Computing with D-Algebraic Sequences

Bertrand Teguia Tabuguia

Abstract

A sequence is difference algebraic (or D-algebraic) if finitely many shifts of its general term satisfy a polynomial relationship; that is, they are the coordinates of a generic point on an affine hypersurface. The corresponding equations are called algebraic difference equations (ADE). We show that subsequences of D-algebraic sequences, indexed by arithmetic progressions, satisfy ADEs of the same orders as the original sequences. Additionally, we provide algorithms for operations with D-algebraic sequences and discuss the difference-algebraic nature of holonomic and $C^{2}$ -finite sequences.

keywords:

Affine D-algebraic sequence, subsequence, radical-rational dynamical system

1 Introduction

Some well-known nonlinear recurrence relations arose with the Babylonian (or Heron’s) method and Aitken extrapolation. The latter was introduced to estimate limits of convergent sequences. To a sequence of general term $s(n)$ , the $\Delta^{2}$ process, or Aitken’s transformation [Ait27], associates the term

t(n)=s(n)-\frac{(\Delta s(n))^{2}}{\Delta^{2}s(n)}=s(n)-\frac{\left(s(n+1)-s(n)\right)^{2}}{s(n+2)-2\,s(n+1)+s(n)},

(1)

to accelerate the rate of convergence to $\lim_{n\to\infty}s(n)<\infty$ . As discussed in [Wen89], this transformation has numerous applications in theoretical physics and other sciences [BJ75, Bre85]. This paper presents a general study of a class of sequences that remains unchanged by such transformations. These sequences satisfy nonlinear rational or polynomial recursions, similar to the one presented in (1).

As an example, consider approximating $\sqrt{3}$ using the Babylonian method. The iteration is governed by the recursion

s(n+1)=\frac{1}{2}\left(s(n)+\frac{3}{s(n)}\right),

(2)

with the chosen initial term $s(0)=1$ . Using our result, we can systematically show that the Aitken transformation yields the sequence with initial term $t(0)=\frac{9}{5}$ and recursion

t(n+1)=\frac{6\,t(n)}{t(n)^{2}+3}=\frac{6}{t(n)+\frac{3}{t(n)}}.

(3)

The above may be generalized to any square root computation. Equations similar to (2) also appear in modeling, as illustrated with the discrete May-Leonard model by Roeger and Allen [RA04].

Following J. F. Ritt’s algebraic approach to differential equations [Rit50], the study of associated objects such as differential polynomials and differential varieties was further developed by Raudenbush [Rau34], Rosenfeld [Ros59], and, more comprehensively with modern mathematics, by Kolchin [Kol73]. For difference equations, which relate to the classical $\Delta$ operator defined as $\Delta f(x)\coloneqq f(x+1)-f(x)$ , R. Cohn developed much of the theory [Coh65].

A key result of R. Cohn states that every nontrivial ordinary algebraically irreducible difference polynomial admits an abstract solution [Coh48]. This theorem has important implications for sequences over an algebraically closed field, say $\mathbb{K}$ , of characteristic zero. Indeed, specialized to the difference ring $\mathbb{K}^{\mathbb{N}}$ (with the monoid of natural integers $\mathbb{N}=\{0,1,\ldots\}$ ), Cohn’s theorem ensures the existence of sequences that are zeros of ordinary difference polynomial. We call them difference-algebraic (or D-algebraic) sequences and denote their set by $\mathbb{K}^{\aleph_{0}}$ (see Definition 2.8).

D-algebraic functions [AEMSTT24, TT25] generalize functions defined by polynomial differential equations, encompassing classes such as D-finite functions [Kau23] and differentially definable functions [JPPS20] (see also [TTK21] for related power series). Similarly, D-algebraic sequences generalize sequences defined by polynomial equations involving their shifts and the index variable. These sequences arise naturally in computer science, for example, in the context of cost-register automata [ADD⁺13] and polynomial automata [BDSW17] over a unary alphabet. A particularly important subclass of D-algebraic sequences consists of the simple-rational recursive sequences, which are the zeros of difference polynomials linear in their highest shifts (l.h.s. difference equations) [CDBMP23]. It was shown in [TTW24] that any D-finite (holonomic or P-recursive) sequence satisfies an l.h.s. difference equation of order bounded by the sum of its holonomic degree and order. We provide a corrected proof of this statement here, as an error was inadvertently introduced in the final version of [TTW24]. Another notable subclass of D-algebraic sequences is that of $C^{2}$ -finite sequences, introduced by Jiminez-Pastor, Nuspl, and Pillwein in [JPNP23]. We will give a constructive proof of their D-algebraicity.

This paper demonstrates that difference-algebraic sequences behave well under field operations (with a mild assumption for division) thanks to our algorithm for performing arithmetic operations on them. Our method is based on the differential elimination-prolongation method developed by Ovchinnikov, Pogudin, and Vo [OPV16], which was adapted to the difference algebra setting in [OPS20] (see also [GVDHYZ09]). Wibmer introduced a notion of dimension for systems of algebraic difference equations with solutions in sequence rings, showing that this dimension need not be an integer [Wib21]. We emphasize that the consistency of the systems of algebraic difference equations resulting from our arithmetic operations follows from Cohn’s existence theorem applied to the individual input difference polynomials, thereby avoiding the undecidability results of [PSW20]. While the computational theory for the differential case is relatively well-established (see, e.g., [BLOP95, Rob14, vDH19, TT25] and references therein), an effective theory for the arithmetic of (nonlinear) difference-algebraic sequences has received less attention. This motivates the present work. In addition to field operations, we prove that D-algebraic sequences are closed under partial sums, partial products, radicals, and composition with arithmetic progressions. This last result can be extended to composition with broader classes of strictly increasing integer sequences. We are unaware of comparable results for such compositions, and we consider our method for computing equations for D-algebraic subsequences a key contribution.

We conclude this introduction with a simple example illustrating our algorithm for the arithmetic of D-algebraic sequences.

Example 1.1.

We here compute an algebraic difference equation satisfied by the product of a Fibonacci sequence, i.e., a solution of the equation $s(n+2)=s(n+1)+s(n)$ , and $k^{2^{n}}$ , for some constant $k$ , i.e., a solution of the equation $s(n+1)=s(n)^{2}$ . Using our algorithm, we find the following third-order algebraic difference equation

-s(n+3)^{3}s(n)^{6}-s(n+3)^{2}s(n+2)^{2}s(n+1)s(n)^{4}+s(n+2)^{7}s(n)^{2}-3s(n+3)s(n+2)^{4}s(n+1)^{2}s(n)^{2}-5s(n+3)^{2}s(n+2)s(n+1)^{4}s(n)^{2}+s(n+3)^{2}s(n+1)^{7}-4s(n+3)s(n+2)^{3}s(n+1)^{5}+4s(n+2)^{6}s(n+1)^{3}=0.

(4)

The above equation provides three possible continuations for the sequence, but the original two equations determine the correct one, thus defining a recursion for the entire sequence. The results in [CDBMP23] imply that whenever we have two initial equations that are linear in their highest shifts (l.h.s.), a higher-order l.h.s. difference equation can be derived for the arithmetic of their zeros. In this case, we obtain a fourth-order l.h.s. equation. $\blacksquare$

The NLDE package [TT23] contains a sub-package DalgSeq dedicated to the use of nonlinear algebra for difference equations. Most of the algorithms highlighted in this paper are implemented in DalgSeq available at https://github.com/T3gu1a/D-algebraic-functions.

2 Definitions

In this section, we fix some notations and establish a formal definition of a difference-algebraic sequence from the difference algebra setting. For further details on this theoretical perspective, we refer the reader to [Coh65, Lev08].

Definition 2.1.

A difference ring is a pair $(R,\Sigma)$ where $R$ is a commutative ring and $\Sigma$ is a finite set of pairwise commuting $R$ -endomorphisms.

A difference ideal of $(R,\Sigma)$ is an ideal $J$ of $R$ such that $\sigma(J)\subset J$ for all $\sigma\in\Sigma$ . The difference ring $(R,\Sigma)$ is also denoted $\Sigma$ -ring $R$ . When $\Sigma=\{\sigma\}$ , the $\Sigma$ -ring $R$ is simply denoted $\sigma$ -ring and said to be ordinary. When $\left|\Sigma\right|>1$ , $R$ is a partial difference ring. In this paper, we restrict ourselves to ordinary difference rings. We denote by $\mathbb{N}$ the set of non-negative integers.

We primarily consider the $\sigma$ -ring $(\mathbb{K},\sigma)$ , where $\mathbb{K}$ is an algebraically closed field of characteristic zero and $\sigma$ is the identity on $\mathbb{K}$ .

Definition 2.2.

