On bi-variate poly-Bernoulli polynomials

Bernoulli numbers are one of the most important mathematical objects that have been studied by mathemathicians since they appeared in the 18-th century (see [Ma]). A recent important generalization of the Bernoulli numbers $B_{n}$ and Bernoulli polynomials $B_{n}(x)$ is about the so-called poly-Bernoulli numbers and poly-Bernoulli polynomials. Poly-Bernoulli numbers $B_{n}^{\left(k\right)}$, where $k$ is a given positive integer, were introduced by M. Kaneko [Ka] in 1997, by means of the generating function

where Li${}_{k}(z)=\sum_{j=1}^{\infty}z^{j}/j^{k}$ is the polylogarithm function. The case $k=1$ corresponds to the standard Bernoulli numbers $B_{n}$ (except the sign of $B_{1}^{\left(1\right)}$). Poly-Bernoulli polynomials $B_{n}^{\left(k\right)}(x)$ can be defined by the generating function

(see [Ce-Ko]). The case $k=1$ corresponds to $(-1)^{n}B_{n}(x)$, and the case $x=0$ corresponds to the poly-Bernoulli numbers $B_{n}^{\left(k\right)}$ mentioned before. Some slightly different definitions of poly-Bernoulli polynomials $B_{n}^{\left(k\right)}(x)$, with $x$ replaced by $-x$, and/or with an additional factor $(-1)^{n}$, can be found in some related papers (see [Ba-Ha], [Ce-Ko], [Co-Ca], [Ha]). In this work we use the following explicit formula

as our definition of poly-Bernoulli polynomials (see formula (1.8) in [Ba-Ha]). It is important to mention that the notation $B_{n}^{\left(k\right)}(x)$ is also used for a different kind of mathematical objects, namely, Bernoulli polynomials of $k$-th order (see [Ca1]). A different generalization of Bernoulli polynomials, studied in the past few years, is about considering Bernoulli polynomials in several variables $B_{p_{1},\ldots,p_{t}}(x_{1},\ldots,x_{t})$, that is, polynomials of degree $p_{i}$ in the variable $x_{i}$, with

seeking that reasonable generalizations of the known properties in the one-variable case, remain valid. This kind of work is done in [Shib], with a flavor of multivariable analysis and working with Jack polynomials. A different approach is presented in [Shishk] (see also [Bre], [DiNar]). In this work we study poly-Bernoulli polynomials in two variables (bi-variate poly-Bernoulli polynomials). We define the bi-variate poly-Bernoulli polynomials by using a generalization of Stirling numbers of the second kind we studied in [Pi], and then we use the results in [Pi] to obtain results for the bi-variate poly-Bernoulli polynomials considered in this work. We present now the definitions and results in [Pi] that we will use in the remaining sections. The generalized Stirling numbers of the second kind (GSN, for short), denoted as $S_{a_{1},b_{1}}^{\left(a_{2},b_{2},p_{2}\right)}\left(p_{1},k\right)$, where $a_{j},b_{j}\in\mathbb{C}$, $a_{j}\neq 0$, $j=1,2$, and $p_{1},p_{2}$ non-negative integers, are defined by means of the expansion

($S_{a_{1},b_{1}}^{a_{2},b_{2},p_{2}}(p_{1},k)=0$ if $k<0$ or $k>p_{1}+p_{2}$ ). An explicit formula for these numbers is

If $p_{2}=0$, we write the GSN $S_{a,b}^{a_{2},b_{2},0}(p,k)$ as $S_{a,b}(p,k)$. We have

In the case $a=1,b=0$, the corresponding GSN $S_{1,0}(p,k)=\frac{1}{k!}\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}\left(k-j\right)^{p}$ are the known Stirling numbers of the second kind. We will refer to them as “standard Stirling numbers”, and in this case we use the known notation $S(p,k)$. From (3) it is clear that $S_{a,b}^{a,b,p_{2}}\left(p_{1},k\right)=S_{a,b}\left(p_{1}+p_{2},k\right)$. We can see directly from (4) that

In this work we will use GSN of the form $S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k)$. Some important facts about the GSN $S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k)$ are the following:

• •

  Some values of the GSN $S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k)$ are

  	$\displaystyle S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},0)$	$\displaystyle=$	$\displaystyle\!\!x_{1}^{p_{1}}\!x_{2}^{p_{2}}\!,$		(7)
  	$\displaystyle S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},1)$	$\displaystyle=$	$\displaystyle(x_{1}+1)^{p_{1}}\!(x_{2}+1)^{p_{2}}\!-x_{1}^{p_{1}}\!x_{2}^{p_{2}}\!,$
  		$\displaystyle\vdots$
  	$\displaystyle S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},p_{1}+p_{2})$	$\displaystyle=$	$\displaystyle 1.$

• •

  The GSN $S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k)$ can be written in terms of the GSN $S_{1,y_{1}}^{1,y_{2},p_{2}}(p_{1},k)$ as follows

  $$S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k)=\sum_{j_{1}=0}^{p_{1}}\sum_{j_{2}=0}^{p_{2}}\binom{p_{1}}{j_{1}}\binom{p_{2}}{j_{2}}(x_{1}-y_{1})^{p_{1}-j_{1}}(x_{2}-y_{2})^{p_{2}-j_{2}}S_{1,y_{1}}^{1,y_{2},j_{2}}(j_{1},k).$$

  (8)

• •

  The GSN $S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k)$ can be written in terms of standard Stirling numbers as follows

  	$\displaystyle k!S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k)$	$\displaystyle=$	$\displaystyle\sum_{j_{1}=0}^{p_{1}}\sum_{j_{2}=0}^{p_{2}}\binom{p_{1}}{j_{1}}\binom{p_{2}}{j_{2}}(x_{1}-m)^{p_{1}-j_{1}}(x_{2}-m)^{p_{2}-j_{2}}$		(9)
  			$\displaystyle\times\sum_{i=0}^{m}\binom{m}{i}(k+i)!S(j_{1}+j_{2},k+i),$

  where $m$ is an arbitrary non-negative integer.

