Abstract:

The Arakawa-Kaneko zeta function has been introduced ten years ago
by T. Arakawa and M. Kaneko in [22]. In [22], Arakawa and Kaneko have
expressed the special values of this function at negative integers with
the help of generalized Bernoulli numbers [B.sup.(k)] called
poly-Bernoulli numbers. Kim-Kim [4] introduced Multi poly- Bernoulli
numbers and proved that special values of certain zeta functions at
non-positive integers can be described in terms of these numbers. The
study of Multi poly-Bernoulli and Euler numbers and their combinatorial
relations has received much attention [2,4,6,7,12,13,14,19,22,27]. In
this paper we introduce the generalization of Multi poly-Bernoulli and
Euler numbers and consider some combinatorial relationships of the
Generalized Multi poly- Bernoulli and Euler numbers of higher order. The
present paper deals with Generalization of Multi poly-Bernouli numbers
and polynomials of higher order. In 2002, Q. M. Luo and et al (see [11,
23, 24]) defined the generalization of Bernoulli polynomials and Euler
numbers. Some earlier results of Luo in terms of generalized Multi
poly-Bernoulli and Euler numbers, can be deduced. Also we investigate
some relationships between Multi poly-Bernoulli and Euler polynomials.

Key Words: Generalized Multi poly-Bernoulli polynomials, generalized Multi poly- Euler polynomials, stirling numbers, polylogarithm, Multi-polylogarithm.

AMS(20 10): 05A10, 05A19

Key Words: Generalized Multi poly-Bernoulli polynomials, generalized Multi poly- Euler polynomials, stirling numbers, polylogarithm, Multi-polylogarithm.

AMS(20 10): 05A10, 05A19

Article Type:

Report

Subject:

Functions, Zeta
(Research)

Polynomials (Research)

Polynomials (Research)

Authors:

Jolany, Hassan

Mohebbi, Hossein

Alikelaye, R. Eizadi

Mohebbi, Hossein

Alikelaye, R. Eizadi

Pub Date:

07/01/2011

Publication:

Name: International Journal of Mathematical Combinatorics Publisher: American Research Press Audience: Academic Format: Magazine/Journal Subject: Mathematics Copyright: COPYRIGHT 2011 American Research Press ISSN: 1937-1055

Issue:

Date: July, 2011 Source Volume: 2

Topic:

Event Code: 310 Science & research

Geographic:

Geographic Scope: Iran Geographic Code: 7IRAN Iran

Accession Number:

268601861

Full Text:

[section] 1. Introduction

Bernoulli numbers are the signs of a very strong bond between elementary number theory, complex analytic number theory, homotopy theory(the J-homomorphism, and stable homotopy groups of spheres), differential topology(differential structures on spheres), the theory of modular forms(Eisenstein series) and p-adic analytic number theory(the p-adic L- function) of

Mathematics. For n [member of] Z, n [less than or equal to] 0, Bernulli numbers [B.sub.n] originally arise in the study of finite sums of a given power of consecutive integers. They are given by [B.sub.0] = 1, [B.sub.1] = -1/2, [B.sub.2] = 1/6, [B.sub.3] = 0, [B.sub.4] = -1/30,with [B.sub.2n+1] = 0 for n > 1, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The modern definition of Bernoulli numbers Bn can be defined by the contour integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where the contour encloses the origin, has radius less than 2[pi].

Also Bernoulli polynomials [B.sub.n](x) are usualy defined(see[1], [4], [5])by the generating function

G(x,t) = [te.sup.xt] / [e.sup.t]-1 = [[infinity].summation over (n=0)] [B.sub.n](x) [t.sup.n] / n!, [absolute value of t] < 2[pi] (3)

and consequently, Bernoulli numbers [B.sub.n](0) := [B.sub.n] can be obtained by the generating function

t / [e.sup.t]-1 = [[infinity].summation over (n=0)] [B.sub.n] [t.sup.n] / n!

