}}\rho_{0}^{m}\frac{Q_{m}(1;a,b\|q)}{(q;q)_{m}}$
	$\displaystyle=N\left(1-\frac{{\mathsf{c}}}{\sqrt{N}}\right)^{2}\,\frac{\left\lfloor x\sqrt{N}\right\rfloor}{N}\,\left(1-\frac{{\mathsf{c}}}{\sqrt{N}}\right)^{\left\lfloor x\sqrt{N}\right\rfloor}\,\frac{(q\rho_{0};q)_{\infty}^{2}}{(q;q)_{m}}\;\frac{Q_{m}(1;a,b\|q)}{m\,(a\rho_{0},b\rho_{0};q)_{\infty}}\to{\mathsf{c}}^{2}x\,e^{-{\mathsf{c}}x}$

as $N\to\infty$ ∎

Next we prove a local limit theorem with $q\nearrow 1$ . Note that if parameter $q$ varies with $N$ , then the law of the Markov process $\{X_{k}\}$ from Theorem 1.1 also varies with $N$ , so we denote this process by $\{X_{k}^{(N)}\}$ .

For $z\in\mathds{R}$ we denote

(4.20)

m_{N}(z)=\left\lfloor z\sqrt{N}\right\rfloor+\left\lfloor\sqrt{N}\,\log\,\sqrt{N\,2\log(1+\sigma)}\right\rfloor.

With this notation we have the following version of Theorem 4.2 for $q\nearrow 1$ :

Theorem 4.3.

If $0<\sigma\leq 1$ is fixed and $q=e^{-2/\sqrt{N}}$ then for $x,y\in\mathds{R}$ , $t>0$ we have

(4.21)

\lim_{N\to\infty}\sqrt{N}\,\mathds{P}\Big{(}X_{\left\lfloor Nt\right\rfloor}^{(N)}=m_{N}(y)\Big{|}\;X_{0}^{(N)}=m_{N}(x)\Big{)}=\frac{K_{0}(e^{-y})}{K_{0}(e^{-x})}\;{\mathsf{p}}_{\frac{t}{1+\sigma}}(x,y),

where ${\mathsf{p}}_{t}$ , $t>0$ , is given in (1.9).

In addition, if $\rho_{0}=e^{-{\mathsf{c}}/\sqrt{N}}=q^{{\mathsf{c}}/2}$ for some fixed constant ${\mathsf{c}}>0$ , then

(4.22)

\lim_{N\to\infty}\sqrt{N}\mathds{P}\left(X_{0}^{(N)}=m_{N}(x)\right)=\frac{4}{2^{\mathsf{c}}\Gamma({\mathsf{c}}/2)^{2}}K_{0}(e^{-x}).

Proof.

We write the parameters (4.5) in trigonometric form $a=-qe^{{\textnormal{i}}\alpha}$ , $b=-qe^{-{\textnormal{i}}\alpha}$ with $\sigma=\cos\alpha$ , $\alpha\in(-\pi/2,\pi/2)$ .

First, we prove (4.22). As in (4.19) we have

\sqrt{N}\,\mathds{P}(X_{0}^{(N)}=m_{N}(x))=\sqrt{N}\frac{(\rho_{0};q)_{\infty}^{2}}{(a\rho_{0},b\rho_{0};q)_{\infty}}\rho_{0}^{m_{N}(x)}\frac{Q_{m_{N}(x)}(1;a,b|q)}{(q;q)_{m_{N}(x)}}.

Denote $M=\sqrt{N}$ . Then with $\widetilde{a}=e^{i\alpha}$ and $\widetilde{b}=e^{-i\alpha}$ we see that $m$ as defined in Theorem 3.3 assumes the form $m=m_{N}(x)$ . Consequently, we can rewrite this expression as follows

\sqrt{N}\,\mathds{P}(X_{0}^{(N)}=m_{N}(x))=\left(Me^{-\frac{m}{M}}\right)^{{\mathsf{c}}}\,M^{2-{\mathsf{c}}}\,\left(\frac{(\rho_{0};q)_{\infty}}{(q;q)_{\infty}}\right)^{2}\frac{(a,b;q)_{\infty}}{(a\rho_{0},b\rho_{0};q)_{\infty}}\times J_{M},

where

(4.23)

J_{M}=\frac{(q;q)_{\infty}^{2}}{M(-q\widetilde{a},-q\widetilde{b};q)_{\infty}}\frac{Q_{m}(1;-q\widetilde{a},-q\widetilde{b}|q)}{(q;q)_{m}}\to K_{0}(e^{-x})\quad\mbox{as }\;N\to\infty

with the limit deduced from (3.15). Clearly,

(4.24)

Me^{-\frac{m}{M}}=\sqrt{N}e^{-\frac{m_{N}(x)}{\sqrt{N}}}\to\frac{e^{-x}}{\sqrt{2(1+\sigma)}}\quad\mbox{as }\;N\to\infty.

Moreover, recalling (5.2), we can write

(4.25)

M^{2-{\mathsf{c}}}\,\left(\frac{(\rho_{0};q)_{\infty}}{(q;q)_{\infty}}\right)^{2}=M^{2-{\mathsf{c}}}\,\left(\frac{(q^{\frac{{\mathsf{c}}}{2}};q)_{\infty}}{(q;q)_{\infty}}\right)^{2}=\frac{(M(1-q))^{2-{\mathsf{c}}}}{\Gamma_{q}\left({\mathsf{c}}/2\right)^{2}}\to\frac{2^{2-{\mathsf{c}}}}{\Gamma\left({\mathsf{c}}/2\right)^{2}}

as $N\to\infty$ , where the last limit follows from (5.3).

Finally, relying on (5.1), we get

(4.26)

\frac{(a,b;q)_{\infty}}{(a\rho_{0},b\rho_{0};q)_{\infty}}=\frac{(-\widetilde{a}q,-\widetilde{b}q;q)_{\infty}}{(-\widetilde{a}q^{1+{\mathsf{c}}/2},-\widetilde{b}q^{1+{\mathsf{c}}/2};q)_{\infty}}\\ =\frac{(1+\widetilde{a}q^{\frac{{\mathsf{c}}}{2}})(1+\widetilde{b}q^{\frac{{\mathsf{c}}}{2}})}{(1+\widetilde{a})(1+\widetilde{b})}\,\frac{(-\widetilde{a};q)_{\infty}}{(-\widetilde{a}q^{\frac{{\mathsf{c}}}{2}};q)_{\infty}}\,\frac{(-\widetilde{b};q)_{\infty}}{(-\widetilde{b}q^{\frac{{\mathsf{c}}}{2}};q)_{\infty}}\to(1+e^{{\textnormal{i}}\alpha})^{{\mathsf{c}}/2}(1+e^{-{\textnormal{i}}\alpha})^{{\mathsf{c}}/2}=2^{{\mathsf{c}}/2}(1+\sigma)^{{\mathsf{c}}/2}

as $(1+e^{{\textnormal{i}}\alpha})^{{\mathsf{c}}/2}(1+e^{-{\textnormal{i}}\alpha})^{{\mathsf{c}}/2}=(2+2\cos\alpha)^{{\mathsf{c}}/2}$ . Combining (4.23), (4.24), (4.25) and (4.26) we get (4.22).

Second, we prove (4.21). Our starting point is formula (2.25) which we use with

(4.27)

m:=m_{N}(x),\quad n:=m_{N}(y),\quad k=\left\lfloor Nt\right\rfloor.

Even though $x,y\in\mathds{R}$ , note that $m,n$ are non-negative integers for large enough $N$ . In particular, we note that $m/n\to 1$ and $m/k\to 0$ as $N\to\infty$ . It is also clear that $[n+1]_{q}\sim[m+1]_{q}\sim\sqrt{N}/2$ . (To avoid notation clash, we use $w$ and $\widetilde{w}$ instead of $x,y$ for the variables of integration, or in formulas like (4.7).)

From (2.25), we get

(4.28)

\sqrt{N}\mathds{P}(X_{k}=m_{N}(y)|X_{0}=m_{N}(x))=\frac{\pi_{n}}{\pi_{m}}\frac{\sqrt{N}}{B^{k}}\int_{A}^{B}\widetilde{w}^{k}p_{m}(\widetilde{w})[n+1]_{q}p_{n}(\widetilde{w})g(\widetilde{w})d\widetilde{w}\\ =\frac{Q_{n}(1;a,b|q)}{(q;q)_{n}}\;\frac{(q;q)_{m}}{Q_{m}(1;a,b|q)}\;\frac{\sqrt{N}}{[n+1]_{q}}\int_{-1}^{1}\frac{\left(w+\sigma\right)^{k}}{(1+\sigma)^{k}}\frac{Q_{m}\left(w;a,b|q\right)}{(q;q)_{m}}\,\frac{Q_{n}\left(w;a,b|q\right)}{(q;q)_{n}}\,g(w)\,dw,

where the latter equation follows from (4.9) and (4.7).