The ring of difference polynomials in one difference indeterminate $x$ over $\mathbb{K}$ , also called the free difference $\mathbb{K}$ -algebra in $x$ , is the difference ring $\left(\mathbb{K}[x_{0},x_{1},x_{2},\ldots],\tilde{\sigma}\right)$ , where $\tilde{\sigma}$ extends $\sigma$ as follows: $\tilde{\sigma}(a)=\sigma(a)=a$ , for any $a\in\mathbb{K}$ , and $\tilde{\sigma}(x_{j})=x_{j+1}$ , $j\in\mathbb{N}$ . For simplicity, this difference ring is denoted by $\mathcal{D}_{\sigma}(\mathbb{K},x)\coloneqq\mathbb{K}\left[\sigma^{j}(x)\mid j\in\mathbb{N}\right]$ or $\mathbb{K}\left[\sigma^{\infty}(x)\right]$ with the property $\sigma^{j+1}(x)=\sigma(\sigma^{j}(x)),j\in\mathbb{N}$ .

Definition 2.3.

A difference polynomial $p\in\mathcal{D}_{\sigma}(\mathbb{K},x)$ is said to be in normal form when $x$ effectively occurs as the smallest shift in $p$ . The order of a difference polynomial $p\in\mathcal{D}_{\sigma}(\mathbb{K},x)$ , denoted $\operatorname{ord}(p)$ , is the maximum $j\in\mathbb{N}$ such that $\sigma^{j}(x)$ appears in the normal form of $p$ . The degree of $p$ denoted $\deg(p)$ is the total degree of $p$ (in normal form) as a polynomial in the algebraic polynomial ring $\mathbb{K}[x,\sigma(x),\ldots,\sigma^{\operatorname{ord}(p)}(x)]$ . We say that $p$ is l.h.s. (linear in its highest shift) when the degree of $p$ with respect to $\sigma^{\operatorname{ord}(p)}(x)$ is $1$ .

Observe that the order of a difference polynomial is unchanged by application of $\sigma$ . To any difference polynomial $p\in\mathcal{D}_{\sigma}(\mathbb{K},x)$ , we associate a recurrence equation called algebraic difference equation obtained by the equality $p=0$ .

Example 2.4.

We give the orders and degrees of the input recurrence equations in Example 1.1. We can always assume that $\mathbb{K}$ is the field of complex number $\mathbb{C}$ . For $p\coloneqq\sigma^{2}(x)-\sigma(x)-x,\operatorname{ord}(p)=2$ and $\deg(p)=1$ . For $q\coloneqq\sigma^{2}(x)-\sigma(x)^{2}$ , $\operatorname{ord}(q)=1$ and $\deg(q)=2$ . $\blacksquare$

In classical algebraic geometry, zeros of polynomials over $\mathbb{K}$ correspond to points in $\mathbb{K}^{N}$ , the affine $N$ -space over $\mathbb{K}$ , for some fixed $N\in\mathbb{N}\setminus\{0\}$ , defining algebraic sets or varieties with the ideals of those polynomials. In our setting, the corresponding points may be regarded with denumerable coordinates as we want any zero of $p\in\mathcal{D}_{\sigma}(\mathbb{K},x)$ to have coordinates that vanish $\sigma^{j}(p)$ , for all $j\in\mathbb{N}$ . Every such point defines a sequence over $\mathbb{K}$ . We denote the set of such sequences over $\mathbb{K}$ by $\mathbb{K}^{\aleph_{0}}$ , where $\aleph_{0}$ is aleph zero, the cardinality of $\mathbb{N}$ . This makes sense of the passage from a finite to an infinite (countable) number of coordinates. Note that $\mathbb{K}^{\aleph_{0}}\subset\mathbb{K}^{\mathbb{N}}$ , and the inclusion is strict since $(n^{n})_{n\in\mathbb{N}}$ is not a zero of a difference polynomial. In Section 3.1, we will see that $\mathbb{K}^{\aleph_{0}}$ has the structure of a difference ring. This fact is used in Definition 2.6. This means that ring operations are defined componentwise, and for $(s(n))_{n\in\mathbb{N}}=(s(0),s(1),s(2),\ldots)\in\mathbb{K}^{\aleph_{0}}$ (we also use the notation $(s(n))_{n}$ ), we consider the endomorphism $\sigma^{\prime}$ such that $\sigma^{\prime}((s(n))_{n\in\mathbb{N}})=(s(n+1))_{n\in\mathbb{N}}$ . This $\sigma^{\prime}$ is often known as the shift operator. Observe that $\Delta=\sigma^{\prime}-\text{id}$ . The following definition helps to clarify the correspondence between the zeros of difference polynomials and $\mathbb{K}^{\aleph_{0}}$ .

Definition 2.5.

A homomorphism of difference rings $(R_{1},\sigma_{1})$ and $(R_{2},\sigma_{2})$ is a ring-homomorphism $\varphi\colon R_{1}\longrightarrow R_{2}$ such that $\varphi\circ\sigma_{1}=\sigma_{2}\circ\varphi$ .

Definition 2.6.

Let $p\in\mathcal{D}_{\sigma}(\mathbb{K},x)$ and $(s(n))_{n\in\mathbb{N}}\in(\mathbb{K}^{\aleph_{0}},\sigma^{\prime})$ as previously defined. We say that $(s(n))_{n\in\mathbb{N}}$ is a generic zero of $p$ if, under the unique homomorphism of difference rings $\mathcal{D}_{\sigma}(\mathbb{K},x)\longrightarrow~{}(\mathbb{K}^{\aleph_{0}},\sigma^{\prime})$ given by the extension that sends $x$ to $(s(n))_{n\in\mathbb{N}}$ , $p$ is sent to $0$ .

We emphasize that our choice of the ground difference field $\mathbb{K}$ with the identity endomorphism puts no distinction between the concept of solutions, formal solutions, or generic zeros of difference polynomials over $\mathbb{K}$ ; see the details in [Lev08, Section 2.2].

Finally, we can define a difference-algebraic sequence based on the following corollary of R. Cohn’s existence theorem for ordinary difference polynomials.

Theorem 2.7.

[Coh48, Theorem IV] Every difference polynomial in $\mathcal{D}_{\sigma}(\mathbb{K},x)\setminus\mathbb{K}$ has a generic zero.

Definition 2.8.

A difference-algebraic sequence over $\mathbb{K}$ is a generic zero of some non-constant difference polynomial over $\mathbb{K}$ .

We view $\mathbb{K}^{\aleph_{0}}$ as the set of affine D-algebraic sequences, which we will simply refer to as D-algebraic sequences, as these are our sole focus. Consider a difference polynomial $p\in\mathcal{D}_{\sigma}(\mathbb{K},x)$ of order $r$ and degree $d$ . When $p$ is algebraically irreducible over $\mathbb{K}[x,\sigma(x),\ldots,\sigma^{r}(x)]$ , its zeros are said to be defined by it. In this case, the D-algebraic sequences defined by $p$ are also said to be of order $r$ and degree $d$ . Note that if $r>0$ or $d>1$ , then there are infinitely many points in $\mathbb{K}^{\aleph_{0}}$ that are zeros of $p$ .

Remark 2.9.

•

With the above definition of a difference-algebraic sequence, it is convenient to look at a difference polynomial $p\in\mathcal{D}_{\sigma}(\mathbb{K},x)$ as the specialized (see [PSW20]) difference polynomial $p\in\mathbb{D}_{s}\coloneqq\mathbb{K}[\sigma^{j}(s(n))|j\in\mathbb{N}]$ , where $s(n)$ (symbolically) represents the general term of a generic zero of $p$ . This will be used to avoid lengthy notations. We will often use the automorphism $\sigma$ with the assumption that it is extended accordingly for $\mathbb{K}^{\aleph_{0}}$ .
•

Let $n_{1},n_{2},n_{3}\in\mathbb{N}$ , and $g$ be a bivariate rational function. To perform operations with the zeros of $p\coloneqq P(x,\sigma(x),\ldots,\sigma^{n_{1}}(x))$ , $q\coloneqq Q(x,\sigma(x),\ldots,\sigma^{n_{2}}(x))\in\mathcal{D}_{\sigma}(\mathbb{K},x)$ , we write $p=P(s_{1}(n),s_{1}(n+1),\ldots,s_{1}(n+n_{1}))\in\mathbb{D}_{s_{1}}$ and $q=Q(s_{2}(n),s_{2}(n+1),\ldots,s_{2}(n+n_{2}))~{}\in~{}\mathbb{D}_{s_{2}}$ , with $(s_{1}(n))_{n\in\mathbb{N}}$ and $(s_{2}(n))_{n\in\mathbb{N}}$ standing for D-algebraic sequences defined by $p$ and $q$ , respectively. We aim to build $t\coloneqq T(x,\sigma(x),\ldots,\sigma^{n_{3}}(x))$ (which may be regarded as $T(s_{3}(n),s_{3}(n+1),\ldots,s_{3}(n+n_{3}))\in\mathbb{D}_{s_{3}}$ ) such that $g(s_{1}(n),s_{2}(n))$ is a generic zero of $t$ . We refer to the difference rings $\mathbb{D}_{s_{1}},\mathbb{D}_{s_{2}},\mathbb{D}_{s_{3}}$ as the corresponding specializations of $\mathcal{D}_{\sigma}(\mathbb{K},x)$ . This notation is generalized for arbitrarily many difference polynomials.
•

As the order of a difference polynomial is unchanged by application of $\sigma$ , it follows that if $(s(n))_{n}$ is difference algebraic with defining difference polynomial $p\in\mathcal{D}_{\sigma}(\mathbb{K},x)$ , then any shift of $(s(n))_{n}$ is also D-algebraic and defined by $p$ . The distinction between $(s(n))_{n}$ and $(s(n+i))_{n},i\in\mathbb{N}$ is apparent in their initial terms.