Bernoulli polynomials, first studied by Euler (see[1]), are employed in the integral representation of differentiable periodic functions, and play an important role in the approximation of such functions by means of polynomials (see[14]-[18]).

Euler polynomials [E.sub.n](x) are defined by the generating function

2[e.sup.xt] / [e.sup.t]-1 = [[infinity].summation over (n=0)] [E.sub.n](x) [t.sup.n] / n!, [absolute value of t] < [pi] (4)

Euler numbers [E.sub.n] can be obtained by the generating function

2 / [e.sup.t]+1 = [[infinity].summation over (n=0)] [E.sub.n](x) [t.sup.n] / n! (5)

The first four such polynomials, are

[B.sub.0](x) = 1, [B.sub.1](x) = x - 1/2, [B.sub.2](x) = [x.sup.2] - x +1/6 [B.sub.3](x) = [x.sup.3] - 3/2[x.sup.2] + 1/2x,...

and

[E.sub.o](x) = 1, [E.sub.1](x) = x - 1/2, [E.sub.2](x) = [x.sup.2] - x, [E.sub.3](x) = [x.sup.3] - 3/2[x.sup.2] + 1/4,...

Euler polynomials are strictly connected with Bernoulli ones, and are used in the Taylor expansion in a neighborhood of the origin of trigonometric and hyperbolic secant functions.

In the sequel, we list some properties of Bernoulli and Euler numbers and polynomials as well as recurrence relations and identities.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[B.sub.n](x +1) - [B.sub.n](x)= [nx.sub.n-1], (8)

[E.sub.n](x +1) + [E.sub.n](x) = [2.sub.xn.]. (9)

Lemma 1.1(see[20],[21]) For any integer n [greater than or equal to] 0, we have

MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Consequently, from (8), (9) and lemma 1.1, we obtain,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Lemma 1.3 For any positive integer n [greater than or equal to] 0, we have

[B.sub.n](px) = [p.sup.n-1] / [p-1.summation over (r=0)] [B.sub.n](x+ r/p) (p is a positive integer) (14)

[E.sub.n](px) = [p.sup.n] / [p-1.summation over (r=0)] [(-1).sup.r][E.sub.n](x+ r/p) (p is a positive integer) (15)

Let us briefly recall k - th polylogarithm. The polylogarithm is a special function [Li.sub.k](z), that is defined by the sum

[Li.sub.k](z):= [[infinity].summation over (s=1)] [z.sup.s] / [s.sup.k] (16)

For formal power series [Li.sub.k](z) is the k - th polylogarithm if k [greater than or equal to] 1, and a rational function if k [less than or equal to] 0. The name of the function come from the fact that it may alternatively be defined as the repeated integral of itself, namely that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (l7)

for integer values of k, we have the following explicit expressions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The integral of the Bose-Einstein distribution is expressed in terms of a polylogarithm,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Lemma 1.3(see[18]) For n [member of] N [union] {0}, we have an explicit formula for [Li.sub.-n](z) as follow

[Li.sub.-n](z)=[n+1.summation over (k=1)] [(-1).sub.n+k+1](k-1)!S(n+1,k) / [(1- z).sub.k] (19)

where s(n, k) are Stirling numbers of the second kind.

Now, we introduce the generalization of [Li.sub.k](z). Let r be an integer with a value greater than one.

Definition 1.1 Let [k.sub.1],[k.sub.2],...[k.sub.r] be integers. The generalization of polylogarithm are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

The rational numbers [B.sup.(k).sub.n], (n = 0,1, 2,...) are said to be poly- Bernoulli numbers if they satisfy

[li.sub.k](1-[e.sup.-x]) / 1-[e.sup.-x] = [[infinity].summation over (n=0] [B.sup.(k).sub.n] [x.sup.n]/n! (21)

In addition, for any n [greater than or equal to] 0, [B.sup.(1).sub.n] is the classical Bernoulli number, [B.sub.n] (see[7], [12]). Also , the rational numbers [H.sup.(k).sub.n,] (n = 0,1, 2,...)are said to be poly-Euler numbers if they satisfy