By Theorem 3.3 used with $M=\sqrt{N}$ , $u=0$ , and $m=m_{N}(y)$ or $m=m_{N}(x)$ we get

(4.29)

\lim_{N\to\infty}\frac{Q_{n}(1;a,b|q)}{(q;q)_{n}}\;\frac{(q;q)_{m}}{Q_{m}(1;a,b|q)}\\ =\lim_{N\to\infty}\frac{(q;q)_{\infty}^{2}Q_{m_{N}(y)}\,(1;a,b|q)}{\sqrt{N}(a,b;q)_{\infty}\,(q;q)_{m_{N}(y)}}\times\lim_{N\to\infty}\frac{\sqrt{N}(a,b;q)_{\infty}(q;q)_{m_{N}(x)}}{(q;q)_{\infty}^{2}Q_{m_{N}(x)}(1;a,b|q)}=\frac{K_{0}(e^{-y})}{K_{0}(e^{-x})}.

This gives the pair of Bessel functions $K_{0}$ on the right-hand side of (4.21).

Next, we show that the contribution from the integral in (4.28) over $[-1,1/2]$ is negligible. Noting that for $w$ from this interval $|w+\sigma|<(1/2+\sigma)\vee(1-\sigma)=r(1+\sigma)$ , similarly as in the proof of Theorem 4.2, the contribution of this integral is bounded by

Cr^{k}\,\frac{\sqrt{N}}{[n+1]_{q}}\int_{-1}^{1/2}\frac{|Q_{m}\left(w;a,b|q\right)|}{(q;q)_{m}}\frac{|Q_{n}\left(w;a,b|q\right)|}{(q;q)_{n}}g\left(w\right)dw\leq\\ Cr^{k}\,\frac{\sqrt{N}}{2[n+1]_{q}}\int_{-1}^{1}\left(\frac{Q_{m}^{2}\left(w;a,b|q\right)}{(q;q)_{m}^{2}}+\frac{Q_{n}^{2}\left(w;a,b|q\right)}{(q;q)_{n}^{2}}\right)g\left(w\right)dw\\ =Cr^{k}\,\frac{\sqrt{N}}{2[n+1]_{q}}\left([m+1]_{q}+[n+1]_{q}\right)\to 0.

Here $C$ is a constant from boundedness of the convergent sequence in (4.29), and the last equality follows from (4.2) and (4.7). To compute the limit we used the observation that $[n+1]_{q}\sim[m+1]_{q}\sim\sqrt{N}/2$ and $r\in[0,1)$ .

Substituting $w=\cos(u/\sqrt{N})$ into (4.28) and discarding the non-contributing part of the integral, we have

(4.30)

\sqrt{N}\mathds{P}(X_{k}=m_{N}(y)|X_{0}=m_{N}(x))\sim\frac{K_{0}(e^{-y})}{K_{0}(e^{-x})}\int_{0}^{\infty}\,\mathbf{1}_{(0,\,\frac{\pi}{3}\sqrt{N})}(u)f_{N}(u)\,du,

where

f_{N}(u)=\frac{1}{[n+1]_{q}}\left(\tfrac{\sigma+\cos\frac{u}{\sqrt{N}}}{1+\sigma}\right)^{k}\frac{Q_{m}\left(\cos\frac{u}{\sqrt{N}};a,b|q\right)}{(q;q)_{m}}\,\frac{Q_{n}\left(\cos\frac{u}{\sqrt{N}};a,b|q\right)}{(q;q)_{n}}g\left(\cos(\tfrac{u}{\sqrt{N}})\right)\sin(\tfrac{u}{\sqrt{N}}).

Referring first to (3.2) and then using $(1-q)(ab;q)_{\infty}=(q;q)_{\infty}$ and (5.2) we get

g\left(\cos(\tfrac{u}{\sqrt{N}})\right)\sin(\tfrac{u}{\sqrt{N}})=\frac{(q,ab;q)_{\infty}}{2\pi}\,\frac{|(q^{{\textnormal{i}}u};q)_{\infty}|^{2}}{|(aq^{{\textnormal{i}}u/2},bq^{{\textnormal{i}}u/2};q)_{\infty}|^{2}}=\frac{(q;q)_{\infty}^{4}(1-q)}{2\pi|\Gamma_{q}({\textnormal{i}}u)|^{2}\,|(aq^{{\textnormal{i}}u/2},bq^{{\textnormal{i}}u/2};q)_{\infty}|^{2}}.

Thus $f_{N}$ can be written as

(4.31)

f_{N}(u)=\frac{N(1-q)}{[n+1]_{q}}\,\left(\frac{\sigma+\cos\frac{u}{\sqrt{N}}}{1+\sigma}\right)^{\left\lfloor Nt\right\rfloor}\,\frac{1}{2\pi\,|\Gamma_{q}({\textnormal{i}}u)|^{2}}\\ \times\frac{(q;q)_{\infty}^{2}Q_{m_{N}(x)}(\cos\frac{u}{\sqrt{N}};a,b|q)}{\sqrt{N}(a,b;q)_{\infty}\,(q;q)_{m_{N}(x)}}\,\frac{(q;q)_{\infty}^{2}Q_{m_{N}(y)}(\cos\frac{u}{\sqrt{N}};a,b|q)}{\sqrt{N}(a,b;q)_{\infty}\,(q;q)_{m_{N}(y)}}\,\frac{(a,b;q)_{\infty}^{2}}{|(aq^{{\textnormal{i}}u/2},bq^{{\textnormal{i}}u/2};q)_{\infty}|^{2}}.

Using $[n+1]_{q}\sim\sqrt{N}/2$ again, we see that the first factor is asymptotically constant, ${N(1-q)}/{[n+1]_{q}}\sim 4$ . Furthermore,

\left(\frac{\sigma+\cos\frac{u}{\sqrt{N}}}{1+\sigma}\right)^{\left\lfloor Nt\right\rfloor}\to e^{-\frac{u^{2}t}{2(1+\sigma)}}.

By Lemma 5.1, $\Gamma_{q}({\textnormal{i}}u)\to\Gamma({\textnormal{i}}u)$ . From Theorem 3.3, we see that for real $z$ , recall (4.20),

(4.32)

\frac{(q;q)_{\infty}^{2}}{\sqrt{N}(a,b;q)_{\infty}}\,\frac{Q_{m_{N}(z)}(\cos\frac{u}{\sqrt{N}};a,b|q)}{(q;q)_{m_{N}(z)}}\to K_{{\textnormal{i}}u}(e^{-z}).

In view of Lemma 5.1

\frac{(a,b;q)_{\infty}}{(aq^{{\textnormal{i}}u/2},bq^{{\textnormal{i}}u/2};q)_{\infty}}=\frac{(-e^{{\textnormal{i}}\alpha}q;q)_{\infty}}{(-e^{{\textnormal{i}}\alpha}q^{1+{\textnormal{i}}u/2};q)_{\infty}}\frac{(-e^{-{\textnormal{i}}\alpha}q;q)_{\infty}}{(-e^{-{\textnormal{i}}\alpha}q^{1+{\textnormal{i}}u/2};q)_{\infty}}\\ \to(1+e^{{\textnormal{i}}\alpha})^{{\textnormal{i}}u/2}(1+e^{-{\textnormal{i}}\alpha})^{{\textnormal{i}}u/2}=(2+2\sigma)^{{\textnormal{i}}\frac{u}{2}}.

Note that $|(2+2\sigma)^{{\textnormal{i}}\frac{u}{2}}|=1$ , so this factor does not contribute to the limit of $f_{N}$ .

To summarize, we see that

(4.33)

\lim_{N\to\infty}\,\mathbf{1}_{(0,\,\frac{\pi}{3}\sqrt{N})}(u)\,f_{N}(u)=\mathbf{1}_{(0,\infty)}(u)\;\frac{2}{\pi}\,e^{-\frac{u^{2}t}{2(1+\sigma)}}\,K_{{\textnormal{i}}u}(e^{-x})\,K_{{\textnormal{i}}u}(e^{-y})\,\frac{1}{|\Gamma({\textnormal{i}}u)|^{2}}.

