Remarkable relations between the central binomial series, Eulerian polynomials, and poly-Bernoulli numbers

The central binomial series is a Dirichlet series defined by

Borwein, Broadhurst, and Kamnitzer [6] studied special values $\zeta_{CB}(k)$ at positive integers and recovered some remarkable connections. A classical evaluation is $\zeta_{CB}(4)=\frac{17\pi^{4}}{3240}=\frac{17}{36}\zeta(4)$. In particular, for $k\geq 2$ Borwein–Broadhurst–Kamnitzer showed that $\zeta_{CB}(k)$ can be written as a $\mathbb{Q}$-linear combination of multiple zeta values and multiple Clausen and Glaisher values.

On the other hand, Lehmer [15] proved that for $k\leq 1$, $\zeta_{CB}(k)$ is a $\mathbb{Q}$-linear combination of $1$ and $\pi/\sqrt{3}$. For example, we have

He considered the general sum

and its derivatives to derive interesting series evaluations. More precisely, Lehmer provided the following explicit formula for the special values $\zeta_{CB}(k)$ at negative integers. Define two sequences of polynomials $(p_{k}(x))_{k\geq-1}$ and $(q_{k}(x))_{k\geq-1}$ by the initial values $p_{-1}(x)=0,q_{-1}(x)=1$ and the recursion

Then for $k\geq-1$, we have

Consequently,

The first few polynomials are: $p_{0}(x)=1$, $p_{1}(x)=3$, $p_{2}(x)=8x+7$; and $q_{0}(x)=1$, $q_{1}(x)=2x+1$, $q_{2}(x)=4x^{2}+10x+1$.

In 2004, Stephan [23, A098830] observed that the rational part of (1.4) is nothing but (a third of) a sum of poly-Bernoulli numbers of negative indices. Poly-Bernoulli numbers $B_{n}^{(k)}$ are a generalization of classical Bernoulli numbers using polylogarithm functions and were introduced by Kaneko [12]. We will give a precise definition in Section 3.

In this article, we connect both polynomials $p_{n}(x)$ and $q_{n}(x)$ to known numbers and polynomials. More precisely, we prove Stephan’s conjecture (relating this way $p_{n}(x)$ to the poly-Bernoulli numbers) using the fact that the polynomial sequence $q_{n}(x)$ is a generalization of the classical Eulerian polynomials.

Eulerian polynomials have been studied by Euler himself. Since then they have been studied and became classical. Several extensions, generalizations and applications are known today.

Let $\mathfrak{S}_{n}$ denote the set of permutations $\pi=\pi_{1}\pi_{2}\dots\pi_{n}$ of $[n]=\{1,2,\dots,n\}$. For each $\pi\in\mathfrak{S}_{n}$, the excedance set is defined as $\mathrm{Exc}(\pi)=\{i\in[n]\mid\pi_{i}>i\}$. We set $\mathrm{exc}(\pi)=|\mathrm{Exc}(\pi)|$. It is well-known that the Eulerian number $A(n,k)$ counts the number of permutations $\pi\in\mathfrak{S}_{n}$ with $\mathrm{exc}(\pi)=k$. For instance, $A(3,1)=4$ because there are $4$ permutations of $\{1,2,3\}$ with $\mathrm{exc}(\pi)=1$, namely, $132,213,312,321$. A map $f:\mathfrak{S}_{n}\to{\mathbb{Z}}_{\geq 0}$ satisfying $|\{\pi\in\mathfrak{S}_{n}\mid f(\pi)=k\}|=A(n,k)$ is often called an Eulerian statistic. The map $\mathrm{exc}$ is an example of Eulerian statistics. By Foata’s fundamental transformation, it is also known that the number of permutations with $k$ excedances is the same as the number of permutations with $k$ descents, or equivalently formulated, with $k+1$ ascending runs, (see Bóna’s book [5]).

The Eulerian polynomial is defined by

The generating function of the Eulerian polynomials is given as

For a more detailed history and properties on the Eulerian numbers (polynomials) and Eulerian statistics, the articles [5, 9, 19] are good references.

We recall now a generalization of the Eulerian polynomial introduced by Foata–Schützenberger [9, Chapter IV-3]. Here we define a shifted version. Let $\mathrm{cyc}(\pi)$ denote the number of cycles in the disjoint cycle representation of $\pi\in\mathfrak{S}_{n}$.

The generating function of the bivariate Eulerian polynomials is given by

Savage–Viswanathan [22] derived several identities for the polynomials. Here we recall their recursion formula.

Note that by the definition, we have $F_{n}(x,1)=A_{n}(x)$. Moreover, for $y=r\in{\mathbb{Z}}_{\geq 2}$, the polynomials $F_{n}(x,r)$ are the $r$-Eulerian polynomials originally studied by Riordan [20]. In addition, we have $F_{n+1}(x,-1)=-(x-1)^{n}$ and $F_{n+1}(1,y)=y(y+1)\cdots(y+n)$ for any $n\geq 0$.