[Li.sub.k] (1-[e.sup.(1-u)]) / u-[e.sup.t] = [[infinity].summation over (n=0)] [H.sup.(k).sub.n](u) [t.sup.n]/n! (22)

where u is an algebraic real number and k > 1.(see[13],[19])

Let us now introduce a generalization of poly-Bernoulli numbers, making use of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 1.1(see[7]) Multi poly-Bernoulli numbers [B.sup.(k0,...,kr).sub.n,] (n = 0, 1, 2, ...) are defined for each integer ki, k2,kr by the generating series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

By Definition 1.2, the left hand side of (23) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

hence we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Definition 1.3 Multi poly-Euler numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , (n = 0,1, ...) are defined for each integer [k.sub.1],[k.sub.r] by the generating series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Kaneko [6] presented the following recurrence formulae for poly-Bernoulli numbers which we state hear.

Theorem 1.1(Kaneko)([2,6,14,22]) For any k [member of] Z and n [greater than or equal to] 0,we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

called the second type stirling numbers.

Y. Hamahata and H. Masubuchi in [12], presented the following recurrence formulae for Multi poly-Bernoulli numbers.

Theorem 1.2 (H. Masubuchi & Y. Hamahata) For n [greater than or equal to] 0 and ([k.sub.1],...,[k.sub.r] [member of] Z) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

If [k.sub.r] [not eqaul to] 1and n [greater than or equal to] 1, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

If [k.sub.r] = 1 and n [greater than or equal to] 1 ,then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

Also, they proved (see[1]) if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

then for n, k [greater than or equal to] 0, we have

B[[r].sup.(-k).sub.n] = B[[r].sup.(-n).sub.k] (38)

In [23], [24], Q.M.Luo, F.Oi and L.Debnath defined the generalization of Bernoulli and Euler polynomials [B.sub.n](x, a, b, c) and En(x, a, b, c) respectively, which are expressed as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

In this paper, by the method of Q.M.Luo and et al [11], we give some properties on generalized Multi poly-Bernoulli and Euler polynomials

Definition 1.4 Let a, b > 0 and a [not equal to] b. The generalized Multi poly- Bernoulli numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (a,b) , the generalized Multi poly-Bernoulli polynomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are defined by the following generating functions, respectively;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (42)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

Definition 1.5 Let a, b > 0, and a [not equal to] b, the generalized Multipoly-Euler numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (u; a, b),the generalized multi poly-Euler polynomial [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are defined by the following generating functions, respectively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

[section]2. Main Theorems

In this section, we introduce our main results. We give some theorems and corollaries which are related to generalized Multi poly-Bernoulli numbers and generalized Multi poly-Euler polynomials. We present some recurrence formulae for generalized Multi-poly-Bernoulli and Euler polynomials.

Theorem 2.1 Let a, b > 0 and a [not equal to] b, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47)

proof By applying Definition 1.4, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, by comparing the coefficients of [t.sup.n]/n! on both sides, proof will be complete

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The generalized Multi poly-Bernoulli and Euler numbers process a number of interesting properties which we state here

Theorem 2.2 Let a, b > 0 and a = b. For real algebraic u we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (48)

Next, we investigate a strong relationships between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 2.3 Let a, b > 0,a [not equal to] band a > b > 0, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

By applying Definition 1.4, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By comparing the coefficient of [t.sup.n]/n! on both sides, we get.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 2.4 Let a, b > 0, and b > a > 0. For algebraic real number u, we have

By the same method proceeded in the proof of Theorem 2.3, we obtained similar relations for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 2.4 Let a, b > 0, and b > a > 0. For algebraic real number u, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

Theorem 2.5 Let x [member of] R and conditions of Theorem 2.3 holds true, then we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (51)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52)

Proof By applying Definitions 1.4 and 1.5, proof will be complete. []

Theorem 2.6 Let conditions of Theorem 2.5 holds true, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (53)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (54)

Proof By applying Theorems 2.1 and 2.5, we get (53), and Obviously, the result of (54) is similar with (53). []

Theorem 2.7 Let conditions of Theorem 2.5 holds true, then we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (55)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (56)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (58)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (59)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (60)

Proof We only prove (59) and (55)-(60) can be derived by Definitions 1.4 and 1.5.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So by comparing the coefficients of [t.sup.n] / n! in the two expressions, we obtain the desired result 2.13. [??]