3 Arithmetic

We adapt the construction from [TT25, Section 2.2] to the difference case. The effectiveness of the underlying elimination theory was established in [OPS20]. The main contribution of this section is the development of systematic algorithms that automate the often tedious arithmetic of solutions to algebraic difference equations, thereby enhancing their accessibility.

3.1 Algorithm

Let $N\in\mathbb{N}\setminus\{0\}$ , and consider $N$ difference-algebraic sequences as generic zeros of $N$ given difference polynomials $p_{i}\in\mathbb{D}_{s_{i}}$ of order $n_{i}\in\mathbb{N}$ . Let $f\coloneqq f(s_{1},\ldots,s_{N})$ , where $f\in\mathbb{K}(X_{1},\ldots,X_{N})$ . We stretch that the sequence $n\mapsto f(s_{1}(n),\ldots,s_{N}(n))$ is well defined for sufficiently large integers $n$ . Set $M\coloneqq\sum_{j=1}^{N}n_{i}$ . We define the indeterminates

\begin{split}&w_{j+1}(n)=s_{1}(n+j),\,\,\text{ for }\,\,0\leq j<n_{1}\\ &w_{j+1}(n)=s_{2}(n+j),\,\,\text{ for }\,\,n_{1}\leq j<n_{1}+n_{2}\\ &\vdots\\ &w_{j+1}(n)=s_{N}(n+j),\,\,\text{ for }\,\,M-n_{N}\leq j<M.\end{split}

(5)

Observe that for $j\in{1,\ldots,M}\setminus\{n_{1},n_{1}+n_{2},\ldots,M\}$ , $\sigma(w_{j}(n))=w_{j+1}(n)$ , and $\sigma(w_{M_{i+1}}(n))$ satisfies

p_{i+1}\left(w_{M_{i}+1}(n),\ldots,w_{M_{i+1}}(n),\sigma(w_{M_{i+1}}(n))\right)=0,

(6)

where $M_{i}\coloneqq\sum_{j=0}^{i}n_{i}$ , for $i=0,\ldots,N-1$ and $n_{0}=0.$ The index variable can now be regarded implicitly, i.e., we can write $w_{j}$ for $w_{j}(n)$ since its shifts are well understood.

For $i=0\ldots,N-1$ , we write the difference polynomial $p_{i+1}$ as

c_{m_{i+1}}\,{\sigma(w_{M_{i+1}})}^{m_{j}}+p_{i+1,1}\left(w_{M_{i}+1},\ldots,w_{M_{i+1}},\sigma(w_{M_{i+1}})\right),

(7)

$c_{m_{i+1}}\in L\coloneqq\mathbb{K}[w_{M_{i}+1},\ldots,w_{M_{i+1}}]$ , $p_{i+1,1}\in L[\sigma(w_{M_{i+1}})]$ such that $\deg_{\sigma(w_{M_{i+1}})}(p_{i+1,1})<m_{i+1}$ . The coefficient $c_{m_{i+1}}$ is often called the initial of $p_{i+1}$ . We can now view the problem as resulting from the following radical-rational dynamical system

\begin{cases}\sigma(\mathbf{w})^{\mu}=\mathbf{A}(\mathbf{w})+\mathbf{E}_{\mathbf{A}}(\sigma(\mathbf{w}))\\ z=B(\mathbf{w})\end{cases}\coloneqq\begin{cases}&{\sigma(w_{1})}^{\mu_{1}}=A_{1}(w_{1},\ldots,w_{M})+E_{A_{1}}(\sigma(w_{1}))\\ &\vdots\\ &{\sigma(w_{M})}^{\mu_{M}}=A_{M}(w_{1},\ldots,w_{M})+E_{A_{M}}(\sigma(w_{M}))\\ &z=B(w_{1},\ldots,w_{M})\end{cases},\,\quad\,(\mathcal{M}_{f})

(8)

where

•

$\mu_{i},i=1,\ldots,M$ , is either $1$ or one of the $m_{i},i=1,\ldots,N$ ;
•

$A_{i}$ (with numerator $a_{i}$ ) is either $w_{i+1}$ or the part of $p_{i,1}/c_{m_{i}}$ that is free of $\sigma(w_{i})$ $i=1,\ldots,N$ ;
•

$E_{A_{i}}(\sigma(w_{i}))$ (with numerator $e_{a_{i}}$ ) is either $0$ or the part of $p_{i,1}/c_{m_{i}}$ that contains $\sigma(w_{i})$ (in its numerator), $i=1,\ldots,N$ ;
•

$B(w_{1},\ldots,w_{M})=f(w_{1},w_{n_{1}+1},\ldots,w_{\sum_{i=1}^{N-1}n_{i}+1})$ .

We refer to $M$ as the dimension of $(\mathcal{M}_{f})$ .

Remark 3.1.

When the $N$ given difference polynomials are l.h.s., i.e., $m_{i}=1,$ $i=1,\ldots,N$ , and $E_{A_{i}}=0$ , $(\mathcal{M}_{f})$ is a classical rational dynamical system often considered in theoretical computer science [CDBMP23].

Let $Q$ be the least common multiple of all the denominators in the system. Without loss of generality, we assume that all $A_{i}$ and $E_{A_{i}}$ are written with $Q$ as the denominator and consider that $B=b/Q$ . In the difference-algebra context, $(\mathcal{M}_{f})$ may be regarded as a system of difference polynomials on the multivariate ring of difference polynomials $\mathbb{D}_{\mathbf{w},z}\coloneqq\mathbb{D}_{w_{1},\ldots,w_{M},z}$ . In this case, we consider the difference ideal

I_{\mathcal{M}_{f}}\coloneqq\langle Q{\sigma(\mathbf{w})}^{\mu}-\mathbf{a}(\mathbf{w})-\mathbf{e}_{\mathbf{a}}(\sigma(\mathbf{w})),\,Q\,z-b(\mathbf{w})\rangle\colon H^{\infty}\subset\mathbb{D}_{\mathbf{w},z},

(9)

where $H\coloneqq\{Q,\sigma(Q),\sigma^{2}(Q),\ldots\}$ , and “ $\colon H^{\infty}$ ” denotes the saturation with $H$ .

Algorithm 1 gives the steps of our approach for the arithmetic of D-algebraic sequences.

Algorithm 1 Arithmetic of difference-algebraic sequences

N

difference polynomials

p_{i}\in\mathbb{D}_{s_{i}}

of order

n_{i}

and a function

f\in\mathbb{K}(X_{1},\ldots,X_{n})

A difference polynomial of order at most

M\coloneqq n_{1}+\cdots+n_{N}

that vanishes at

f(f_{1},\ldots,f_{N})

for appropriate values of

n

, where each

f_{i}

is a generic zero of

p_{i}

1.

Construct $(\mathcal{M}_{f})$ from the input $p_{1},\ldots,p_{N}$ as in (8).

Denote by $\mathcal{E}$ the set

	$\displaystyle\mathcal{E}$	$\displaystyle\coloneqq$	$\displaystyle\{Q\,{\sigma(\mathbf{w})}^{\mu}-\mathbf{a}(\mathbf{w})-\mathbf{e}_{\mathbf{a}}(\mathbf{w}^{\prime}),Q\,z-b(\mathbf{w})\}$
		$\displaystyle=$	$\displaystyle\big{\{}Q\,{\sigma(w_{i})}^{\mu_{i}}-a_{i}(w_{1},\ldots,w_{M})-e_{a_{i}}(\sigma(w_{i})),i=1,\ldots,M,\,Q\,z-b(w_{1},\ldots,w_{M})\big{\}},$

Algorithm 2 Arithmetic of difference-algebraic sequences

\mathcal{E}\subset\mathbb{K}[w_{1},\ldots,w_{M},\sigma(w_{1}),\ldots,\sigma(w_{M}),z].

3.