The surprising fact is however that the values at $y=1/k$, for any positive integer $k$, have also nice combinatorial interpretations. Namely, for a sequence $\mathbf{s}=(s_{i})_{i\geq 1}$ of positive integers, let the $\mathbf{s}$-inversion sequence of length $n$ be defined as

The ascent statistic on $e\in I_{n}^{(\mathbf{s})}$ is

with the convention that $e_{0}/s_{0}=0$. Then the $\mathbf{s}$-Eulerian polynomials are defined by

For more properties of the $\mathbf{s}$-Eulerian polynomials, see also Savage–Visontai [21]. Note that for $\mathbf{s}=(i)_{i\geq 1}=(1,2,3,\dots)$, the $\mathbf{s}$-Eulerian polynomials are the classical Eulerian polynomials, $E_{n}^{(\mathbf{s})}=A_{n}(x)$. Savage–Viswanathan [22] showed that for $\mathbf{s}=((i-1)k+1)_{i\geq 1}=(1,k+1,2k+1,3k+1,\dots)$ where $k$ is a positive integer, it holds

They called the coefficients in this special case the $1/k$-Eulerian numbers. The $1/k$-Eulerian numbers play role in the theory of $k$-lecture hall polytopes [22] and enumerate certain statistics in $k$-Stirling permutations [17]. We now show that for $k=2$, the $E_{n}^{(1,3,5,\dots)}(x)=2^{n}F_{n}(x,1/2)$ is the same as the $q_{n}(x)$ polynomial sequence in Lehmer’s identity.

The generating function $Q(x,t)=\mathscr{F}(x,1/2;2t)$ tells us that the coefficients of $q_{n}(x)$ count perfect matchings with the restriction on the number of matching pairs have odd smaller entries (see [16] and [23, A185411]) and $q_{n}(1)=(2n+1)!!$.

The relation between the polynomials $F_{n}(x,1/2)$ and $q_{n}(x)$ shed light on a proof of Stephan’s conjecture which follows in the next section.

In this section, we focus on the polynomial sequence $p_{n}(x)$ in the expression of Lehmer (1.3). We prove the observation of Stephan who noticed a relation of the sequence with the poly-Bernoulli numbers. Poly-Bernoulli numbers were introduced by Kaneko [12] by the polylogarithm function ($\mathrm{Li}_{k}(z)=\sum_{m=1}^{\infty}z^{m}/m^{k}$ for any integer $k$) as a generalization of the classical Bernoulli numbers. The poly-Bernoulli numbers $B_{n}^{(k)}\in{\mathbb{Q}}$ are defined by

Poly-Bernoulli numbers have attractive properties. In particular, the values with negative indices $k$ enumerate several combinatorial objects, (see for instance, [3, 4, 8, 11] and the references therein).

As one of the most basic properties, Arakawa and Kaneko [1] showed that

holds for any positive integer $n$. Since then, several authors have generalized the formula for the alternating anti-diagonal sum in [14, 18], but not much is known about the anti-diagonal sum in 1.1.

In most of the combinatorial interpretations, the roles of $n$ and $k$ are separately significant, hence it is not natural to consider the anti-diagonal sum. However, one of the interpretations, where this is natural, is the set of permutations with ascending-to-max property [10]. A permutation $\pi\in\mathfrak{S}_{n}$ is called ascending-to-max, if for any integer $i$, $1\leq i\leq n-2$

1. a.

   if $\pi^{-1}(i)<\pi^{-1}(n)$ and $\pi^{-1}(i+1)<\pi^{-1}(n)$ then $\pi^{-1}(i)<\pi^{-1}(i+1)$, and

2. b.

   if $\pi^{-1}(i)>\pi^{-1}(n)$ and $\pi^{-1}(i+1)>\pi^{-1}(n)$, then $\pi^{-1}(i)>\pi^{-1}(i+1)$.

In other words: record a permutation in one-line notation and draw an arrow from value $i$ to $i+1$ for each $i$. Then, the permutation has the ascending-to-max property if all the arrows starting from the left of $n$ point forward and all the arrows starting from an element to the right of $n$ point backward. For instance, $47518362$ has the property, but $41385762$ has not. It follows from the results of Bényi and Hajnal [2] that the number of permutations $\pi\in\mathfrak{S}_{n+1}$ with the ascending-to-max property is given by the anti-diagonal sum $b_{n}=\sum_{k=0}^{n}B_{n-k}^{(-k)}$. However, no explicit formula or recursion was known about the sequence $b_{n}$.

Our first result is a recursion for the sequence $b_{n}$.

In order to prove this theorem, we need some preparations. Recall that by [1, p.163], we have

where $(x)_{j}=x(x+1)(x+2)\cdots(x+j-1)$ is the Pochhammer symbol and we put

By a direct calculation, we have

Thus, the desired recursion in (3.1) is equivalent to

To prove (3.3), we derive a useful equation.

To relate the sequence $(b_{n})_{n\geq 0}$ to the polynomial sequence $(p_{n}(x))_{n\geq-1}$, we next derive the generating function for $p_{n}(x)$.

We define the sequence $a_{n}$ as special values of $p_{n}(x)$,

Using the generating function of $p_{n}(x)$, we obtain the recurrence formula that the sequence $(a_{n})_{n\geq 0}$ satisfies.

In conclusion, we have the main theorem.

In the course of our proof, we obtain two types of generating functions in (3.2) and (3.5) for the sequences $(a_{n})_{n\geq 0}=(b_{n})_{n\geq 0}$. As a corollary, we have an explicit formula for the anti-diagonal sum, (see [3, p.24]).

As a final remark, we show that the polynomial $p_{n}(x)$ can also be expressed in terms of bivariate Eulerian polynomials.