Theorem 2.8 By the same method proceeded in the proof of previous Theorems, we find similar relations for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (61)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (62)

Now, we present formulae which show a deeper motivation of generalized poly-Bernoulli and Euler polynomials.

Theorem 2.9 Let x, y [member of] R and conditions of Theorem 2.5 holds true, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (63)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (64)

Proof We can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By comparing the coefficients of [t.sup.n] / [n.sup.!] on both sides, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

GI-Sang Cheon and H.M.Srivastava in [8],[10] investigated the classical relationship between Bernoulli and Euler polynomials . Now we present a relationship between generalized Multi poly-Bernoulli and generalized Euler polynomials. The following relation (65) are given by Q.M.Luo, So by applying this recurrence formula, we obtain Theorem 2.10,

[E.sub.k](x + 1,1, b, b) + [E.sub.k](x, 1, b, b) = 2[x.sup.k][(lnb).sup.k] (65)

Theorem 2. 10 Let a, b > 0, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (66)

Proof We know

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore we obtain the desired result (66). []

The following corollary is a straightforward consequence of Theorem 2.10

Corollary 2.1(see [8],[10]) In Theorem 2.10, if we set r =1, k = 1 and b = e, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (67)

Further work: In [25], Jang et al. gave new formulae on Genocchi numbers. They defined poly-Genocchi numbers to give the relation between Genocchi numbers, Euler numbers, and poly-Genocchi numbers. After Y. Simsek [26], gave a new generating functions which produce Genocchi zeta functions. So by applying a similar method of Kim-Kim [4], we can introduce generalized Genocchi Zeta functions and next define Multi poly-Genocchi numbers and obtain several properties in this area.

Acknowledgments: The authors wishes to express his sincere gratitude to the referee for his/her valuable suggestions and comments and Professor Mohammad Maleki for his cooperations and helps.

(1) Received December 29, 2010. Accepted June 8, 2011.

References

[1] M.Abramowits, I. A.Stegun, Handbook of Mathematical functions with formulas, Graphs and Mathematical tables,National Bureau of standards, Washington, DC, 1964.

[2] T.Arakawa and M.Kaneko, On poly-Bernoulli numbers, Comment Math. univ. st. Pauli, 48, (1999), 159-167.

[3] B. N.Gue and F.Qi, Generalization of Bernoulli polynomials, I. J. Math. Ed. Sci. Tech, 33, (2002), N0 3, 428-431

[4] M.-S.-Kim and T.Kim, An explicit formula on the generalized Bernoulli number with other n, Indian. J. Pure and applied Math., 31, (2000), 1455-1466.

[5] Hassan jolany and M.R.Darafsheh, Some other remarks on the generalization of Bernoulli and Euler numbers, Journal's name??, Vol. 5 (2009), No. 3, 118-129.

[6] M.Kaneko, Poly-bernoulli numbers, Journal de Theorides de bordeaux, 9(1997), 221-228

[7] Y.Hamahata, H.Masubuch, Special multi-poly-Bernoulli numbers, Journal of Integer sequences, Vol 10, (2007).

[8] H. M.Srivastava and A.Pinter, Remarks on some relationships between the Bernoulli and Euler polynomials, Applied. Math. Letter, 17,(2004)375-380.

[9] Chad Berwbaker, A combinatorial interpretion of the poly-Bernoulli numbers and two fermat analogues, Integers Journal, 8, (2008).

[10] GI-Sang Cheon, A note on the Bernoulli and Euler polynomials, Applied. Math. Letter, 16, (2003), 365-368.

[11] Q. M.Lue, F.Oi and L.Debnath, Generalization of Euler numbers and polynomials, Int. J. Math. Sci., (2003) 3893-3901.