Compute the first $M-1$ shifts (application of $\sigma$ ) of all polynomials in $\mathcal{E}$ and add them to $\mathcal{E}$ .
4.

Compute the $M$ th shift of $Q\,z-b(w_{1},\ldots,w_{M})$ and add it to $\mathcal{E}$ . We are now in the ring $\mathbb{K}[\sigma^{\leq M}(w_{1}),\ldots,\sigma^{\leq M}(w_{M}),\sigma^{\leq M}(z)]$ , which contains all differential polynomials of order at most $M$ in $\mathbb{D}_{\mathbf{w},z}$ .
5.

Let $I\coloneqq\langle\mathcal{E}\rangle\subset\mathbb{K}[\sigma^{\leq M}(w_{1}),\ldots,\sigma^{\leq M}(w_{M}),\sigma^{\leq M}(z)]$ be the ideal generated by the elements of $\mathcal{E}$ .
6.

Let $H\coloneqq\{Q,\sigma(Q),\ldots,\sigma^{M}(Q)\}$ .
7.

Update $I$ by its saturation with $H$ , i.e, $I\coloneqq I\colon H^{\infty}$ .
8.

Compute the elimination ideal $I\cap\mathbb{K}[\sigma^{\leq M}(z)]$ . From the resulting Gröbner basis, choose a polynomial $q$ of the lowest degree among those of the lowest order.
9.

Return $q$ .

To prove the correctness of Algorithm 1, we must show that $\langle\sigma^{\leq M}\left(I_{\mathcal{M}_{f}}\right)\rangle\cap\mathbb{K}[\sigma^{\infty}(z)]$ in step 8 is a non-trivial elimination ideal. The following theorem establishes this fact.

Theorem 3.2.

On the commutative ring $\mathbb{K}[\sigma^{\infty}(\mathbf{w}),\sigma^{\infty}(z)]$ , seen as a polynomial ring in infinitely many variables, consider the lexicographic monomial ordering corresponding to any ordering on the variables such that

(i.)

$\sigma^{j_{1}}(z)\succ\sigma^{j_{2}}(w_{i})$ , $i,j_{1},j_{2}\in\mathbb{N}$ ,
(ii.)

$\sigma^{j+1}(z)\succ\sigma^{j}(z)$ , $\sigma^{j+1}(w_{i_{1}})\succ\sigma^{j}(w_{i_{2}})$ , $i_{1},i_{2},j\in\mathbb{N}$ .

Then the set $\mathcal{E}\coloneqq\left\{Q{\sigma(\mathbf{w})}^{\mu}-\mathbf{a}(\mathbf{w})-\mathbf{e}_{\mathbf{a}}(\sigma(\mathbf{w})),\,Qz-b(\mathbf{y})\right\}$ is a triangular set with respect to this ordering. Moreover,

\langle\sigma^{\leq M}\left(I_{\mathcal{M}_{f}}\right)\rangle\cap\mathbb{K}[\sigma^{\infty}(z)]\neq\langle 0\rangle.

(10)

Proof.

First, let us mention that the system $(\mathcal{M}_{f})$ is consistent by Theorem 3.2 from the individual input difference polynomials. The leading monomials of

\displaystyle\sigma^{j}\left(Q\,\sigma(w_{i})^{\mu_{i}}-a_{i}(\mathbf{w})-e_{a_{i}}(\sigma(w_{i}))\right)\,\text{ and }\,\sigma^{j}\left(Q\,z-b(\mathbf{w})\right)

in the ring $\mathbb{K}[\sigma^{\infty}(\mathbf{w}),\sigma^{\infty}(z)]$ have highest variables $\sigma^{j+1}(w_{i})$ and $\sigma^{j}(z)$ , respectively. Since these variables are all distinct, by definition (see [Hub03, Definition 4.1]), we deduce that $\mathcal{E}$ is a consistent triangular set with coefficients in the field $\mathbb{K}$ . As a triangular set, $\mathcal{E}$ defines the ideal $\langle\mathcal{E}\rangle:H^{\infty}=I_{\mathcal{M}_{f}}$ , where $H~{}\coloneqq~{}\{Q,\sigma(Q),\ldots,\sigma^{M}(Q)\}$ . Therefore by [Hub03, Theorem 4.4]), all associated primes of $\sigma^{\leq M}(I_{\mathcal{M}_{f}})$ share the same transcendence basis given by the non-leading variables $\{w_{1},\ldots,w_{M}\}$ in $\sigma^{\leq M}(I_{\mathcal{M}_{f}})$ . Thus the transcendence degree of $\mathbb{K}[\sigma^{\leq M}(\mathbf{w}),\sigma^{\leq M}(z)]/\langle\sigma^{\leq M}\left(I_{\mathcal{M}_{f}}\right)\rangle$ over $\mathbb{K}$ is $M$ . However, the transcendence degree of $\mathbb{K}(x)[\sigma^{\leq M}(z)]$ is $M+1$ . Hence we must have $\langle\sigma^{\leq M}\left(I_{\mathcal{M}_{f}}\right)\rangle\cap\mathbb{K}[\sigma^{\infty}(z)]\neq\langle 0\rangle$ . ∎

Observe that $M$ is the minimal integer for which the argument of the proof of Theorem 3.2 holds.

3.2 Some closure properties and examples

Let us present some immediate consequences of the result from the previous subsection, including the ring structure of $\mathbb{K}^{\aleph_{0}}$ , induced by addition and multiplication.

Corollary 3.3.

Let $(u(n))_{n}$ and $(v(n))_{n}$ be two difference-algebraic sequences of order $r_{1}$ and $r_{2}$ , respectively. Then, the following sequences are difference-algebraic of order $r$ :

1.

addition: $(u(n)+v(n))_{n}$ , with $r\leq r_{1}+r_{2}$ ;
2.

multiplication: $(u(n)\,v(n))_{n}$ , with $r\leq r_{1}+r_{2}$ ;
3.

division: $(1/v(n))_{n\geq n_{0}}$ , with $r\leq r_{2}$ , provided that $(v(n))_{n}$ is non-zero and $v(n)\neq 0$ for all $n\geq n_{0}\in\mathbb{N}$ ;
4.

taking radicals: $(\sqrt[N]{u(n)}\,)_{n},N\in\mathbb{N}$ , with $r\leq r_{1}$ , provided that $u(n)\geq 0$ for all $n\geq n_{0}\in\mathbb{N}$ ;
5.

partial product: $(\prod_{k=k_{0}}^{n}u(k))_{n},k_{0}\in\mathbb{N}$ , with $r\leq r_{1}+1$ .
6.

partial sum: $(\sum_{k=k_{0}}^{n}u(k))_{n},\,k_{0}\in\mathbb{N}$ , with $r\leq r_{1}+1$ ;

Proof.

The proofs of these properties are deduced from constructions of radical-rational dynamical systems $(\mathcal{M}_{f})$ as in (8) for some rational functions, here denoted $f$ .

1.

$f(X,Y)\coloneqq X+Y$ ;
2.

$f(X,Y)\coloneqq X\,Y$ ;
3.

$f(X)\coloneqq 1/X$ and $(\mathcal{M}_{f})$ built with the defining difference polynomial of $(v(n))_{n}$ only;
4.

This follows from the fact that the correctness of Algorithm 1 is unchanged with having $z$ replaced by $z^{N}$ , $N\in\mathbb{N}$ . Thus, $(\mathcal{M}_{f})$ is built with $f(X)=X$ such that the output equation writes $z^{N}=w_{1}$ . Note, however, that this is an elementary fact. Indeed, one verifies that the difference polynomial obtained by substituting $(\sigma^{j}(x))^{i}$ by $(\sigma^{j}(x))^{i\,N}$ in $p$ has $(\sqrt[N]{u(n)}\,)_{n}$ as a zero.
5.

Let us denote by $(t(n))_{n}$ , the partial product of $(u(n))_{n}$ . It is defined by the recursion $t(n+1)=t(n)\,u(n)$ , with $t(0)=1$ . First, consider the radical-rational dynamical system $(\mathcal{M}_{f})$ constructed from the difference polynomial of $(u(n))_{n}$ , with $f$ unspecified. At this stage, we know that for any $f$ , we have $r_{1}$ components in the system. To take $(t(n))_{n}$ into account, we add a new variable $w_{r_{1}+1}$ , such that $\sigma(w_{r_{1}+1})=w_{r_{1}+1}\,w_{1}$ , which represents the recursion. Then, we choose the function $b$ of the system as $b=w_{r_{1}+1}$ , which tells Algorithm 1 that we want a difference polynomial for $(t(n))_{n}$ . Hence, by construction, $(t(n))_{n}$ is D-algebraic of order at most $r_{1}+1$ as claimed. Some constraints of the resulting difference polynomial may define the value of $n_{0}$ .
6.