We now justify that one can pass to the limit under the integral (4.30). For $0<u<\frac{\pi}{3}\,\sqrt{N}$ we have

\left(\frac{\cos(u/\sqrt{N})+\sigma}{1+\sigma}\right)^{k}\leq\left(1-\frac{u^{2}}{2(1+\sigma)N}\right)^{Nt}\leq Ce^{-\frac{u^{2}t}{2(1+\sigma)}}

for some $C<2.5$ . Recall that $t,\sigma$ are fixed.

By Proposition 3.1 and Theorem 3.3 we know that for real $z$ , recall (4.32), we have

\left|\frac{(q;q)_{\infty}^{2}}{\sqrt{N}(a,b;q)_{\infty}}\,\frac{Q_{m_{N}(z)}(\cos\frac{u}{M};a,b|q)}{(q;q)_{m_{N}(z)}}\right|\leq\frac{(q;q)_{\infty}^{2}}{\sqrt{N}(a,b;q)_{\infty}}\,\frac{Q_{m_{N}(z)}(1;a,b|q)}{(q;q)_{m_{N}(z)}}\to K_{0}(e^{-z}),

therefore the expression on the left-hand side above is uniformly bounded in $u,N$ . By Lemma 5.9, there exist $A>0$ , $B>0$ and $M^{*}$ such that

{\mathbf{1}}_{\{|u|<\pi\sqrt{N}/2\}}\frac{(a,b;q)_{\infty}^{2}}{|(aq^{{\textnormal{i}}u/2},bq^{{\textnormal{i}}u/2};q)_{\infty}|^{2}}\leq Ae^{B|u|}

for all $N\geq M_{*}^{2}$ and all $u\in\mathds{R}$ . By Lemma 5.7, $1/|\Gamma_{q}({\textnormal{i}}u)|^{2}\leq Au^{2}e^{B|u|}$ for $|u|<\sqrt{N}\pi/2$ . Moreover, recalling that $n=m_{N}(y)$ , the convergent sequence $N(1-q)/[n+1]_{q}$ is bounded. Combining all these bounds, we see that there are constants $A,B,M_{*}>0$ such that for all $N>M_{*}^{2}$ function $f_{N}(u)\mathbf{1}_{(0,\frac{\pi}{3}\sqrt{N})}(u)$ is bounded on $[0,\infty)$ by the integrable function

Au^{2}e^{B|u|}e^{-\frac{u^{2}t}{2(1+\sigma)}},

so we can invoke the dominated convergence theorem. Taking the limit (4.33) inside the integral in (4.30), we see that

\lim_{N\to\infty}\sqrt{N}\mathds{P}(X_{k}=m_{N}(y)|X_{0}=m_{N}(x))=\frac{K_{0}(e^{-y})}{K_{0}(e^{-x})}\,\frac{2}{\pi}\int_{0}^{\infty}\,e^{-\frac{u^{2}t}{2(1+\sigma)}}\,K_{{\textnormal{i}}u}(e^{-x})\,K_{{\textnormal{i}}u}(e^{-y})\,\frac{du}{|\Gamma({\textnormal{i}}u)|^{2}},

which ends the proof by (1.9). ∎

4.2. Proofs of Theorems 1.2 and 1.3

Both proofs are very similar.

Proof of Theorem 1.2.

Denote by $\{X_{k}^{(N)}\}$ the Markov process from the conclusion of Theorem 1.1 with $\rho_{0}=e^{-{\mathsf{c}}/\sqrt{N}}$ . For fixed $t_{0}=0<t_{1}<\dots<t_{d}$ and $x_{0},\dots,x_{d}>0$ by (4.11), (4.10) of Theorem 4.2 and the Markov property, we have

\lim_{N\to\infty}N^{(d+1)/2}\,\mathds{P}\left(\frac{X_{\left\lfloor t_{i}N\right\rfloor}^{(N)}}{\sqrt{N}}=\tfrac{\left\lfloor x_{i}\sqrt{N}\right\rfloor}{\sqrt{N}},\,i=0,1,\ldots,d\right)={\mathsf{c}}^{2}e^{-{\mathsf{c}}x_{0}}\,x_{d}\,\prod_{j=1}^{d}\,{\mathsf{q}}_{\frac{t_{j}-t_{j-1}}{1+\sigma}}(x_{j-1},x_{j}),

where we recall that ${\mathsf{q}}_{t}$ is defined in (1.5). This proves local convergence to the joint density of the vector $(\xi_{0},\xi_{t_{1}},\dots,\xi_{t_{d}})$ , see (1.4) and (1.6).

By (Billingsley,, 1999, Theorem 3.3), the convergence of finite-dimensional distributions as stated in (1.3) follows. ∎

Proof of Theorem 1.3.

Denote by $\{X_{k}^{(N)}\}$ the Markov process from the conclusion of Theorem 1.1 with $\rho_{0}=e^{-{\mathsf{c}}/\sqrt{N}}$ and $q=e^{-2/\sqrt{N}}$ .

Clearly, the centering sequence at the left-hand side of (1.7) in Theorem 1.3 can be replaced by $\left\lfloor\sqrt{N}\log\sqrt{2N(1+\sigma)}\right\rfloor$ , $N\geq 1$ .

For fixed $t_{0}=0<t_{1}<\dots<t_{d}$ and $x_{0},\dots,x_{d}\in\mathds{R}$ by (4.22), (4.21) of Theorem 4.3 and the Markov property we have

\lim_{N\to\infty}N^{\frac{d+1}{2}}\mathds{P}\Big{(}\frac{X_{\left\lfloor t_{i}N\right\rfloor}^{(N)}-\left\lfloor\sqrt{N}\log\sqrt{2N(1+\sigma)}\right\rfloor}{\sqrt{N}}=\frac{\left\lfloor x_{i}\sqrt{N}\right\rfloor}{\sqrt{N}},\,i=0,1,\ldots,d\Big{)}\\ \stackrel{{\scriptstyle\eqref{zN}}}{{=}}\lim_{N\to\infty}N^{\frac{d+1}{2}}\mathds{P}\Big{(}X_{\left\lfloor t_{i}N\right\rfloor}^{(N)}=m_{N}(x_{i}),\,i=0,1,\ldots,d\Big{)}\\ =\frac{4}{2^{\mathsf{c}}\Gamma({\mathsf{c}}/2)^{2}}K_{0}(e^{-x})\prod_{j=1}^{d}{\mathsf{p}}_{\frac{t_{j}-t_{j-1}}{1+\sigma}}(x_{j-1},x_{j}),

where we recall that ${\mathsf{p}}_{t}$ is defined in (1.9). This proves local convergence to the joint density of the vector $(\zeta_{0},\zeta_{t_{1}},\dots,\zeta_{t_{d}})$ , see (1.8) and (1.10).

By (Billingsley,, 1999, Theorem 3.3) the convergence of finite-dimensional distributions as stated in (1.7) follows. ∎

5. Auxiliary results on special functions

5.1. Some useful limits for $q\nearrow 1$

The following result appears in Ch. 16 of Ramanujan’s notebook, see (Adiga et al.,, 1985, Entry 1), (Berndt,, 1991, p. 13, Entry 1(i)) or (Gasper and Rahman,, 2004, (I34)) for the statement. The proof in (Koornwinder,, 1990, Proposition A.2) is for real $\lambda$ , which is not enough for our purposes. The proof in (Berndt,, 1991, p. 13) uses analytic continuation from $|z|<1$ . We note that we always use principal branch of the logarithm and of the power function.

Lemma 5.1 (Ramanujan).

For all $\lambda\in{\mathbb{C}}$ and $z$ in cut complex plane ${\mathbb{C}}\setminus[1,\infty)$ we have

(5.1)

\frac{(z;q)_{\infty}}{(zq^{\lambda};q)_{\infty}}\to(1-z)^{\lambda}

as $q\to 1^{-}$ . The convergence is uniform in $z$ on compact subsets of ${\mathbb{C}}\setminus[1,\infty)$ .

Proof.

Denote the left-hand side of (5.1) by $f_{q}(z)$ . Then

h_{q}(z):=\frac{f_{q}^{\prime}(z)}{f_{q}(z)}=\sum\limits_{n\geq 0}\bigg{[}\frac{q^{\lambda+n}}{1-zq^{\lambda+n}}-\frac{q^{n}}{1-zq^{n}}\bigg{]}=-\frac{1-q^{\lambda}}{1-q}\times(1-q)\sum\limits_{n\geq 0}\frac{{q^{n}}}{(1-zq^{\lambda+n})(1-zq^{n})}.