[12] Y.Hamahata, H.Masubuchi, Recurrence formulae for multi-poly-Bernoulli numbers, Integers Journal, 7(2007).

[13] Jin-Woo Son and Min Soo kim, On Poly-Eulerian numbers, Bull.Korean Math. Sco., 1999 36:47-61.

[14] M.Kaneko, N.Kurokawa, M.Wakayama, A variation of Euler's approach to the Riemann

zeta function, Kyushu J.Math., 57(2003) 175-192. [15] M.Ward, A calculus of sequences, Amer.J.Math., 58(1936) 255-266.

[16] H.Tsumura, A note on q-analogues of the Dirichlet series and q-Bernoulli numbers, J.Number Theory, 39(1991) 251-256.

[17] T.Kim, On a q-analogue of the p-adic log gamma functions and related integrals, J.Number Theory, 76(1999)320-329.

[18] H.M.Srivastava and J.Choi, Series Associated with The zeta and Related functions, Kluwer Academic Publisher, Dordrechet,Boston, London, 2001.

[19] Hassan Jolany and M.R.Darafsheh, Generalization on poly-Bernoulli numbers and polynomials, International J.Math. Combin., Vol.2 (2010), 07-14.

[20] Q-M-Luo, H.M.Srivastava, Some relationships between the Apostol-Bernoulli and ApostolEuler polynomials, Comput.Math.Appl., 51(2006) 631-642.

[21] L.Carlitz,q-Bernoulli and Eulerian numbers, Trans.Amer.math.Soc., 76 (1954) 332-350

[22] T.Arakawa, M.Kaneko, Multiple zeta values, Poly-Bernoulli numbers and related zeta functions, Nagoya Math. J., 153 (1999), 189-209.

[23] Q.M.Luo, B. N.Guo, F.Qi, and L.Debnath,Generalization of Bernoulli numbers and polynomials, IJMMS, Vol. 2003, Issue 59, 2003, 3769-3776.

[24] Q.M.Luo, F.Qi, and L.Debnath, Generalizations of Euler numbers and polynomials, IJMMS, Vol. 2003, Issue 61, 2003, 3893-3901.

[25] L.C.Jang, T.Kim, D.H. Lee and D.W. Park, An application of polylogarithms in the analogs of Genocchi numbers, Notes on Number Theory and Discrete Mathematics, Vol.7, No. 3, pp. 65-69, 2001.

[26] Y.Simsek, q-Hardy-Berndt type sums associated with q-Genocchi type zeta and l-functions, http://arxiv. org/abs/0710.5681v1.

[27] Hassan Jolany and Hamed Faramarzi, Generalizations on poly-Eulerian numbers and polynomials, Scientia Magna, Vol. 6 (2010), No. 1, 9-18.

Hassan Jolany, Hossein Mohebbi

(Department of Mathematics, statistics, and computer Science, University of

Tehran, Iran)

R.Eizadi Alikelaye

(Islamic Azad University of QAZVIN, Iran)

E-mail: jolany@ut.ac.ir, re.eizadi@gmail.com

Bernoulli numbers are the signs of a very strong bond between elementary number theory, complex analytic number theory, homotopy theory(the J-homomorphism, and stable homotopy groups of spheres), differential topology(differential structures on spheres), the theory of modular forms(Eisenstein series) and p-adic analytic number theory(the p-adic L- function) of

Mathematics. For n [member of] Z, n [less than or equal to] 0, Bernulli numbers [B.sub.n] originally arise in the study of finite sums of a given power of consecutive integers. They are given by [B.sub.0] = 1, [B.sub.1] = -1/2, [B.sub.2] = 1/6, [B.sub.3] = 0, [B.sub.4] = -1/30,with [B.sub.2n+1] = 0 for n > 1, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The modern definition of Bernoulli numbers Bn can be defined by the contour integral

where the contour encloses the origin, has radius less than 2[pi].

