1

Explicit formula for generalization of Poly-Bernoulli numbers

and polynomials with parameters

*HASSAN JOLANY

*School of Mathematics, Statistics and Computer Science,

University of Tehran, Iran.

*Centre de Mathématiques et Informatique UFR-MIM UNIVERSITÉ DE PROVENCE

(Aix-Marseille I), France

hassan.jolany@khayam.ut.ac.ir

ABSTRACT. In this paper we investigate special generalized Bernoulli polynomials

with parameters that generalize classical Bernoulli numbers and polynomials.

The present paper deals with some recurrence formulae for the generalization of

poly-Bernoulli numbers and polynomials with parameters. Poly-Bernoulli

numbers satisfy certain recurrence relationships which are used in many

computations involving poly-Bernoulli numbers. Obtaining a closed formula for

generalization of poly-Bernoulli numbers with paramerers therefore seems to

be a natural and important problem. By using the generalization of poly-Bernoulli

polynomials with parameters of negative index we define symmetrized

generalization of poly-Bernoulli polynomials with parameters of two variables

and we prove duality property for them. Also by stirling numbers of the second kind

we will find a closed formula for them. Furthermore we generalize the Arakawa-

Kaneko Zeta functions and by using the Laplace-Mellin integral, we define

generalization of Arakawa-Kaneko Zeta functions with parameters and we

obtain an interpolation formula for the generalization of poly-Bernoulli numbers

and polynomials with parameters. Furthermore we present a link between this

type of Zeta functions and Dirichlet series. By our interpolation formula, we will

interpolate the generalization of Arakawa-Kaneko Zeta functions with

parameters.

Mathematical Subject Classification (2000) : 11B73, 11A07

Keywords: Bernoulli numbers and polynomials, Arakawa–Kaneko Zeta functions, Poly-Bernoulli numbers and polynomials, generalization of Poly-Bernoulli numbers and polynomials with paprameters, generalization of Arakawa–Kaneko Zeta functions with parameters.

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1. Introduction, Preliminaries and motivation The poly–Bernoulli polynomials have been studied by many researchers in recent decade. The history of these polynomials goes back to Kaneko [1]. The poly-Bernoulli polynomials have wide-ranging application from number theory and combinatorics and other fields of applied mathematics. One of applications of poly-Bernoulli numbers that was investigated by Chad Brewbaker in [9, 14], is about the number of (0,1)-matrices with n-rows and k columns. He showed the number of (0,1)-matrices with n-rows and k columns uniquely reconstructable from their row and column sums are the poly-Bernoulli numbers of negative

index . Another application of poly-Bernoulli numbers is in Zeta function theory.

Multiple Zeta functions at non-positive integers can be described in terms of these numbers. A third application of poly-Bernoulli numbers that was proposed by Stephane Launois in [23, 24], is about cardinality of some subsets of . He proved the cardinality of sub-poset of the reverse Bruhat ordering of is equal to the poly-Bernoulli numbers. Also one of other applications of poly-Bernoulli numbers is about skew Ferrers boards. In [22], Jonas Sjöstran found a relation between poly-Bernoulli numbers and the number of elements in a Bruhat interval. Also he showed the Poincaré polynomial (for value ) of some particularly interesting intervals in the finite Weyl group can be written in terms of poly-Bernoulli numbers. One of generalizations of poly-Bernoulli numbers that was first proposed by Y.Hamahata in [12], is Multi-poly-Bernoulli numbers and he derived a closed formula for them. Abdelmejid Bayad in [16], introduced a new generalization of poly-Bernoulli numbers and polynomials. He by using Dirichlet character, defined generalized poly-Bernoulli numbers associated to and also he introduced generalized Arakawa-Kaneko L-functions and showed that the non-positive integer values of the complex variable of these L-functions can be written rationally in terms of generalized poly-Bernoulli polynomials associated to . In [21], Hassan Jolany et al, by using real parameters introduced generalization of poly-Bernoulli polynomials with parameters and found a closed relationships between generalized poly-Bernoulli polynomials with parameters and generalized Euler polynomials with parameters.

Let us briefly recall poly-Bernoulli numbers and polynomials. For an integer , put

which is the -th polylogarithm if , and a rational function if . The name of the function come from the fact that it may alternatively be defined as the repeated integral of itself namely that

One knows that Also if k is a negative integer, say , then the polylogarithmic function converges for and equals

(1)

(2)

(3)

3

where the , are the Eulerian numbers. The Eulerian numbers

is the number of permut-

ations of , with permutation ascents. One has

The formal power series can be used to define poly-Bernoulli numbers and

polynomials. The polynomials are said to be poly-Bernoulli

polynomials if they satisfy

where . By (2), the left-hand side of (5) can be written in the form of iterated integrals

For any , we have

where are the classical Bernoulli polynomials given by

For in (5), we have

, where are called poly-Bernoulli numbers

(for more information see [2, 5, 8, 11, 13, 15, 18, 19, 20, 25]). The list of poly-Bernoulli

numbers , with and are given by Arakawa and Kaneko [3, 16].