Thus as $q\to 1^{-}$ we have

h_{q}(z)\to-\lambda\int_{0}^{1}\frac{dy}{(1-zy)^{2}}=-\frac{\lambda}{1-z}.

This convergence clearly holds pointwise, for all $z\in{\mathbb{C}}\setminus[1,\infty)$ . To prove uniform convergence on compact subsets, we use Montel’s theorem, which requires us to show that the functions $h_{q}$ are uniformly bounded on compact subsets of ${\mathbb{C}}\setminus[1,\infty)$ as $q\to 1^{-}$ . For small $\epsilon>0$ , we define $D_{\epsilon}$ to be the closed domain

D_{\epsilon}=\{z\in{\mathds{C}}:|\arg(z)|\leq\epsilon\}\setminus\{z\in{\mathds{C}}:|z|<1-\epsilon\}.

The domain $D_{\epsilon}$ is shown on Figure 2.

Figure 2. Domain

D_{\epsilon}

Now we take an arbitrary compact set $K\subset{\mathbb{C}}\setminus[1,\infty)$ . There exists $\epsilon>0$ small enough such that $K\subset{\mathbb{C}}\setminus D_{\epsilon}$ . Since $q\in[0,1)$ , for any $z\in K$ (in fact, for any $z\not\in D_{\varepsilon}$ ) and any $n=0,1,\dots$ we have $q^{n}z\in{\mathbb{C}}\setminus D_{\epsilon}\subset{\mathbb{C}}\setminus D_{\epsilon/2}$ .

Next, noting that $\lim_{q\to 1^{-}}q^{\lambda}=1$ , there is $q_{0}\in(0,1)$ such that $|q^{\lambda}|<1-\epsilon/2$ and $|\arg(q^{\lambda})|<\epsilon/2$ for all $q\in(q_{0},1)$ . This implies that for any $z\not\in D_{\epsilon}$ we have $q^{\lambda}z\in{\mathbb{C}}\setminus D_{\epsilon/2}$ . Indeed, if $z\not\in D_{\epsilon}$ then either $|z|<1-\epsilon$ or $|\arg(z)|>\epsilon$ . In the first case we have $|zq^{\lambda}|<(1-\epsilon)(1+\epsilon/2)<1-\epsilon/2$ . In the second case, $|\arg(q^{\lambda}z)|\geq|\arg(z)|-|\arg(q^{\lambda})|>\epsilon/2$ .

We have thus shown that there exists $q_{0}$ such that if $z\in K$ then, with $zq^{n}\not\in D_{\epsilon}$ , both $q^{n}z$ and $q^{n+\lambda}z$ are in ${\mathbb{C}}\setminus D_{\epsilon/2}$ for all $n=0,1,\dots$ and all $q\in(q_{0},1)$ . But the distance from $1$ to the set ${\mathbb{C}}\setminus D_{\epsilon/2}$ is strictly positive. This means that there is $\delta>0$ such that for all $z\in K$ , $q\in(q_{0},1)$ and every $n\geq 0$ we have

|1-zq^{\lambda+n}|\geq\delta,\;\;\;|1-zq^{n}|\geq\delta,

which implies an upper bound

|h_{q}(z)|<\frac{|1-q^{\lambda}|}{|1-q|}\times(1-q)\sum\limits_{n\geq 0}q^{n}\delta^{-2}=\frac{|1-q^{\lambda}|}{|1-q|}\times\delta^{-2}<C|\lambda|\delta^{-2},

for some constant $C>0$ .

Thus we have proved that the functions $h_{q}(z)$ converge to $-\lambda/(1-z)$ as $q\to 1^{-}$ , uniformly on compact subsets of ${\mathbb{C}}\setminus[1,\infty)$ . Then, integrating over the segment from $0$ to $z$

f_{q}(z)=\exp\Big{(}\int_{0}^{z}h_{q}(w)dw\Big{)}\to(1-z)^{\lambda}

as $q\to 1^{-}$ , also uniformly on compact subsets of ${\mathbb{C}}\setminus[1,\infty)$ . ∎

Recall (Gasper and Rahman,, 2004, Section 1.10) that for $z\in\mathds{C}\setminus\{0,-1,-2,\dots\}$ and $q\in[0,1)$ , the $q$ -Gamma function is defined by

(5.2)

\Gamma_{q}(z):=(1-q)^{1-z}\;\frac{(q;q)_{\infty}}{(q^{z};q)_{\infty}}.

The following result appears Ch. 16 of Ramanujan notebook. For the statement, see (Adiga et al.,, 1985, Entry 1), or (Berndt,, 1991, p. 13, Entry 1(ii)). For the proof, see (Koornwinder,, 1990, Theorem B.2).

Lemma 5.2.

If $z\neq 0,-1,-2,\dots$ then

(5.3)

\lim_{q\nearrow 1}\Gamma_{q}(z)=\Gamma(z).

We will also need the following.

Lemma 5.3.

Let $q=e^{-2/M}$ . For any $c>0$ the function

t\mapsto|\Gamma_{q}(c+{\textnormal{i}}t)|

is periodic with period $M\pi$ , it is increasing for $t\in[-M\pi/2,0]$ and decreasing for $t\in[0,M\pi/2]$ .

Proof.

Referring to (5.2) we note that

|\Gamma_{q}(c+{\textnormal{i}}t)|=(1-q)^{1-c}(q;q)_{\infty}\prod\limits_{n\geq 0}\frac{1}{|1-q^{n+c+{\textnormal{i}}t}|}

As $t$ changes over the interval $[-M\pi/2,M\pi/2]$ , the point $w=w(t)=q^{n+c+{\textnormal{i}}t}$ goes around a circle $|w|=q^{n+c}$ , thus $1-w$ goes around a circle centered at $1$ and having radius $r=q^{n+c}$ . Thus, $|1-w(t)|=\sqrt{1+q^{2(n+c)}-2q^{n+c}\cos(\frac{2t}{M})}$ is periodic with period $M\pi$ , has minimum at $t=0$ and maximum at $t=\pm M\pi/2$ and it is clearly monotone between these points, which means that for every $n\geq 0$ , the function

t\in[-M\pi/2,M\pi/2]\mapsto\frac{1}{|1-q^{n+c+{\textnormal{i}}t}|}

is increasing on $[-M\pi/2,0]$ and decreasing on $[0,M\pi/2]$ . Taking the product of positive increasing/decreasing functions gives us an increasing/decreasing function. ∎

5.2. Bounds on $\Gamma_{q}$

Following numerous references, such as Corwin and Knizel, (2021); Zhang, (2008) we will use the Jacobi theta functions to derive the bounds we need.

We assume that $v\in{\mathds{C}}$ , $\textnormal{Im}(\tau)>0$ and set $q:=e^{\pi{\textnormal{i}}\tau}$ . The Jacobi theta functions are defined as

(5.4)		$\displaystyle{\theta}_{1}(v\|\tau)$	$\displaystyle=2q^{1/4}\sum\limits_{n\geq 0}(-1)^{n}q^{n(n+1)}\sin((2n+1)\pi v),$
(5.5)		$\displaystyle{\theta}_{4}(v\|\tau)$	$\displaystyle=1+2\sum\limits_{n\geq 1}(-1)^{n}q^{n^{2}}\cos(2n\pi v).$

We will need the following modular transformations of theta functions ${\theta}_{1}(v|\tau)$ and ${\theta}_{4}(v|\tau)$ .

(5.6)		$\displaystyle{\theta}_{1}(v\|\tau)$	$\displaystyle={\textnormal{i}}q^{1/4}e^{-\pi{\textnormal{i}}v}{\theta}_{4}(v-\frac{\tau}{2}\|\tau),$
(5.7)		$\displaystyle{\theta}_{1}(v\|\tau)$	$\displaystyle=i\sqrt{\frac{i}{\tau}}e^{-\pi{\textnormal{i}}v^{2}/\tau}{\theta}_{1}(\frac{v}{\tau}\|-\frac{1}{\tau}).$

The Jacobi’s triple product identity for ${\theta}_{1}(v|\tau)$ is

(5.8)

{\theta}_{1}(v|\tau)=2q^{1/4}\sin(\pi v)\prod\limits_{n\geq 1}(1-q^{2n})(1-q^{2n}e^{2\pi{\textnormal{i}}v})(1-q^{2n}e^{-2\pi{\textnormal{i}}v}).