Also Bernoulli polynomials [B.sub.n](x) are usualy defined(see[1], [4], [5])by the generating function

G(x,t) = [te.sup.xt] / [e.sup.t]-1 = [[infinity].summation over (n=0)] [B.sub.n](x) [t.sup.n] / n!, [absolute value of t] < 2[pi] (3)

and consequently, Bernoulli numbers [B.sub.n](0) := [B.sub.n] can be obtained by the generating function

t / [e.sup.t]-1 = [[infinity].summation over (n=0)] [B.sub.n] [t.sup.n] / n!

Bernoulli polynomials, first studied by Euler (see[1]), are employed in the integral representation of differentiable periodic functions, and play an important role in the approximation of such functions by means of polynomials (see[14]-[18]).

Euler polynomials [E.sub.n](x) are defined by the generating function

2[e.sup.xt] / [e.sup.t]-1 = [[infinity].summation over (n=0)] [E.sub.n](x) [t.sup.n] / n!, [absolute value of t] < [pi] (4)

Euler numbers [E.sub.n] can be obtained by the generating function

2 / [e.sup.t]+1 = [[infinity].summation over (n=0)] [E.sub.n](x) [t.sup.n] / n! (5)

The first four such polynomials, are

[B.sub.0](x) = 1, [B.sub.1](x) = x - 1/2, [B.sub.2](x) = [x.sup.2] - x +1/6 [B.sub.3](x) = [x.sup.3] - 3/2[x.sup.2] + 1/2x,...

and

[E.sub.o](x) = 1, [E.sub.1](x) = x - 1/2, [E.sub.2](x) = [x.sup.2] - x, [E.sub.3](x) = [x.sup.3] - 3/2[x.sup.2] + 1/4,...

Euler polynomials are strictly connected with Bernoulli ones, and are used in the Taylor expansion in a neighborhood of the origin of trigonometric and hyperbolic secant functions.

In the sequel, we list some properties of Bernoulli and Euler numbers and polynomials as well as recurrence relations and identities.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[B.sub.n](x +1) - [B.sub.n](x)= [nx.sub.n-1], (8)

[E.sub.n](x +1) + [E.sub.n](x) = [2.sub.xn.]. (9)

Lemma 1.1(see[20],[21]) For any integer n [greater than or equal to] 0, we have

MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Consequently, from (8), (9) and lemma 1.1, we obtain,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Lemma 1.3 For any positive integer n [greater than or equal to] 0, we have

[B.sub.n](px) = [p.sup.n-1] / [p-1.summation over (r=0)] [B.sub.n](x+ r/p) (p is a positive integer) (14)

[E.sub.n](px) = [p.sup.n] / [p-1.summation over (r=0)] [(-1).sup.r][E.sub.n](x+ r/p) (p is a positive integer) (15)

Let us briefly recall k - th polylogarithm. The polylogarithm is a special function [Li.sub.k](z), that is defined by the sum

[Li.sub.k](z):= [[infinity].summation over (s=1)] [z.sup.s] / [s.sup.k] (16)

For formal power series [Li.sub.k](z) is the k - th polylogarithm if k [greater than or equal to] 1, and a rational function if k [less than or equal to] 0. The name of the function come from the fact that it may alternatively be defined as the repeated integral of itself, namely that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (l7)

for integer values of k, we have the following explicit expressions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The integral of the Bose-Einstein distribution is expressed in terms of a polylogarithm,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Lemma 1.3(see[18]) For n [member of] N [union] {0}, we have an explicit formula for [Li.sub.-n](z) as follow

[Li.sub.-n](z)=[n+1.summation over (k=1)] [(-1).sub.n+k+1](k-1)!S(n+1,k) / [(1- z).sub.k] (19)

where s(n, k) are Stirling numbers of the second kind.

Now, we introduce the generalization of [Li.sub.k](z). Let r be an integer with a value greater than one.