For the partial sum (also known as the partial sum), say $(s(n))_{n}$ , one considers the recursion $s(n+1)=s(n)+u(n)$ , with $s(0)=0$ , and proceeds in the same way as with the partial product. One can verify that the difference polynomial obtained by substituting $x$ by $\sigma(x)-x$ in the defining difference polynomial of $(u(n))_{n}$ belongs to the difference ideal of the output of the algorithm. The obtained difference polynomial may need to be shifted to get its normal form.

∎

Applications of Algorithm 1 reveal that a dynamical system in the form of $(\mathcal{M}_{f})$ from (8) serves as a certificate for computing an algebraic difference equation (or difference polynomial) associated with a given problem.

Example 3.4.

Let $p,q\in\mathcal{D}_{\sigma}(\mathbb{K},x)$ , such that $p=\sigma^{2}(x)-\sigma(x)\,x$ , $q=\sigma(x)-x^{2}-x$ , specialized with the generic solution $u(n)$ and $v(n)$ , respectively. With the initial values $u(0)=1$ , $u(1)=k\in\mathbb{K}$ , the algebraic difference equation associated to $p$ yields the sequence of general term $k^{F_{n}}$ , where $F_{n}$ is the $n$ th Fibonacci number (see, for instance, A000301 from [S⁺03]). With $v(0)=1,v(1)=2$ , v(n) is known to denote the number of ordered trees having nodes of outdegree $0,1,2$ and such that all leaves are at level $n$ (see A007018). Let us find algebraic difference equations satisfied by $(u(n)/v(n))_{n}$ , $(u(n)\,v(n))_{n}$ , and $(\sum_{k=0}^{n}u(k))_{n}$ . In all these cases, we will write the resulting equation with the undetermined term $s(n)$ .

$(u(n)/v(n))_{n}$ .

We use the same notations from the previous subsection. The corresponding dynamical system is given by

\begin{cases}\sigma(w_{1})=w_{2}\\ \sigma(w_{2})=w_{1}\,w_{2}\\ \sigma(w_{3})=w_{3}^{2}+w_{3}\\ z=\frac{w_{1}}{w_{3}}\end{cases}.

(11)

After elimination, we obtain the following principal ideal

\begin{split}&\Big{\langle}\sigma(z)^{4}\sigma^{3}(z)^{2}z^{4}-z^{3}\sigma(z)^{4}\sigma^{2}(z)^{2}\sigma^{3}(z)-z^{3}\sigma(z)^{3}\sigma^{2}(z)\sigma^{3}(z)^{2}-z^{2}\sigma(z)^{3}\sigma^{2}(z)^{3}\sigma^{3}(z)\\ &+z\sigma(z)^{3}\sigma^{2}(z)^{5}-z^{2}\sigma(z)^{2}\sigma^{2}(z)^{2}\sigma^{3}(z)^{2}+2z\sigma(z)^{2}\sigma^{2}(z)^{4}\sigma^{3}(z)+\sigma(z)^{2}\sigma^{2}(z)^{6}\\ &+z\sigma(z)\sigma^{2}(z)^{3}\sigma^{3}(z)^{2}+2\sigma(z)\sigma^{2}(z)^{5}\sigma^{3}(z)+\sigma^{2}(z)^{4}\sigma^{3}(z)^{2}\Big{\rangle},\end{split}

(12)

where $\sigma^{j}(z)^{i}$ is understood as $(\sigma^{j}(z))^{i}$ . Hence the algebraic difference equation

\begin{split}&s(n+1)^{4}s(n+3)^{2}s(n)^{4}-s(n+1)^{4}s(n+3)s(n)^{3}s(n+2)^{2}+s(n+3)^{2}s(n+2)^{4}\\ &-s(n+1)^{3}s(n+3)^{2}s(n)^{3}s(n+2)+s(n+1)^{3}s(n)s(n+2)^{5}+s(n+1)^{2}s(n+2)^{6}\\ &+2s(n+1)^{2}s(n+3)s(n)s(n+2)^{4}-s(n+1)^{2}s(n+3)^{2}s(n)^{2}s(n+2)^{2}\\ &+s(n+1)s(n+3)^{2}s(n)s(n+2)^{3}+2s(n+1)s(n+3)s(n+2)^{5}\\ &-s(n+1)^{3}s(n+3)s(n)^{2}s(n+2)^{3}=0,\\ \end{split}

(13)

of order $3$ and degree $10$ .

$(u(n)\,v(n))_{n}$ .

The system is similar to (11) with the last equation replaced by $z=w_{1}\,w_{3}$ . We obtain a principal ideal with the following associated equation:

\begin{split}&s(n+2)^{2}s(n+1)^{4}s(n)^{4}+2s(n+3)s(n+2)s(n+1)^{3}s(n)^{4}+s(n+2)^{3}s(n+1)^{3}s(n)^{3}\\ &+s(n+3)^{2}s(n+1)^{2}s(n)^{4}+2s(n+3)s(n+2)^{2}s(n+1)^{2}s(n)^{3}-s(n+2)^{4}s(n+1)^{2}s(n)^{2}\\ &+s(n+3)^{2}s(n+2)s(n+1)s(n)^{3}-s(n+3)s(n+2)^{3}s(n+1)s(n)^{2}-s(n+2)^{5}s(n+1)s(n)\\ &-s(n+3)s(n+2)^{4}s(n)+s(n+2)^{6}=0.\end{split}

(14)

Observe that (14) is also satisfied by the sequence $(v(n)/u(n))_{n}$ since $(1/u(n))_{n}$ is also a zero of $p$ .

3.

$(\sum_{k=0}^{n}u(k))_{n}$ .

The corresponding system writes

$\begin{cases}\sigma(w_{1})=w_{2}\\ \sigma(w_{2})=w_{1}\,w_{2}\\ \sigma(w_{3})=w_{3}+w_{1}\\ z=w_{3}\end{cases}.$ (15)

Here again, we obtain a principal ideal. The associated equation is given by

$s(n)s(n+1)-s(n)s(n+2)-s(n+1)^{2}+s(n+1)s(n+2)+s(n+2)-s(n+3)=0.$ (16)

The fact that we obtain principal ideals in all three examples is a common situation encountered in the differential case. It is explained in [DGHP23, Remark 4] that such an ideal is generally “almost principal”. $\blacksquare$

4 L.h.s. D-algebraic sequences

In this section, we focus on two important subclasses of D-algebraic sequences. We show that sequences from both classes are zeros of difference polynomials that are linear in their highest shift terms.

4.1 Holonomic sequences

Holonomic sequences are solutions to linear recurrence equations with polynomial coefficients in the index variable. These sequences are ubiquitous in the sciences. One reason may be that they share similar properties with their generating functions. Some interesting applications can be found in [BCR24, KP11].

In [TTW24], Worrell and the author proposed an algorithm to convert any holonomic equation into an l.h.s. algebraic difference equation. We revisit this result and complete its proof.

For this part, we replace the field $\mathbb{K}$ with $\mathbb{K}(n)$ and work on the specialized ring of difference polynomials $\mathbb{D}_{s}(n)\coloneqq\mathbb{K}(n)[\sigma^{\infty}(s(n))]$ . This enables us to introduce the index variable $n$ in our difference polynomials. However, the aim is to derive a difference polynomial $p\in\mathbb{D}_{s}=\mathbb{K}[\sigma^{\infty}(s(n))]$ where the term $s(n+\operatorname{ord}(p))$ appears linearly.

Definition 4.1.

The holonomic degree of a difference polynomial $p\in\mathbb{D}_{s}(n)$ of degree $1$ is the degree of $p$ viewed as a univariate polynomial in $n$ .

When working with holonomic equations or sequences, we often use the word “degree” to refer to the holonomic degree.

Theorem 4.2.

Every holonomic sequence of order $l$ and degree $d$ is a zero of an l.h.s. difference polynomial of order at most $l+d$ .

Proof.

Let $p\in\mathbb{D}_{s}(n)$ be holonomic of degree $d$ and order $l$ . If $d=0$ , then we are done. For the rest of the proof, we assume that $d>0$ . This assumption implies that $(s(n))_{n}$ (as a sequence) does not vanish any holonomic difference polynomial of order $\leq l$ and degree $\leq d$ . In particular, no constant coefficient linear recurrence of order $\leq l$ is satisfied by $(s(n))_{n}$ .

We look at $p$ and its shifts as polynomials in $n$ such that

\sigma^{j}(p)=\sum_{k=0}^{d}\gamma_{j,k}\,n^{k},\,\,j=0,\ldots,d,

(17)

where $\gamma_{j,k}\in\mathbb{K}[s(n+j),\ldots,s(n+j+l)]$ . We consider the associated equations for $j<d$ and write them as follows

\sum_{k=1}^{d}\gamma_{j,k}\,n^{k}=-\gamma_{j,0},\,\,j=0,\ldots,d-1.