That is,

(5.9)

{\theta}_{1}(v|\tau)=2q^{1/4}\sin(\pi v)(q^{2},q^{2}e^{2\pi{\textnormal{i}}v},q^{2}e^{-2\pi{\textnormal{i}}v};q^{2})_{\infty}.\\

All of these results can be found in (Rademacher,, 1973, Chapter 10).

The next results also require the following elementary bounds: for all $u\in{\mathds{R}}$

(5.10)

e^{-|u|}\leq\frac{u}{\sinh(u)}\leq 10e^{-4|u|/5}\;\;{\textnormal{ and }}\;\;\Big{|}\frac{u}{1-e^{u}}\Big{|}\leq 1+|u|.

We leave their proofs to the reader.

Lemma 5.4.

Let $q=e^{-2/M}$ . There exists $M^{*}$ and a universal constant $0<C<\infty$ such that

(5.11)

\frac{1}{C}e^{-\frac{\pi}{2}|x|}<|\Gamma_{q}(1+{\textnormal{i}}x)|<Ce^{-\frac{3\pi}{20}|x|}

for all $M\geq M^{*}$ and all $x\in[-\pi M/2,\pi M/2]$ .

Proof.

In view of the product formula (5.8) combined with (5.2) we have the following representation

(5.12)

|\Gamma_{q}(1+{\textnormal{i}}x)|^{2}=2\sin(\tfrac{x}{M})\frac{e^{-\frac{1}{4M}}(q;q)_{\infty}^{3}}{\theta_{1}\left(\frac{x}{M\pi}|\frac{{\textnormal{i}}}{M\pi}\right)}.

We use (5.7) to rewrite (5.12) as follows

|\Gamma_{q}(1+{\textnormal{i}}x)|^{2}=\left(\frac{(q;q)_{\infty}\,\exp(\frac{\pi^{2}M}{12})}{\sqrt{M\pi}}\right)^{3}\,e^{-\frac{1}{4M}}\,\frac{\sin(x/M)}{x/M}\,\frac{2\sinh(\pi x)}{v(x,M)}\,\frac{\pi x}{\sinh(\pi x)}\,e^{\frac{x^{2}}{M}},

where

(5.13)

v(x,M):=-{\textnormal{i}}e^{\pi^{2}M/4}\theta_{1}({\textnormal{i}}x|{\textnormal{i}}\pi M)=2\sinh(\pi x)\left(1+S(x,M)\right),

with

S(x,M):=\sum\limits_{n\geq 1}(-1)^{n}\times\frac{\sinh((2n+1)\pi x)}{\sinh(\pi x)}\;e^{-\pi^{2}Mn(n+1)}.

In view of formula (35) in Zhang, (2008) with $\gamma=\pi$ , $n=M^{2}$ and $a=1/2$ (or (Corwin and Knizel,, 2021, Proposition 2.3)) we have

\frac{(q;q)_{\infty}\,\exp(\frac{\pi^{2}M}{12})}{\sqrt{M\pi}}\to 1\quad\mbox{as $M\to\infty$}.

Consequently, for $\delta>0$ and $M$ large enough we have

(5.14)

1-\delta<\left(\frac{(q;q)_{\infty}\,\exp(\tfrac{\pi^{2}M}{12})}{\sqrt{M\pi}}\right)^{3}\,e^{-\frac{1}{4M}}<1+\delta.

For $x\in[-\frac{\pi M}{2},\frac{\pi M}{2}]$ , applying bounds (5.10) we obtain:

	$\displaystyle\bigg{\|}\frac{\sinh((2n+1)\pi x)}{\sinh(\pi x)}\bigg{\|}=(2n+1)\bigg{\|}\frac{\sinh((2n+1)\pi x)}{(2n+1)\pi x}\bigg{\|}\times\bigg{\|}\frac{\pi x}{\sinh(\pi x)}\bigg{\|}$
	$\displaystyle\;\;\;\leq(2n+1)\times e^{(2n+1)\pi\|x\|}\times 10e^{-4\pi\|x\|/5}\leq 10(2n+1)e^{\pi^{2}M(n+1/2)}.$

Consequently,

	$\displaystyle\|S(x,M)\|$	$\displaystyle\leq 10\sum\limits_{n\geq 1}(2n+1)e^{-\pi^{2}M(n^{2}-1/2)}$
		$\displaystyle\leq 10e^{-\pi^{2}M/2}\times\Big{[}\sum\limits_{n\geq 1}(2n+1)e^{-\pi^{2}(n^{2}-1)}\Big{]}=C_{1}e^{-\pi^{2}M/2}$

for an absolute constant $C_{1}>0$ . Therefore, for $\varepsilon>0$ and for $M$ large enough we have

(5.15)

1-\varepsilon\leq\frac{v(x,M)}{2\sinh(\pi x)}\leq 1+\varepsilon,

for all $x\in[-\pi M/2,\pi M/2]$ . In other words, we proved that

(5.16)

-{\textnormal{i}}e^{\pi^{2}M/4}\theta_{1}({\textnormal{i}}x|{\textnormal{i}}\pi M)=2\sinh(\pi x)(1+o(1)),\;\;\;M\to\infty,\;\;

uniformly in $x\in[-\pi M/2,\pi M/2]$ .

Taking into account (5.14), (5.15) and the trivial estimate

\frac{2}{\pi}\leq\frac{\sin(x/M)}{x/M}\leq 1,\quad x\in[-\tfrac{\pi M}{2},\tfrac{\pi M}{2}],

we get

\frac{2}{\pi}\frac{1-\delta}{1+\varepsilon}\,e^{x^{2}/M}\frac{\pi|x|}{|\sinh(\pi x)|}\leq|\Gamma_{q}(1+{\textnormal{i}}x)|^{2}\leq\frac{1+\delta}{1-\varepsilon}\,e^{x^{2}/M}\frac{\pi|x|}{|\sinh(\pi x)|}

which holds for $M$ large enough and for all $x\in[-{\pi M}/2,{\pi M}/2]$ . Setting $\varepsilon=\delta=1/2$ we get

(5.17)

\frac{2}{3\pi}\frac{\pi|x|}{|\sinh(\pi x)|}<\frac{2}{3\pi}e^{x^{2}/M}\frac{\pi|x|}{|\sinh(\pi x)|}<|\Gamma_{q}(1+{\textnormal{i}}x)|^{2}<3e^{x^{2}/M}\frac{\pi|x|}{|\sinh(\pi x)|}.

In the range $x\in[-\pi M/2,\pi M/2]$ we have $x^{2}/M\leq\pi|x|/2$ . This fact, the bounds (5.10) and (5.17) imply that for all $M$ large enough

\frac{2}{3\pi}e^{-\pi|x|}<|\Gamma_{q}(1+{\textnormal{i}}x)|^{2}<30e^{-\frac{3}{10}\pi|x|},

from which the desired result (5.11) follows by taking the square root. ∎

Lemma 5.5.

Let $q=e^{-2/M}$ and assume that $\beta\in(\pi/4,3\pi/4)$ . There exist $A>0$ and $M^{*}$ (which may depend on $\beta$ ) such that

(5.18)

\frac{|\Gamma_{q}(1-{\textnormal{i}}\beta M+{\textnormal{i}}x)|}{|\Gamma_{q}(1-{\textnormal{i}}\beta M)|}<Ae^{\pi x/2}

for all $M\geq M^{*}$ and $x\in[0,\pi M/2]$ .

Proof.

Expression (5.12) with modular transformation (5.7) applied twice gives

(5.19)

\frac{|\Gamma_{q}(1-{\textnormal{i}}\beta M+{\textnormal{i}}x)|^{2}}{|\Gamma_{q}(1-{\textnormal{i}}\beta M)|^{2}}=\frac{\sin(\beta-x/M)}{\sin(\beta)}\,\frac{\theta_{1}({\textnormal{i}}\beta M|{\textnormal{i}}\pi M)}{\theta_{1}({\textnormal{i}}(\beta M-x)|{\textnormal{i}}\pi M)}\,\,e^{x^{2}/M-2\beta x}.