Definition 1.1 Let [k.sub.1],[k.sub.2],...[k.sub.r] be integers. The generalization of polylogarithm are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

The rational numbers [B.sup.(k).sub.n], (n = 0,1, 2,...) are said to be poly- Bernoulli numbers if they satisfy

[li.sub.k](1-[e.sup.-x]) / 1-[e.sup.-x] = [[infinity].summation over (n=0] [B.sup.(k).sub.n] [x.sup.n]/n! (21)

In addition, for any n [greater than or equal to] 0, [B.sup.(1).sub.n] is the classical Bernoulli number, [B.sub.n] (see[7], [12]). Also , the rational numbers [H.sup.(k).sub.n,] (n = 0,1, 2,...)are said to be poly-Euler numbers if they satisfy

[Li.sub.k] (1-[e.sup.(1-u)]) / u-[e.sup.t] = [[infinity].summation over (n=0)] [H.sup.(k).sub.n](u) [t.sup.n]/n! (22)

where u is an algebraic real number and k > 1.(see[13],[19])

Let us now introduce a generalization of poly-Bernoulli numbers, making use of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 1.1(see[7]) Multi poly-Bernoulli numbers [B.sup.(k0,...,kr).sub.n,] (n = 0, 1, 2, ...) are defined for each integer ki, k2,kr by the generating series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

By Definition 1.2, the left hand side of (23) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

hence we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Definition 1.3 Multi poly-Euler numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , (n = 0,1, ...) are defined for each integer [k.sub.1],[k.sub.r] by the generating series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Kaneko [6] presented the following recurrence formulae for poly-Bernoulli numbers which we state hear.

Theorem 1.1(Kaneko)([2,6,14,22]) For any k [member of] Z and n [greater than or equal to] 0,we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

called the second type stirling numbers.

Y. Hamahata and H. Masubuchi in [12], presented the following recurrence formulae for Multi poly-Bernoulli numbers.

Theorem 1.2 (H. Masubuchi & Y. Hamahata) For n [greater than or equal to] 0 and ([k.sub.1],...,[k.sub.r] [member of] Z) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

If [k.sub.r] [not eqaul to] 1and n [greater than or equal to] 1, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

If [k.sub.r] = 1 and n [greater than or equal to] 1 ,then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

Also, they proved (see[1]) if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

then for n, k [greater than or equal to] 0, we have

B[[r].sup.(-k).sub.n] = B[[r].sup.(-n).sub.k] (38)

In [23], [24], Q.M.Luo, F.Oi and L.Debnath defined the generalization of Bernoulli and Euler polynomials [B.sub.n](x, a, b, c) and En(x, a, b, c) respectively, which are expressed as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

In this paper, by the method of Q.M.Luo and et al [11], we give some properties on generalized Multi poly-Bernoulli and Euler polynomials

Definition 1.4 Let a, b > 0 and a [not equal to] b. The generalized Multi poly- Bernoulli numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (a,b) , the generalized Multi poly-Bernoulli polynomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are defined by the following generating functions, respectively;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (42)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

Definition 1.5 Let a, b > 0, and a [not equal to] b, the generalized Multipoly-Euler numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (u; a, b),the generalized multi poly-Euler polynomial [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are defined by the following generating functions, respectively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

[section]2. Main Theorems

In this section, we introduce our main results. We give some theorems and corollaries which are related to generalized Multi poly-Bernoulli numbers and generalized Multi poly-Euler polynomials. We present some recurrence formulae for generalized Multi-poly-Bernoulli and Euler polynomials.

Theorem 2.1 Let a, b > 0 and a [not equal to] b, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47)

proof By applying Definition 1.4, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, by comparing the coefficients of [t.sup.n]/n! on both sides, proof will be complete

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The generalized Multi poly-Bernoulli and Euler numbers process a number of interesting properties which we state here

Theorem 2.2 Let a, b > 0 and a = b. For real algebraic u we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (48)

Next, we investigate a strong relationships between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 2.3 Let a, b > 0,a [not equal to] band a > b > 0, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

By applying Definition 1.4, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By comparing the coefficient of [t.sup.n]/n! on both sides, we get.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 2.4 Let a, b > 0, and b > a > 0. For algebraic real number u, we have