(18)

This is a linear system of $d$ equations in $(n,n^{2},\ldots,n^{d})^{T}$ . We claim that the matrix of the system has full rank due to the shifts involved in the $\gamma_{j,k}$ ’s. Indeed, each $\gamma_{j,k}$ has the following form:

\gamma_{j,k}\coloneqq\sum_{i=0}^{l}c_{j,i,k}\,s(n+j+i)=C_{j,k}\cdot S_{l+j},

(19)

where $c_{j,i,k}$ is the constant coefficient of $n^{k}$ in the polynomial coefficient of $s(n+j+i)$ in $\sigma^{j}(p)$ , $i=0,\ldots,l$ , $j=0,\ldots,d-1$ ; $S_{l+j}=(s(n+j),\ldots,s(n+j+l))^{T}$ and $C_{j,k}=(c_{j,0,k},c_{j,1,k},\ldots,c_{j,l,k})$ . Note that all $c_{j,i,k}$ ’s are linear combinations of the $c_{0,i,k}$ ’s, and $c_{j,i,d}=c_{0,i,d}$ for all $j=0,\ldots,d-1$ .

Let $\Gamma_{0},\ldots,\Gamma_{d-1}$ be the rows of the matrix from (18). Observe that

\Gamma_{j}=(\gamma_{j,1},\ldots,\gamma_{j,d})=(C_{j,1}\cdot S_{l+j},\ldots,C_{j,d}\cdot S_{l+j}).

(20)

The components of the $\Gamma_{j}$ ’s are all nonzero as they all yield $l$ th-order linear recurrence equations with constant coefficients.

Let $\lambda_{0},\lambda_{1},\ldots,\lambda_{d-1}\in\mathbb{K}(s(n),\ldots,s(n+l+d-1))$ such that $\sum_{j=0}^{d-1}\lambda_{j}\,\Gamma_{j}=0$ . The latter is a homogeneous linear system of linearly independent equations in the $\lambda$ ’s and can only have the trivial solution. To see that, fix $k$ and $k^{\prime}$ , $1\leq k\neq k^{\prime}\leq d$ . Notice that for $0\leq j_{1}\neq j_{2}\leq d-1$ , $\gamma_{j_{1},k}$ and $\gamma_{j_{2},k}$ can be seen as two independent variables because $S_{l+j_{1}}$ and $S_{l+j_{2}}$ are two linearly independent vectors over $\mathbb{K}(s(n),\ldots,s(n+l+d-1))$ . Thus, while it might be possible to find $\lambda_{0},\lambda_{1},\ldots,\lambda_{d-1}\in\mathbb{K}(s(n),\ldots,s(n+l+d-1))$ so that

\lambda_{0}\,\gamma_{0,k}+\lambda_{1}\,\gamma_{1,k}+\cdots+\lambda_{d-1}\,\gamma_{d-1,k}=0,

(21)

it is impossible for the same $\lambda$ ’s to verify

\lambda_{0}\,\gamma_{0,k^{\prime}}+\lambda_{1}\,\gamma_{1,k^{\prime}}+\cdots+\lambda_{d-1}\,\gamma_{d-1,k^{\prime}}=0,

(22)

unless $\lambda_{0}=\lambda_{1}=\cdots=\lambda_{d-1}=0$ .

Hence (18) is non-singular and for all positive integers $k\leq d$ , $n^{k}\in\mathbb{K}(s(n),\ldots,s(n+d+l-1))$ . Finally, we plug these rational expressions into $\sigma^{d}(p)$ . Since all the $n^{k}$ ’s are free of $s(n+l+d)$ , the resulting difference polynomial is l.h.s. of order at most $l+d$ . ∎

Our implementation of the algorithm from the proof of Theorem 4.2 is now part of the DalgSeq subpackage of NLDE. The corresponding procedure is HoloToSimpleRatrec.

Example 4.3 (Generating Somos-like sequences [Mal92, EZ14]).

A Somos-like sequence is an integral sequence defined with a rational recursion. Using Theorem 4.2, one can generate a Somos-like sequence as follows:

1.

Take a holonomic equation and choose integral initial values such that all the following terms are also integers. A natural choice is to take an equation of the form

$s(n+k+1)=P_{0}(n)s(n)+\cdots+P_{k}(n)s(n+k),$ (23)

$P_{i}(n)\in\mathbb{K}[n],i=0,\ldots,k$ , with any set of $k+1$ integers for the initial values.
2.

Then use the algorithm from the proof of Theorem 4.2 to convert (23) into an l.h.s. difference polynomial.

Concretely, let us take

s(n+3)=ns(n)+(n+1)s(n+1)+(n+2)s(n+2).

(24)

We choose the initial values $s(0)=s(1)=s(2)=1$ . This is A122752 from [S⁺03]. We have $s(3)=3$ . Thus, the sequence defined by (24) and these initial values is an integer sequence. Our procedure HoloToSimpleRatrec produces the following rational recursion from (24):

s(n+4)=\frac{1}{{s(n)+s(n+1)+s(n+2)}}\Big{(}s(n)s(n+1)+2s(n)s(n+2)+3s(n)s(n+3)\\ +3s(n+1)s(n+3)+2s(n+2)s(n+3)+s(n+3)^{2}\Big{)}.

(25)

Using (25), we generate the next $7$ terms of $(s(n))_{n}$ below:

12,59,352,2455,19592,176033,1758218.

$\blacksquare$

Example 4.4.

Catalan numbers are defined by the formula $C(n)\coloneqq\frac{1}{n+1}\binom{2n}{n}$ , and satisfy the equation $(n+2)s(n+1)-(4n+2)s(n)=0$ , which is holonomic. Using HoloToSimpleRatrec, we convert that equation into the rational recursion

s(n+2)=\frac{2s(n+1)\left(8s(n)+s(n+1)\right)}{10s(n)-s(n+1)}.

(26)

The above recursion appeared on the OEIS website for A000108 in 2006.

A noteworthy observation emerges from the geometrical view of Theorem 4.2. For the present example, with $\mathbb{K}=\mathbb{R}$ , let us consider the surface defined by

H_{C}\coloneqq\left\{(x,y,z),\,\mathbb{R}^{3}\,\colon z\,(x+2)-y\,(4x+2)=0\right\}.

(27)

This encodes the holonomic equation of Catalan numbers. On this surface, Catalan numbers are the points $P_{n}\coloneqq(n,C(n),C(n+1)),n\in\mathbb{N}$ . For the D-algebraic representation, we consider

H_{C}^{\prime}\coloneqq\left\{(x,y,z),\,\mathbb{R}^{3}\,\colon z\,(10x-y)-(8x+y)=0\right\},

(28)

where Catalan numbers are the points $P_{n}^{\prime}\coloneqq(C(n),C(n+1),C(n+2)),n\in\mathbb{N}$ . It turns out that $H_{C}$ and $H_{C}^{\prime}$ intersect on a curve surrounded by the points $P_{n}$ and $P_{n}^{\prime}$ , with $P_{1}=P_{0}^{\prime}$ on it. The projection of the algebraic variety thus defined onto the $xy$ -plane is given by the curve

\mathcal{C}_{C}\coloneqq\left\{(x,y),\,\mathbb{R}^{2}\,\colon y\,(x+1)-2x(2x-1)=0\right\}.

(29)

Figure 1 illustrates this geometry from the first quadrant of the $xy$ -plane.

Refer to caption — Figure 1: A geometric view of Theorem 4.2 for Catalan numbers.
$H_{C}$ is in red, $H_{C}^{\prime}$ in blue, and $\mathcal{C}_{C}$ is in magenta.

Further study is needed to elucidate the connection between such varieties and their related holonomic sequences. $\blacksquare$

4.2 $C^{2}$ -Finite sequences

A $C$ -finite sequence solves a linear recurrence equation with constant coefficients. A $C^{2}$ -finite sequence is a solution to a linear recurrence equation with $C$ -finite term coefficients [JPNP23]. As in the differential case with $D^{2}$ -finite functions, $C^{2}$ -finite sequences are D-algebraic. An implementation for computing algebraic difference equations from $C^{2}$ -finite sequences is provided in our package under the name CCfiniteToDalg. The approach is similar to that of $D^{2}$ -finite functions as described in [TT23]. In this subsection, we rely on a result from [CDBMP23] to establish the following theorem.

Theorem 4.5.

If $(s(n))_{n}$ is $C^{2}$ -finite such that the leading $C$ -finite term coefficient of its defining equation is non-zero for all $n\geq N\in\mathbb{N}$ . Then $(s(n))_{n\geq N}$ satisfies an l.h.s. algebraic difference equation.

Proof.