We consider two cases. If $\beta\in(\pi/4,\pi/2]$ , then $1/\sin\beta\leq\sqrt{2}$ and the first arguments in both $\theta_{1}$ functions arguments satisfy the assumptions $|\beta M|\leq M\pi/2$ and $|\beta M-x|\leq M\pi/2$ . Using estimate (5.16) we obtain

\frac{\theta_{1}({\textnormal{i}}\beta M|{\textnormal{i}}\pi M)}{\theta_{1}({\textnormal{i}}(\beta M-x)|{\textnormal{i}}\pi M)}=\frac{\sinh(\pi\beta M)}{\sinh(\pi(\beta M-x))}(1+o(1)),\;\;\;\mbox{as }\;M\to\infty,

uniformly in $x\in[-\frac{M\pi}{2},\,\frac{M\pi}{2}]$ . We also write

\sin(\beta-\tfrac{x}{M})\frac{\sinh(\pi\beta M)}{\sinh(\pi(\beta M-x))}=\frac{-1}{2\pi M}\times\frac{\sin(\beta-x/M)}{\beta-x/M}\times\frac{2\pi(x-\beta M)}{1-e^{2\pi(x-\beta M)}}\times(1-e^{-2\pi\beta M})\times e^{\pi x}.

The second and fourth factors in the right hand side are bounded by one in absolute value, and using the last bound in (5.10) we see that the product of the first and third factors is bounded by $1+\pi$ . Using these results and the bound $x^{2}/M\leq\pi x/2$ on $x\in[0,\pi M/2]$ we see that in the case $\beta\in(\pi/4,\pi/2]$ we have

\frac{|\Gamma_{q}(1-{\textnormal{i}}\beta M+{\textnormal{i}}x)|^{2}}{|\Gamma_{q}(1-{\textnormal{i}}\beta M)|^{2}}<A_{1}e^{\pi x/2-2\beta x+\pi x}=A_{1}e^{x(3\pi/2-2\beta)}\leq A_{1}e^{\pi x}

for some absolute constant $A_{1}$ , and this implies (5.18).

Next we consider the case $\beta\in(\pi/2,3\pi/4)$ . We use formula (5.6) which together with (5.19) gives

(5.20)

\frac{|\Gamma_{q}(1-{\textnormal{i}}\beta M+{\textnormal{i}}x)|^{2}}{|\Gamma_{q}(1-{\textnormal{i}}\beta M)|^{2}}=\frac{\sin(\beta-x/M)}{\sin(\beta)}\,\frac{\theta_{4}({\textnormal{i}}\tilde{\beta}M|{\textnormal{i}}\pi M)}{\theta_{4}({\textnormal{i}}(\tilde{\beta}M-x)|{\textnormal{i}}\pi M)}\,e^{x^{2}/M+(\pi-2\beta)x}

with $\tilde{\beta}=\beta-\pi/2\in(0,\pi/4)$ .

We use the same method as in the proof of Lemma 5.4. In view of (5.5) we write

(5.21)

\theta_{4}({\textnormal{i}}w|{\textnormal{i}}\pi M)=1+\tilde{S}(w,M),

and estimate

\tilde{S}(w,M):=2\sum\limits_{n\geq 1}(-1)^{n}e^{-\pi^{2}Mn^{2}}\cosh(2n\pi w)

as follows: for any small $\varepsilon>0$ , $M\geq 1$ and $|w|\leq(1-\varepsilon){M\pi}/2$ we have

|\tilde{S}(w,M)|\leq 2\sum\limits_{n\geq 1}e^{-\pi^{2}Mn^{2}+2n\pi|w|}\leq 2\sum\limits_{n\geq 1}e^{-\pi^{2}Mn^{2}+n\pi^{2}(1-\varepsilon)M}=\\ =2e^{-\pi^{2}M\varepsilon}\sum\limits_{n\geq 1}e^{-\pi^{2}M(n^{2}-n)-(n-1)\pi^{2}\varepsilon M}\leq 2e^{-\pi^{2}M\varepsilon}\sum\limits_{n\geq 1}e^{-\pi^{2}M(n^{2}-n)}\leq\,C_{2}e^{-\pi^{2}M\varepsilon}.

for an absolute constant $C_{2}>0$ ,

From the above estimate and (5.21) it follows that for arbitrary $\varepsilon\in(0,1)$ , as $M\to+\infty$ ,

(5.22)

\theta_{4}({\textnormal{i}}w|{\textnormal{i}}\pi M)=1+o(1),\quad\mbox{uniformly in}\;\;w\in[-(1-\varepsilon)\tfrac{\pi M}{2},(1-\varepsilon)\tfrac{\pi M}{2}].

Since $\tilde{\beta}\in(0,\pi/4)$ and $x\in[0,\pi M/2]$ , it is easy to see that

\tilde{\beta}M-x,\;\tilde{\beta}M\in\left[-(1-\varepsilon)\tfrac{\pi M}{2},(1-\varepsilon)\tfrac{\pi M}{2}\right]\quad\mbox{with}\;\;\varepsilon=2\tilde{\beta}/\pi.

Therefore, applying (5.22) for $w=\tilde{\beta}M-x$ and $w=\tilde{\beta}M$ to the second factor in (5.20) we conclude that

(5.23)

\displaystyle\frac{|\Gamma_{q}(1-{\textnormal{i}}\beta M+{\textnormal{i}}x)|^{2}}{|\Gamma_{q}(1-{\textnormal{i}}\beta M)|^{2}}

\displaystyle=\frac{\sin(\beta-x/M)}{\sin(\beta)}e^{x^{2}/M+(\pi-2\beta)x}\times(1+o(1))

as $M\to\infty$ uniformly in $x\in[0,\pi M/2]$ . Since $\beta\in(\pi/2,3\pi/4)$ , the right-hand side in (5.23) can be bounded by $C\exp(\pi x)$ for $x\in[0,\,\pi M/2]$ and we again conclude that (5.18) holds. ∎

Lemma 5.6.

Fix $u\in\mathds{R}$ and let $q_{M}=e^{-2/M}$ , $M=1,2\dots$ . Then there exist constants $A,B,M_{*}>0$ that depend only on $u$ such that for all $M>M_{*}$ we have

(5.24)

{\mathbf{1}}_{\{|s|\leq M\pi/2\}}\left|\Gamma_{q_{M}}(1+{\textnormal{i}}(s+u))\right|\leq Ae^{-B|s|}

for all $s\in\mathds{R}$ .

Proof.

We will show that (5.24) holds with $A=Ce^{3\pi|u|/20}$ , $B=3\pi/40$ and $M_{*}=\max\{M^{*},4|u|/\pi\}$ , where $M^{*}$ and $C$ are from Lemma 5.4. Let $M>M_{*}$ and $|s|\leq M\pi/2$ .

We consider two cases. If $|s+u|\leq M\pi/2$ , then (5.24) follows from Lemma 5.4. On the other hand, if $|s+u|>M\pi/2$ , then with $M>M_{*}=\max\{M^{*},4|u|/\pi\}\geq 4|u|/\pi$ we have $|u|\leq M\pi/4$ . Recalling that $|s|\leq M\pi/2$ , we get $M\pi/2<|s+u|\leq M\pi/2+M\pi/4=3M\pi/4$ .

By Lemma 5.3, function $x\mapsto|\Gamma_{q}(1+{\textnormal{i}}x)|$ is periodic with period $M\pi$ and is increasing on the interval $[-M\pi/2,-M\pi/4]$ . With $x=s+u\in[M\pi/2,3M\pi/4]$ we have $x-M\pi\in[-M\pi/2,-M\pi/4]$ so we get

\left|\Gamma_{q}(1+{\textnormal{i}}x)\right|=\left|\Gamma_{q}(1+{\textnormal{i}}(x-M\pi))\right|\leq\left|\Gamma_{q}(1-{\textnormal{i}}M\pi/4)\right|\leq Ce^{-\frac{3\pi^{2}}{80}M}\leq Ce^{-\frac{3\pi}{40}|s|},

where we used Lemma 5.4 and then the bound $M\geq\tfrac{2}{\pi}|s|$ . Thus in this case (5.24) follows, too.

∎

We also need the following result.

Lemma 5.7.

There exist constants $A,B,M_{*}>0$ such that with $q=e^{-2/M}$ , for all $M>M_{*}$ and all $u\in\mathds{R}$ we have

(5.25)

{\mathbf{1}}_{\{|u|<M\pi/2\}}\frac{1}{|\Gamma_{q}({\textnormal{i}}u)|^{2}}\leq Au^{2}e^{B|u|}.

Proof.