In 2002, Q. M. Luo et al. [26], defined the generalization of Bernoulli numbers and poly-nomials with parameters as follows

So, by (7) we get and

that are called generalization of Bernoulli numbers with parameters. Also they proved following expression for this type of polynomials which interpolate generalization of Bernoulli polynomials with parameters.

H. Jolany et al. in [21], defined a new generalization for poly-Bernoulli numbers and poly-nomials. They introduced generalization of poly-Bernoulli polynomials with parameters as follows

Also they extended the definition of generalized poly-Bernoulli polynomials for three parameters as follows

(4)

(5)

(6)

(7)

(8)

(9)

4

where are called, generalization of poly-Bernoulli polynomials with

parameters. These are coefficients of power series expansion of a higher genus algebraic function with respect to suitable variable. In the sequel, we list some closed formulas of poly-Bernoulli numbers and polynomials

Kaneko in [1, 2, 4], presented following explicit formulas for poly-Bernoulli numbers

where

Called the second type stirling numbers. A. Bayad et al. in [16, 17], generalized poly-

Bernoulli poly-nomials. They defined generalized poly-Bernoulli polynomials

associated to . So by applying their method, we introduce a closed formula and also interpolation formula for generalization of poly-Bernoulli numbers and polynomials with parameters which yeilds a deeper insight into the effectiveness of this type of generalizations.

2. Explicit formulas for generalization of poly-Bernoulli polynomials with

parameters

Now, we are in position to state and prove the main results of this paper. In this section, we

obtain some interesting new relations associated to generalization of poly-Bernoulli numbers

and polynomials with parameters. Here we prove a collection of important and funda-

mental identities involving this type of number and polynomials. We also deduce their special

cases which led to the corresponding results for the poly-Bernoulli polynomials.

First of all, we present an explicit formula for generalization of poly-Bernoulli polynomials

with parameters

Theorem 2.1. (Explicit formula) For , , we have

Proof. By applying the expression (8), we get

(10)

5

So, we get

By comparing the coefficients of on both sides, prove is complete.

As a direct result, by applying the same method of Theorem 2.1, we derive following corolla-

ries.

Corollary 2.1. For , , we have

As a direct result, by applying in Corollary 2.1, we get Corollary 2.2

Corollary 2.2. For , , we have

Furthermore, by setting in Corollary 2.2 and because we have,

, we obtain following Explicit formulas for classical Bernoulli numbers and

polynomials

Corollary 2.3. For , we have

Now, we investigate some recursive formulas for the generalization of poly-Bernoulli num-

bers and polynomials with parameters.

Theorem 2.2. (recurrence formula I) For all , , we have

Proof. We know

so

(11)

(12)

(13)

(14)

(15)

6

So, we get

therefore, we obtain

So, by applying the following identity, we obtain

Put

Put

By comparing the coefficients the coefficients of

on both sides, proof will be complete.

As a direct result, by setting , in Theorem 2.2 we obtain following corollary

which is the well known recurrence formula for classical poly-Bernoulli polynomials and is

the one of the main results of [17].

Corollary 2.4. For , , we have

(16)

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Let us consider the extreme recurrence formula for generalization of poly-Bernoulli poly-

nomials with parameters. By using following lemma and some standard techniques

based upon generating function and series rearrangement we present a new recurrence

formula for generalization of poly-Bernoulli polynomials with parameters.

Lemma 2.1. For and , we have

Proof. By applying (8), we have

So, by comparing the coefficients of on both sides, we obtain the desired result.

Now we are ready to present our second recurrence formula for generalization of Poly-

Bernoulli numbers and polynomials with parameters.

Theorem 2.3. For and , we have

Proof. From [16], we have following recurrence formula for poly-Bernoulli numbers

(17)

(18)

(19)

(20)

8

So, by using Lemma 2.1 and replacing by

in (21), we obtain the desired result.

Now, we show that the generalization of poly-Bernoulli polynomials of parameters are in

the set of Appell polynomials.

For a sequence of Appell polynomials, which is a sequence of polynomials

satisfying,

Tremendous properties are well known. Among them, the most important classifications of

Appell polynomials may be the following equivalent conditions ([27, 28, 29])

Theorem 2.4. Let be a sequence of polynomials. Then the following are all

equivalent

is a sequence of Appell polynomials.

has a generating function of the form

Where is a formal power series independent of with

(c) satisfies

Now, in following theorem we prove generalization of poly-Bernoulli polynomials are in the

set of Appell sequence

Theorem 2.5. (Appell sequence) we have

Proof.By differentiating both sides of (8) with respect to gives

(21)

(22)

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from this, we obtain

which yields the results.

Thus, by applying the property of (c) of Theorem 2.4, we obtain following corollary.