It suffices to construct a dynamical system in the form of (8) where $\mu=(1,\ldots,1)^{T}$ and $\mathbf{E}_{\mathbf{A}}(\sigma(\mathbf{w}))=\boldsymbol{0}$ . This is straightforward from the formal definition of a $C^{2}$ -finite sequence. Let $(s(n))_{n}$ be $C^{2}$ -finite of order $l$ with $C$ -finite coefficients $c_{l}(n),\ldots,c_{0}(n)$ , of order $r_{l},\ldots,r_{0}$ , respectively. For simplicity, we assume $N=0$ . From the defining equation of the coefficient $c_{j}(n)$ , we can write

c_{j}(n+r_{i})=\alpha_{0,j}\,c_{i}(n)+\cdots+\alpha_{r_{j}-1,j}\,c_{j}(n+l-1),

(30)

where $\alpha_{i,j}\in\mathbb{K}$ , $i=0,\ldots,r_{j}-1,j=0,\ldots,l$ . Similarly, from the equation of $s(n)$ , we have

s(n+l)=-\frac{1}{c_{l}(n)}\left(c_{0}(n)\,s(n)+\cdots+c_{l-1}(n)\,s(n+l-1)\right).

(31)

Thus, following the construction from Section 3.1, we deduce that each $c_{j}(n)$ adds $r_{j}$ equations to the system, which leads to a rational dynamical system of dimension $l+\sum_{j=0}^{l}r_{j}$ . ∎

In Theorem 4.5, $N$ is known to exist for non-degenerate $C$ -finite sequences [BM76]. In the general case, however, the proper definition of a $C^{2}$ -finite sequence is subject to answering the Skolem problem [OW12] for the leading $C$ -finite term coefficient. Note that this constraint highly depends on the choice of initial values: the sequence $((-1)^{n}-1)_{n}$ satisfies the recurrence $s(n+2)-s(n)=0$ and has infinitely many zeros; however, the sequence $(2\,(-1)^{n}-1)_{n}$ satisfies the same recurrence equation but has no zero. In the following example, we use our algorithm to compute an algebraic difference equation satisfied by a $C^{2}$ -finite sequence for which two not-necessarily-equal solutions of $s(n+2)=s(n)$ appear within the coefficients of its equation.

Example 4.6.

Consider a $C^{2}$ -finite sequence $(s(n))_{n}$ solution of

u(n)\,s(n)+2\,s(n+1)+v(n)\,s(n+2)=0,

(32)

where $(u(n))_{n}$ and $(v(n))_{n}$ are two not-necessarily identical solutions of $s(n+2)-s(n)=0$ . This includes the result of [JPNP23, Example 4.1]. The corresponding dynamical system writes:

\begin{cases}\sigma(w_{1})=w_{2}\\ \sigma(w_{2})=-\frac{w_{5}\,w_{1}+2\,w_{2}}{w_{3}}\\ \sigma(w_{3})=w_{4}\\ \sigma(w_{4})=w_{3}\\ \sigma(w_{5})=w_{6}\\ \sigma(w_{6})=w_{5}\\ z=w_{1}\end{cases}.

(33)

The resulting $6$ th-order algebraic difference equation is l.h.s. and may be written recursively as follows:

s(n+6)=\frac{s(n+5)s(n+4)s(n)+s(n+4)s(n+3)s(n+2)-s(n+4)^{2}s(n+1)-s(n+5)s(n+2)^{2}}{s(n+3)s(n)-s(n+2)s(n+1)}.

(34)

$\blacksquare$

Although we obtained an l.h.s. algebraic difference equation in (34), the underlying adaptation of Algorithm 1 does not guarantee such an output. A bound for the order of the desired l.h.s. equation is provided in [CDBMP23] but without an algorithm.

5 Subsequences

Recall that a subsequence of a sequence $(s(n))_{n}$ is a sequence $(t(n))_{n}$ such that $t(n)=s(\varphi(n))$ , where $\varphi$ is a strictly increasing embedding of $\mathbb{N}$ . Thus, to find a difference polynomial $p$ that vanishes at $(t(n))_{n}$ , we need an endomorphism $\tilde{\sigma}$ that sends $s(\varphi(n))$ to $s(\varphi(n+1))$ , i.e., the minimal gap between the orders of two difference monomials in $p$ is $\delta_{\tilde{\sigma}}=\min_{n\in\mathbb{N}}\{\varphi(n+1)-\varphi(n)\}$ and not the usual $n+1-n=1$ . This section focuses on the case where $\varphi(n)=dn$ . This is the $n$ th term formula of an arithmetic progression of common difference $d$ with initial term $0$ . What makes this case rather natural is that all order gaps are multiple of the common difference $d$ , and here we have $\delta_{\tilde{\sigma}}=d$ . This implies that the normal form of $p$ can be seen as follows:

p(x,\sigma^{d}(x),\sigma^{2d}(x),\sigma^{3d}(x),\ldots,\sigma^{dk}(x),\ldots).

(35)

The desired endomorphism is thus defined by $\tilde{\sigma}=\sigma^{d}$ .

Theorem 5.1.

Let $d$ be a positive integer. If $(s(n))_{n\in\mathbb{N}}$ is D-algebraic then so is $(s(dn))_{n\in\mathbb{N}}$ .

Proof.

For cleaner formulation, we will use the following iterative notation:

For $N$ variables $x_{1},\ldots,x_{N}$ and a function $f$ in $N$ variables, we write

	$\displaystyle x_{1},x_{2},\ldots,x_{N}=\underline{x_{i}}_{i},$
	$\displaystyle f(f(x_{1}),f(x_{2}),\ldots,f(x_{N}))=f\left(\underline{f(x_{i})}_{i}\right),$
	$\displaystyle f(f(x_{2}),f(x_{3}),\ldots,f(x_{N}))=f\left(\underline{f(x_{i\neq 1})}_{i}\right)=f\left(\underline{f(x_{i>1})}_{i}\right).$

Let $\tilde{\sigma}=\sigma^{d}$ and $p\in\mathcal{D}_{\sigma}(x,x)$ of order $r$ such that

p(s(n))=0,\,\text{ for all }n\in\mathbb{N}.

Define

y_{1}=x,\,y_{2}=\sigma(x),\ldots,y_{d}=\sigma^{d-1}(x).

(36)

Let $r_{1},r_{0}$ be the quotient and remainder of the Euclidean division of $r+1$ by $d$ : $r+1=d\,r_{1}+r_{0}$ . We can rewrite

p(x,\sigma(x),\ldots,\sigma^{r}(x)),

p\Big{(}y_{1},y_{2},\ldots,y_{d},\tilde{\sigma}(y_{1}),\tilde{\sigma}(y_{2}),\ldots,\tilde{\sigma}^{r_{1}}(y_{1}),\tilde{\sigma}^{r_{1}}(y_{2}),\ldots,\tilde{\sigma}^{r_{1}}(y_{r_{0}})\Big{)}.

(37)

The latter is a multivariate difference polynomial in

\mathcal{D}_{\tilde{\sigma}}(\mathbb{K},\{y_{1},y_{2},\ldots,y_{d}\}).

Our aim is to eliminate the difference indeterminates $y_{j}$ , $j>1$ . To do so, we build a radical-rational dynamical system and apply Algorithm 1. This will yield a univariate difference polynomial in $y_{1}$ that has $(s(dn))_{n\in\mathbb{N}}$ as a zero. Let $m$ be the degree of $\sigma^{r}(x)$ in $p$ and $R_{p}$ be the rational expression obtained by solving $p=0$ for $(\sigma^{r}(x))^{m}$ . Let $y_{i,j},i=1,\ldots,d$ , $j=0,\ldots,r_{1}$ be new indeterminates such that

	$\displaystyle y_{i,0}=y_{i},i=1,\ldots,d,$	(38)
	$\displaystyle\tilde{\sigma}(y_{i,j})=y_{i,j+1},\,i=1,\ldots,d,\,j=0,\ldots,r_{1}-2,$	(39)
	$\displaystyle\tilde{\sigma}(y_{i,r_{1}-1})=y_{i,r_{1}},\,i=1,\ldots,r_{0}-1,$	(40)
	$\displaystyle\tilde{\sigma}(y_{r_{0},r_{1}-1})^{m}=R_{p}\left(\underline{y_{i,j}}_{i,j},\tilde{\sigma}(y_{r_{0},r_{1}-1})\right),$	(41)
$\displaystyle\begin{split}&\tilde{\sigma}(y_{r_{0}+k,r_{1}-1})^{m}=R_{p}\left(\underline{y_{i\geq k,0}}_{i},\underline{y_{i,j\neq 0}}_{i,j},\underline{\tilde{\sigma}(y_{r_{0}\leq i\leq r_{0}+k,r_{1}-1})}_{i}\right),\\ &k=1,\ldots,d-r_{0}\end{split}$		(42)

and

\begin{split}&\tilde{\sigma}(y_{k,r_{1}})^{m}=R_{p}\Big{(}\underline{y_{i\geq d-r_{0}+k,0}}_{i},\underline{y_{i,j\neq 0}}_{i,j},\underline{\tilde{\sigma}(y_{r_{0}\leq i\leq r_{0}+k,r_{1}})}_{i},\underline{\tilde{\sigma}(y_{i\leq k,r_{1}})}_{i}\Big{)},\\ &k=1,\ldots,r_{0}-1.\end{split}

(43)

The equations in (39)–(43) define the desired dynamical system together with the output equation $z=y_{1,0}$ . ∎

Corollary 5.2.