Definition (5.2) yields

\frac{1}{|\Gamma_{q}({\textnormal{i}}u)|^{2}}=\frac{|1-q^{{\textnormal{i}}u}|^{2}}{(1-q)^{2}}\frac{1}{|\Gamma_{q}(1+{\textnormal{i}}u)|^{2}}.

Since $|1-q^{{\textnormal{i}}u}|^{2}=4\sin^{2}(u/M)$ . we get

\frac{|1-q^{{\textnormal{i}}u}|^{2}}{(1-q)^{2}}=4u^{2}\,\frac{1}{(M(1-e^{-2/M}))^{2}}\,\left(\frac{\sin(u/M)}{u/M}\right)^{2}\leq Cu^{2}

Thus (5.25) follows from Lemma 5.4. ∎

5.3. Additional technical lemmas

The following technical Lemma is needed in proof of Theorem 3.3.

Lemma 5.8.

Let $\widetilde{a},\widetilde{b}$ be either real or complex conjugates. If $\textnormal{Re}(\widetilde{a})\geq 0$ , $\textnormal{Re}(\widetilde{b})\geq 0$ , $c\geq 0$ then for $s\in\mathds{R}$ and all $0\leq q<1$ we have

(5.26)

\left|\frac{(-\widetilde{a}q^{1+c+{\textnormal{i}}s},-\widetilde{b}q^{1+c+{\textnormal{i}}s};q)_{\infty}}{(-\widetilde{a}q,-\widetilde{b}q;q)_{\infty}}\right|\leq 1.

Proof.

We first prove the bound for the case of complex-conjugate parameters. Write $\widetilde{a}=re^{{\textnormal{i}}\alpha}$ , $\widetilde{b}=re^{-{\textnormal{i}}\alpha}$ with $\alpha\in[-\pi/2,\pi/2]$ and $r>0$ . (The case $r=0$ is trivial.) Consider one factor of the infinite product:

\frac{|(1+rq^{n+c+{\textnormal{i}}s}e^{{\textnormal{i}}\alpha})(1+rq^{n+c+{\textnormal{i}}s}e^{-{\textnormal{i}}\alpha})|}{|(1+rq^{n}e^{{\textnormal{i}}\alpha})(1+rq^{n}e^{-{\textnormal{i}}\alpha})|}=\frac{|e^{{\textnormal{i}}\alpha}+rq^{n+c+{\textnormal{i}}s}|\cdot|e^{-{\textnormal{i}}\alpha}+rq^{n+c+{\textnormal{i}}s}|}{|e^{{\textnormal{i}}\alpha}+rq^{n}|\cdot|e^{-{\textnormal{i}}\alpha}+rq^{n}|},\;n\geq 1.

As $s$ varies through the real line, the point $z=-rq^{n+c+{\textnormal{i}}s}$ moves on the circle $|z|=rq^{n+c}$ . One readily checks that the product in the numerator is largest when $\textnormal{Re}(z)$ is minimal, i.e., when $s=0$ . Thus

\frac{|(1+rq^{n+c+{\textnormal{i}}s}e^{{\textnormal{i}}\alpha})(1+rq^{n+c+{\textnormal{i}}s}e^{-{\textnormal{i}}\alpha})|}{|(1+rq^{n}e^{{\textnormal{i}}\alpha})(1+rq^{n}e^{-{\textnormal{i}}\alpha})|}\leq\frac{|(1+rq^{n+c}e^{{\textnormal{i}}\alpha})(1+rq^{n+c}e^{-{\textnormal{i}}\alpha})|}{|(1+rq^{n}e^{{\textnormal{i}}\alpha})(1+rq^{n}e^{-{\textnormal{i}}\alpha})|}.

To end the proof, we note that

\frac{|1+rq^{n+c}e^{{\textnormal{i}}\alpha}|^{2}}{|1+rq^{n}e^{{\textnormal{i}}\alpha}|^{2}}\leq 1.

The proof for the case of real $\widetilde{a},\widetilde{b}\in[0,\infty)$ requires some minor modifications and is omitted. (This case is not used in the proof of Theorem 3.3.) ∎

The proof of Theorem 4.3 uses the following technical estimate:

Lemma 5.9.

Let $q=e^{-2/M}$ . If $a=-qe^{{\textnormal{i}}\alpha}$ , $b=-qe^{-{\textnormal{i}}\alpha}$ and $|\alpha|<\pi/2$ then there exist $A>0$ , $B>0$ and $M_{*}$ such that

(5.27)

{\mathbf{1}}_{\{|u|<\pi M/2\}}\frac{(a,b;q)_{\infty}^{2}}{|(aq^{{\textnormal{i}}u/2},bq^{{\textnormal{i}}u/2};q)_{\infty}|^{2}}\leq Ae^{B|u|}

for all $M\geq M_{*}$ and all $u\in\mathds{R}$ .

Proof.

Replacing $u$ by $-u$ if needed, without loss of generality we may assume $u\geq 0$ . In view of (5.2) the left hand side of (5.27) can be rewritten in terms of $q$ -Gamma functions. We have

\frac{|(a;q)_{\infty}|}{|(aq^{{\textnormal{i}}u/2};q)_{\infty}|}=\frac{|\Gamma_{q}(1-{\textnormal{i}}\beta M+{\textnormal{i}}u/2)|}{|\Gamma_{q}(1-{\textnormal{i}}\beta M)|},

where $\beta=(\alpha+\pi)/2\in(\pi/4,3\pi/4)$ . Similarly,

\frac{|(b;q)_{\infty}|}{|(bq^{{\textnormal{i}}u/2};q)_{\infty}|}=\frac{|\Gamma_{q}(1-{\textnormal{i}}\beta^{\prime}M+{\textnormal{i}}u/2)|}{|\Gamma_{q}(1-{\textnormal{i}}\beta^{\prime}M)|},

where $\beta^{\prime}=(\pi-\alpha)/2\in(\pi/4,3\pi/4)$ .

Therefore inequality (5.27) follows from Lemma 5.5.∎

Acknowledgements

The first author was supported in part by Simons Foundation/SFARI Award Number: 703475, US. The second author was supported in part by Natural Sciences and Engineering Research Council of Canada. The authors acknowledge support from the Taft Research Center at the University of Cincinnati.