• •

  The GSN $S_{1,x_{1}}^{1,x_{2},p_{2}}\left(p_{1},k\right)$ can be written in terms of standard Stirling numbers as follows

  	$\displaystyle S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k)$	$\displaystyle=$	$\displaystyle\sum_{j_{1}=0}^{p_{1}}\sum_{j_{2}=0}^{p_{2}}\binom{p_{1}}{j_{1}}\binom{p_{2}}{j_{2}}(x_{1}-n)^{p_{1}-j_{1}}(x_{2}-n)^{p_{2}-j_{2}}$		(10)
  			$\displaystyle\times\sum_{i=0}^{n-1}(-1)^{i}s(n,n-i)S(j_{1}+j_{2}+n-i,k+n),$

  where $n$ is an arbitrary positive integer, and $s(\cdot,\cdot)$ are the unsigned Stirling numbers of the first kind.

• •

  The GSN $S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k)$ satisfy the identity

  $$S_{1,x_{1}+1}^{1,x_{2}+1,p_{2}}(p_{1},k)=S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k)+(k+1)S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k+1).$$

  (11)

• •

  The GSN $S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k)$ satisfy the recurrence

  $$S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1},k)=S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1}-1,k-1)+(k+x_{1})S_{1,x_{1}}^{1,x_{2},p_{2}}(p_{1}-1,k).$$

  (12)

The relation of Bernoulli (numbers and polynomials) with Stirling (numbers of the second kind) is an old story, that dates back to Worpitsky [W] (see also [G-K-P, p. 560] and [Ke, p. 5]). We have the following formula for Bernoulli numbers

and in the case of Bernoulli polynomials we have

An important observation of formula (1) is that poly-Bernoulli polynomial $B_{p}^{\left(k\right)}(x)$ can be written in terms of the GSN as

The generalization of (15) to the case of two variables comes through the GSN: we define poly-Bernoulli polynomial in the variables $x_{1},x_{2}$, denoted by $B_{p_{1},p_{2}}^{\left(k\right)}(x_{1},x_{2})$, as

If $p_{2}=0$, formula (16) becomes (15). By using (9) we can write $B_{p_{1},p_{2}}^{\left(k\right)}(x_{1},x_{2})$ in terms of standard Stirling numbers as

where $m$ is an arbitrary non-negative integer. Similarly, by using (10) we have that

where $n$ is an arbitrary positive integer. The simplest cases of (17) and (18) are

and

respectively. Two examples are the following

Clearly we have

Observe also that

In particular, we have

From (3) and (16) we have

from where we see that

We can use (8) to write $B_{p_{1},p_{2}}^{\left(k\right)}(x_{1},x_{2})$ in terms of $B_{j_{1},j_{2}}^{\left(k\right)}\left(y_{1},y_{2}\right)$, $0\leq j_{i}\leq p_{i}$, $i=1,2$, as

that generalizes the known addition formula $B_{p}^{\left(k\right)}(x)=\sum_{j=0}^{p}\binom{p}{j}\left(x-y\right)^{p-j}B_{j}^{\left(k\right)}\left(y\right)$ for one-variable poly-Bernoulli polynomials. In fact, we have

as claimed. In particular, if we set $y_{1}=y_{2}=y$ in (25) we obtain an expression for the bi-variate poly-Bernoulli polynomial $B_{p_{1},p_{2}}^{\left(k\right)}(x_{1},x_{2})$ in terms of one-variable poly-Bernoulli polynomials $B_{j}^{\left(k\right)}\left(y\right)$, $0\leq j\leq p_{1}+p_{2}$, namely

and then we can write the bi-variate poly-Bernoulli polynomial $B_{p_{1},p_{2}}^{\left(k\right)}(x_{1},x_{2})$ in terms of poly-Bernoulli numbers $B_{j}^{\left(k\right)}$, $0\leq j\leq p_{1}+p_{2}$, as

The cases $k=0$ and $k=-1$ of (27) are

and

respectively. In fact, according to (16), formulas (28) and (29) are the particular cases $r=0$ and $r=1$ of the identity

where $r$ is a non-negative integer. We leave the proof of (30) to the reader. Observe also that (25) implies that

which generalizes the known binomial formula for standard Bernoulli polynomials

If we set $x_{1}=x_{2}=x$ in (25), we obtain a formula for the standard poly-Bernoulli polynomial $B_{p_{1}+p_{2}}^{\left(k\right)}(x)$ in terms of the bi-variate poly-Bernoulli polynomials $B_{j_{1},j_{2}}^{\left(k\right)}(y_{1},y_{2})$, $0\leq j_{i}\leq p_{i}$, $i=1,2$, namely

Some additional simple observations are the following

and

In this section we obtain some identities involving poly-Bernoulli polynomials, by using the following result:

For example, by using Theorem 3.1 in the trivial identity $x^{p}=\sum_{j=0}^{p}\binom{p}{j}(x-y)^{j}y^{p-j}$ we obtain the addition formula for poly-Bernoulli polynomials

that we can write as

We can use again Theorem 3.1 to obtain from (41) that