Let $d$ be a positive integer. If $(s(n))_{n\in\mathbb{N}}$ is D-algebraic of order $r$ in $\mathcal{D}_{\sigma}(\mathbb{K},x)$ , then $(s(dn))_{n\in\mathbb{N}}$ is D-algebraic of order at most $r$ in $\mathcal{D}_{\sigma^{d}}(\mathbb{K},x)$ . In other words, $(s(dn))_{n\in\mathbb{N}}$ is D-algebraic of order at most $dr$ in $\mathcal{D}_{\sigma}(\mathbb{K},x)$ .

Proof.

The proof reduces to determining the dimension of the dynamical system obtained in the proof of Theorem 5.1. Indeed, we have

	$\displaystyle(r_{1}-1)d\,\text{ equations from }\,\eqref{eq:syssub1},$		(44)
	$\displaystyle r_{0}-1\,\text{ equations from }\,\eqref{eq:syssub2},$		(45)
	$\displaystyle 1\,\text{ equation from }\,\eqref{eq:syssub3},$		(46)
	$\displaystyle d-r_{0}\,\text{ equations from }\,\eqref{eq:syssub4},$		(47)
	$\displaystyle r_{0}-1\,\text{ equations from }\,\eqref{eq:syssubfin},$		(48)

which give a total of

dr_{1}+r_{0}-1=r+1-1=r,

(49)

representing the dimension of the dynamical system and the order of $(s(n))_{n}$ . ∎

Example 5.3 (Illustrative example: $r=4,d=3$ ).

We have $r_{1}=r_{0}=1$ , $\tilde{\sigma}=\sigma^{3}$ and $p=p(x,\sigma(x),\ldots,\sigma^{4}(x))$ . We build our dynamical system with the difference indeterminates $y_{1,0},y_{2,0},y_{3,0}$ and $y_{1,1}$ . This yields

\begin{cases}\tilde{\sigma}(y_{1,0})=y_{1,1}\\ \tilde{\sigma}(y_{2,0})^{m}=R_{p}(y_{1,0},y_{2,0},y_{3,0},y_{1,1},\tilde{\sigma}(y_{2,0}))\\ \tilde{\sigma}(y_{3,0})^{m}=R_{p}(y_{2,0},y_{3,0},y_{1,1},\tilde{\sigma}(y_{2,0}),\tilde{\sigma}(y_{3,0}))\\ \tilde{\sigma}(y_{1,1})^{m}=R_{p}(y_{3,0},y_{1,1},\tilde{\sigma}(y_{2,0}),\tilde{\sigma}(y_{3,0}),\tilde{\sigma}(y_{1,1}))\\ z=y_{1,0}\end{cases}.

(50)

$\blacksquare$

Remark 5.4.

The recurrence equation of the subsequence need not be written in the same difference ring as the equation of the original sequence. In Theorem 5.1, both $(s(n))_{n\in\mathbb{N}}$ and $(s(dn))_{n\in\mathbb{N}}$ are zeros of the constructed difference polynomial of order $dr$ in $\mathcal{D}_{\sigma}(\mathbb{K},x)$ , but $(s(n))_{n}$ is generally not a zero of the corresponding difference polynomial of order $r$ in $\mathcal{D}_{\sigma^{d}}(\mathbb{K},x)$ . This is illustrated in the next example.

Example 5.5 (Explicit example: Catalan numbers at $(3n)_{n\in\mathbb{N}}$ ).

In Section 4.1, Example 4.4, we used the implementation from [TTW24] to convert the holonomic equation of Catalan numbers $(C(n))_{n\in\mathbb{N}}$ to the recursion

s(n+2)=\frac{2s(n+1)\left(8s(n)+s(n+1)\right)}{10s(n)-s(n+1)}=R_{p}(s(n),s(n+1)),

(51)

with $p=(10x-\sigma(x))\sigma^{2}(x)-2\sigma(x)(8x+\sigma(x))$ . Thus, the corresponding dynamical system in $\mathcal{D}_{\tilde{\sigma}}(\mathbb{K},x)$ , with $\tilde{\sigma}=\sigma^{3}$ , is given by

\begin{cases}\tilde{\sigma}(y_{1,0})=\frac{2(16y_{1,0}-y_{2,0})(8y_{1,0}+y_{2,0})y_{2,0}}{(7y_{1,0}-y_{2,0})(10y_{1,0}-y_{2,0})}\\[8.53581pt] \tilde{\sigma}(y_{2,0})=\frac{4(16y_{1,0}-y_{2,0})(8y_{1,0}+y_{2,0})y_{2,0}(8y_{1,0}-y_{2,0})}{(6y_{1,0}-y_{2,0})(7y_{1,0}-y_{2,0})(10y_{1,0}-y_{2,0})}\\ z=y_{1,0}\end{cases}.

(52)

We obtain the equation

343597383680s(n)^{3}s(n+1)^{3}-69004689408s(n)^{3}s(n+1)^{2}s(n+2)+4274823168s(n)^{3}s(n+1)s(n+2)^{2}\\ -83243160s(n)^{3}s(n+2)^{3}-1258291200s(n)^{2}s(n+1)^{4}+266514432s(n)^{2}s(n+1)^{3}s(n+2)\\ -26883000s(n)^{2}s(n+1)^{2}s(n+2)^{2}+1043658s(n)^{2}s(n+1)s(n+2)^{3}-122880s(n)s(n+1)^{5}\\ -101544s(n)s(n+1)^{4}s(n+2)+65067s(n)s(n+1)^{3}s(n+2)^{2}-4113s(n)s(n+1)^{2}s(n+2)^{3}\\ +1400s(n+1)^{6}-30s(n+1)^{5}s(n+2)-75s(n+1)^{4}s(n+2)^{2}+5s(n+1)^{3}s(n+2)^{3}=0.

(53)

One verifies that $(C(n))_{n}$ is not a solution of (53). Note that higher-order equations may be obtained here. Indeed, $(C(3n))_{n}$ is holonomic of order $1$ and degree $3$ and therefore satisfies an l.h.s. algebraic difference equation of order at most $4$ by Theorem 4.2. Our Gröbner bases method from [TTW24] yields an equation of order $3$ . $\blacksquare$

Returning to our introductory motivation—the acceleration of sequences in numerical analysis—we have demonstrated a method for deriving equations for subsequences of D-algebraic sequences, indexed by arithmetic progressions. The method may be generalized to a broader class of subsequences. This work suggests a potential bridge between difference algebra and classical acceleration techniques (see [BDGB83]). Careful selection of subsequences based on these computations may lead to the development of new effective methods for accelerating convergence.

Acknowledgments

The author gratefully acknowledges Filip Mazowiecki and James Worrell for stimulating discussions regarding [CDBMP23]. This work was supported by UKRI Frontier Research Grant EP/X033813/1.

Author’s address:

Bertrand Teguia Tabuguia, University of Oxford [email protected]

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Computing with D-Algebraic Sequences

Abstract

keywords:

1 Introduction

Example 1.1.

2 Definitions

Definition 2.1.

Definition 2.2.

Definition 2.3.

Example 2.4.

Definition 2.5.

Definition 2.6.

Theorem 2.7.

Definition 2.8.

Remark 2.9.

3 Arithmetic

3.1 Algorithm

Remark 3.1.

Theorem 3.2.

Proof.

3.2 Some closure properties and examples

Corollary 3.3.

Proof.

Example 3.4.

4 L.h.s. D-algebraic sequences

4.1 Holonomic sequences

Definition 4.1.

Theorem 4.2.

Proof.

Example 4.3 (Generating Somos-like sequences [Mal92, EZ14]).

Example 4.4.

4.2 C2C^{2}-Finite sequences

Theorem 4.5.

Proof.

Example 4.6.

5 Subsequences

Theorem 5.1.

Proof.

Corollary 5.2.

Proof.

Example 5.3 (Illustrative example: r=4,d=3r=4,d=3).

Remark 5.4.

Example 5.5 (Explicit example: Catalan numbers at (3​n)n∈ℕ(3n)_{n\in\mathbb{N}}).

Acknowledgments

References

4.2 $C^{2}$ -Finite sequences

Example 5.3 (Illustrative example: $r=4,d=3$ ).

Example 5.5 (Explicit example: Catalan numbers at $(3n)_{n\in\mathbb{N}}$ ).