References

Adiga et al., (1985) Adiga, C., Berndt, B. C., Bhargava, S., and Watson, G. N. (1985). Ch 16 of Ramanujan’s Second Notebook: Theta-Functions and $q$ -Series: Theta Functions and Q-series, volume 315. American Mathematical Soc.
Aptekarev, (1993) Aptekarev, A. I. (1993). Asymptotics of orthogonal polynomials in a neighborhood of the endpoints of the interval of orthogonality. Sbornik: Mathematics, 76(1):35.
Baik et al., (2003) Baik, J., Kriecherbauer, T., McLaughlin, K.-R., and Miller, P. D. (2003). Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles: announcement of results. International Mathematics Research Notices, 2003(15):821–858.
Barraquand and Corwin, (2023) Barraquand, G. and Corwin, I. (2023). Stationary measures for the log-gamma polymer and KPZ equation in half-space. Ann. Probab., 51(5):1830–1869.
Barraquand and Le Doussal, (2022) Barraquand, G. and Le Doussal, P. (2022). Steady state of the KPZ equation on an interval and Liouville quantum mechanics. Europhysics Letters, 137(6):61003. ArXiv preprint with Supplementary material: https://arxiv.org/abs/2105.15178.
Barraquand and Le Doussal, (2023) Barraquand, G. and Le Doussal, P. (2023). Stationary measures of the KPZ equation on an interval from Enaud-Derrida’s matrix product ansatz representation. J. Phys. A, 56(14):Paper No. 144003, 14.
Berg and Ismail, (1996) Berg, C. and Ismail, M. E. H. (1996). $q$ -Hermite polynomials and classical orthogonal polynomials. Canad. J. Math., 48(1):43–63.
Berndt, (1991) Berndt, B. C. (1991). Ramanujan’s Notebooks. Part III. Springer-Verlag, New York.
Billingsley, (1999) Billingsley, P. (1999). Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition. A Wiley-Interscience Publication.
Bryc and Kuznetsov, (2022) Bryc, W. and Kuznetsov, A. (2022). Markov limits of steady states of the KPZ equation on an interval. ALEA, Lat. Am. J. Probab. Math. Stat., 19:1329–1351. (http://arxiv.org/abs/2109.04462).
Bryc and Wang, (2023) Bryc, W. and Wang, Y. (2023). Limit theorems for random Motzkin paths near boundary. Bernoulli (to appear). http://arxiv.org/abs/2304.00924.
Bryc and Wang, (2024) Bryc, W. and Wang, Y. (2024). Fluctuations of random Motzkin paths II. ALEA, Lat. Am. J. Probab. Math. Stat., 21:73–94. http://arxiv.org/abs/2304.12975.
Corwin and Knizel, (2021) Corwin, I. and Knizel, A. (2021). Stationary measure for the open KPZ equation. Communications on Pure and Applied Mathematics, to appear. arXiv preprint arXiv:2103.12253, DOI: 10.1002/cpa.22174.
Deift et al., (2001) Deift, P., Kriecherbauer, T., McLaughlin, K.-R., Venakides, S., and Zhou, X. (2001). A Riemann–Hilbert approach to asymptotic questions for orthogonal polynomials. Journal of Computational and Applied Mathematics, 133(1-2):47–63.
Deift et al., (1999) Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S., and Zhou, X. (1999). Strong asymptotics of orthogonal polynomials with respect to exponential weights. Communications on Pure and Applied Mathematics, 52(12):1491–1552.
Flajolet and Sedgewick, (2009) Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press, Cambridge.
Gasper and Rahman, (2004) Gasper, G. and Rahman, M. (2004). Basic Hypergeometric Series, volume 96. Cambridge university press.
Ismail, (2005) Ismail, M. E. (2005). Asymptotics of $q$ -orthogonal polynomials and a $q$ -Airy function. International Mathematics Research Notices, 2005(18):1063–1088.
Ismail and Li, (2013) Ismail, M. E. and Li, X. (2013). Plancherel–Rotach asymptotics for $q$ -orthogonal polynomials. Constructive Approximation, 37:341–356.
Ismail et al., (2022) Ismail, M. E., Zhang, R., and Zhou, K. (2022). Orthogonal polynomials of Askey-Wilson type. arXiv preprint arXiv:2205.05280.
Ismail, (1986) Ismail, M. E. H. (1986). Asymptotics of the Askey-Wilson and $q$ -Jacobi polynomials. SIAM J. Math. Anal., 17(6):1475–1482.
Ismail, (2009) Ismail, M. E. H. (2009). Classical and Quantum Orthogonal Polynomials in One Variable, volume 98 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge.
Ismail and Wilson, (1982) Ismail, M. E. H. and Wilson, J. A. (1982). Asymptotic and generating relations for the $q$ -Jacobi and ${}_{4}\varphi_{3}$ polynomials. J. Approx. Theory, 36(1):43–54.
Kaigh, (1976) Kaigh, W. D. (1976). An invariance principle for random walk conditioned by a late return to zero. The Annals of Probability, pages 115–121.
Karlin and McGregor, (1959) Karlin, S. and McGregor, J. (1959). Random walks. Illinois Journal of Mathematics, 3(1):66–81.
Keener, (1992) Keener, R. W. (1992). Limit theorems for random walks conditioned to stay positive. The Annals of Probability, pages 801–824.
Koekoek et al., (2010) Koekoek, R., Lesky, P. A., Swarttouw, R. F., Koekoek, R., Lesky, P. A., and Swarttouw, R. F. (2010). Hypergeometric Orthogonal Polynomials. Springer.
Koekoek and Swarttouw, (1994) Koekoek, R. and Swarttouw, R. F. (1994). The askey-scheme of hypergeometric orthogonal polynomials and its $q$ -analogue. https://homepage.tudelft.nl/11r49/askey/.
Koornwinder, (1990) Koornwinder, T. H. (1990). Jacobi functions as limit cases of $q$ -ultraspherical polynomials. Journal of Mathematical Analysis and Applications, 148(1):44–54.
Kuijlaars et al., (2004) Kuijlaars, A. B. J., McLaughlin, K. T.-R., Van Assche, W., and Vanlessen, M. (2004). The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on $[-1,1]$ . Adv. Math., 188(2):337–398.
Levin and Lubinsky, (2020) Levin, E. and Lubinsky, D. S. (2020). Local limits for orthogonal polynomials for varying measures via universality. Journal of Approximation Theory, 254:105394.
Lubinsky, (2020) Lubinsky, D. S. (2020). Pointwise asymptotics for orthonormal polynomials at the endpoints of the interval via universality. International Mathematics Research Notices, 2020(4):961–982.
Nevai, (1984) Nevai, P. (1984). Asymptotics for orthogonal polynomials associated with $\exp(-x^{4})$ . SIAM journal on mathematical analysis, 15(6):1177–1187.
Olver et al., (2023) Olver, F. W. J., Daalhuis, A. B. O., Lozier, D. W., Schneider, B. I., Boisvert, R. F., Clark, C. W., Millerand, B. R., Saunders, B. V., Cohl, H. S., and McClain, M. A. (2023). NIST digital library of mathematical functions. https://dlmf.nist.gov/, 2023-12-15 DLMF Update; Version 1.1.12.
Rademacher, (1973) Rademacher, H. (1973). Topics in Analytic Number Theory. Die Grundlehren der mathematischen Wissenschaften, Band 169. Springer-Verlag, New York-Heidelberg. Edited by E. Grosswald, J. Lehner and M. Newman.
Szwarc, (1992) Szwarc, R. (1992). Connection coefficients of orthogonal polynomials. Canadian Mathematical Bulletin, 35(4):548–556.
Viennot, (1985) Viennot, G. (1985). A combinatorial theory for general orthogonal polynomials with extensions and applications. In Brezinski, C., Draux, A., Magnus, A. P., Maroni, P., and Ronveaux, A., editors, Polynômes Orthogonaux et Applications, pages 139–157, Berlin, Heidelberg. Springer Berlin Heidelberg.
Yakubovich, (2011) Yakubovich, S. (2011). The heat kernel and Heisenberg inequalities related to the Kontorovich-Lebedev transform. Commun. Pure Appl. Anal., 10(2):745–760.
Zhang, (2008) Zhang, R. (2008). On asymptotics of $q$ -gamma functions. J. Math. Anal. Appl., 339(2):1313–1321.

(5.6)		$\displaystyle{\theta}_{1}(v\|\tau)$	$\displaystyle={\textnormal{i}}q^{1/4}e^{-\pi{\textnormal{i}}v}{\theta}_{4}(v-\frac{\tau}{2}\|\tau),$
(5.7)		$\displaystyle{\theta}_{1}(v\|\tau)$	$\displaystyle=i\sqrt{\frac{i}{\tau}}e^{-\pi{\textnormal{i}}v^{2}/\tau}{\theta}_{1}(\frac{v}{\tau}\|-\frac{1}{\tau}).$

Random Motzkin paths near boundary

Limits of Random Motzkin paths with KPZ related asymptotics

Abstract.

1. Introduction and main results

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.

2. General limit theorem for random Motzkin paths at the boundary

2.1. Matrix representation and integral representation

Lemma 2.1 (matrix ansatz).

Proof.

Assumption 2.1.

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

2.2. Limit theorem

Theorem 2.4.

Proof.

Proposition 2.5.

Proof.

3. Properties of the Al-Salam–Chihara polynomials

3.1. Maximum of the Al-Salam–Chihara polynomials

Proposition 3.1.

Remark 3.1.

Corollary 3.1.

Proof.

Proof of Proposition 3.1.

3.2. Pointwise asymptotics near the end of the interval of orthogonality

Theorem 3.2.

Remark 3.2.

Proof of Theorem 3.2.

Proof of Remark 3.2.

Theorem 3.3.

Proof.

4. Motzkin paths with qq-number weights

Lemma 4.1.

Proof.

Proof of Theorem 1.1.

4.1. Local limit theorems

Theorem 4.2.

Proof.

Theorem 4.3.

Proof.

4.2. Proofs of Theorems 1.2 and 1.3

Proof of Theorem 1.2.

Proof of Theorem 1.3.

5. Auxiliary results on special functions

5.1. Some useful limits for q↗1q\nearrow 1

Lemma 5.1 (Ramanujan).

Proof.

Lemma 5.2.

Lemma 5.3.

Proof.

5.2. Bounds on Γq\Gamma_{q}

Lemma 5.4.

Proof.

Lemma 5.5.

Proof.

Lemma 5.6.

Proof.

Lemma 5.7.

Proof.

5.3. Additional technical lemmas

Lemma 5.8.

Proof.

Lemma 5.9.

Proof.

Acknowledgements

References

4. Motzkin paths with $q$ -number weights

5.1. Some useful limits for $q\nearrow 1$

5.2. Bounds on $\Gamma_{q}$