Corollary 2.5. (Addition formula) For and , we have

In particular

and by taking , we obtain Multiplication theorem for them

Actually, because generalization of poly-Bernoulli polynomials of parameters are in

the set of Appell polynomials, we can derive numerous properties for them. For instance in

[30], F. A. Costabile and E. Longo presented a new definition by means of a determinantal

form for Appell polynomials by using of linear algebra tools and also E. H. Ismail in [31],

found a differential equation for Appell polynomials.

3. Symmetrized generalization of poly-Bernoulli polynomials with a, b parameters of

two variables

Kaneko et al. in [1, 2, 3, 6], defined symmetrized poly-Bernoulli polynomials with two

variable and by using their method we introduce Symmetrized generalization of poly-

Bernoulli polynomials with parameters of two variables and construct a generating

function for Symmetrized generalization of poly-Bernoulli polynomials with parameters

of two variables. Also we give a closed formula and duality property for this type of

polynomials as well.

Definition 3.1. For , we define

(23)

(24)

(25)

(26)

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Now, in following theorem we introduce a generating function for .

Theorem 3.1. For , we have

Proof. By using the definition of , the left hand side can be rewritten as

By putting , we get

But Kaneko in [1], proved following expression

So, by applying this expression, we obtain the desired result.

As a direct result, we have the following corollary for that is well known to

duality property

Corollary 3.1. (Duality property) For , we have

(27)

(28)

11

Now, we are ready to show a closed formula for which is important and

fundamental.

Theorem 3.2(Closed formula). For , we have

Proof. By applying Theorem 3.1 we have

By applying the generating function of stirling numbers of second kind

the right hand side of the last expression becomes,

which yields the result.

4. Generalization of Arakawa-Kaneko L-functions with parameters

(29)

(30)

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It is well known since the second-half of the 19-th century the Riemann Zeta function may

be represented by the normalized Mellin transformation

After T. Arakawa and M. Kaneko by inspiration of last expression, introduced Arakawa-

Kaneko Zeta function as follows. For any integer ;

It is defined for and if , and for and if .

The function has analytic continuation to an entire function on the whole complex -

plane and

for all non-negative integer and (for more information see[10] )

For , the generalization of Arakawa-Kaneko Zeta function with parameters are

given by

The Laplace-Mellin integral. It is defined for and if , and for

and if . It is easy to see that the generalization of Arakawa-Kaneko zeta

function with a, b parameters include the Arakawa-Kaeko Zeta function and Hurwitz Zeta

function .

Now in this section we derive an interpolation formula of generalization of poly-Bernoulli

polynomials with parameters and investigate fundamental properties of . At

first, in following Lemma we give a relation between generalization of Arakawa-Kaneko Zeta

function with parameters and classical Arakawa-Kaneko Zeta function.

Lemma 4.1. For , we have

Proof. It is easy to see that

(31)

(32)

(33)

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so, by changing by , we obtain

this yields the proof of the lemma.

Theorem4.1(Interpolationformula).The function has analytic continuat-

ion to an entire function on the whole complex -plane and for any positive integer , we

have

Proof. By using Lemma 4.1, to prove that has analytic continuation to an

entire function on the whole complex -plane, it is sufficient to show that has

such property. Since this face comes from the first part of the Theorem 1.10 in [10], we omit

it. By using Lemma 2.1 and expression (31), we get

so we obtain the desired result and proof is complete.

As an immediate consequence of previous theorems in this section, we obtain an explicit

formula for .

Corollary 4.1. For , we have

Proof. By applying Theorem 4.1, we can invert Hence

(34)

(35)

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So, proof is complete.

Raabe’s formula is a fundamental and universal property in the theory of Zeta function and

plays important role in special functions. Raabe’s formula is satisfies for several type of Zeta

function. For instance, Hurwitz Zeta functions, Euler Zeta function, q-Euler Zeta function,

multiple Zeta function. This formula provides a powerful link between zeta integrals and

Dirichlet series. Raabe’s formula can be obtained from the Hurwitz zeta function

via the integral formula

Now, in next theorem, we will present a interesting link between integral of generalization

of Arakawa-Kaneko zeta function with a, b parameters and Dirichlet series. In fact we prove

the Raabe’s formula for our new types of zeta function .

Lemma 4.2 (Difference formula) we have

Proof. By applying the definition of generalization of Arakawa-Kaneko zeta function with a,

b parameters, we get

(36)

15

So, we obtain the desired result.

Now, we are ready to present the Raabe’s formula for our new types of zeta function.

Theorem 4.2(Raabe’s formula). we have

Proof. By using Lemma 4.2, we get

So, we obtain the desired result and proof is complete.

As a direct result of Raabe’s formula and interpolation formula, we obtain following

corollary for generalization of poly-Bernoulli numbers with parameters

Corollary 4.2. Raabe’s formula in terms of generalization of poly-Bernoulli polynomials with

a, b parameters is as follows,

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Acknowledgment

The author would like to express his gratitude to Professor M.R.Darafsheh for his

mathematical supports, careful reading of this paper and warm encouragements.

The author also thanks Frédéric Tissot for his help and warm encouragements.

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