By the same method proceeded in the proof of Theorem 2.3, we obtained similar relations for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 2.4 Let a, b > 0, and b > a > 0. For algebraic real number u, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

Theorem 2.5 Let x [member of] R and conditions of Theorem 2.3 holds true, then we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (51)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52)

Proof By applying Definitions 1.4 and 1.5, proof will be complete. []

Theorem 2.6 Let conditions of Theorem 2.5 holds true, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (53)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (54)

Proof By applying Theorems 2.1 and 2.5, we get (53), and Obviously, the result of (54) is similar with (53). []

Theorem 2.7 Let conditions of Theorem 2.5 holds true, then we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (55)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (56)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (58)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (59)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (60)

Proof We only prove (59) and (55)-(60) can be derived by Definitions 1.4 and 1.5.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So by comparing the coefficients of [t.sup.n] / n! in the two expressions, we obtain the desired result 2.13. [??]

Theorem 2.8 By the same method proceeded in the proof of previous Theorems, we find similar relations for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (61)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (62)

Now, we present formulae which show a deeper motivation of generalized poly-Bernoulli and Euler polynomials.

Theorem 2.9 Let x, y [member of] R and conditions of Theorem 2.5 holds true, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (63)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (64)

Proof We can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By comparing the coefficients of [t.sup.n] / [n.sup.!] on both sides, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

GI-Sang Cheon and H.M.Srivastava in [8],[10] investigated the classical relationship between Bernoulli and Euler polynomials . Now we present a relationship between generalized Multi poly-Bernoulli and generalized Euler polynomials. The following relation (65) are given by Q.M.Luo, So by applying this recurrence formula, we obtain Theorem 2.10,

[E.sub.k](x + 1,1, b, b) + [E.sub.k](x, 1, b, b) = 2[x.sup.k][(lnb).sup.k] (65)

Theorem 2. 10 Let a, b > 0, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (66)

Proof We know

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore we obtain the desired result (66). []

The following corollary is a straightforward consequence of Theorem 2.10

Corollary 2.1(see [8],[10]) In Theorem 2.10, if we set r =1, k = 1 and b = e, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (67)

Further work: In [25], Jang et al. gave new formulae on Genocchi numbers. They defined poly-Genocchi numbers to give the relation between Genocchi numbers, Euler numbers, and poly-Genocchi numbers. After Y. Simsek [26], gave a new generating functions which produce Genocchi zeta functions. So by applying a similar method of Kim-Kim [4], we can introduce generalized Genocchi Zeta functions and next define Multi poly-Genocchi numbers and obtain several properties in this area.

Acknowledgments: The authors wishes to express his sincere gratitude to the referee for his/her valuable suggestions and comments and Professor Mohammad Maleki for his cooperations and helps.

(1) Received December 29, 2010. Accepted June 8, 2011.

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zeta function, Kyushu J.Math., 57(2003) 175-192. [15] M.Ward, A calculus of sequences, Amer.J.Math., 58(1936) 255-266.

[16] H.Tsumura, A note on q-analogues of the Dirichlet series and q-Bernoulli numbers, J.Number Theory, 39(1991) 251-256.

[17] T.Kim, On a q-analogue of the p-adic log gamma functions and related integrals, J.Number Theory, 76(1999)320-329.

[18] H.M.Srivastava and J.Choi, Series Associated with The zeta and Related functions, Kluwer Academic Publisher, Dordrechet,Boston, London, 2001.

[19] Hassan Jolany and M.R.Darafsheh, Generalization on poly-Bernoulli numbers and polynomials, International J.Math. Combin., Vol.2 (2010), 07-14.

[20] Q-M-Luo, H.M.Srivastava, Some relationships between the Apostol-Bernoulli and ApostolEuler polynomials, Comput.Math.Appl., 51(2006) 631-642.

[21] L.Carlitz,q-Bernoulli and Eulerian numbers, Trans.Amer.math.Soc., 76 (1954) 332-350

[22] T.Arakawa, M.Kaneko, Multiple zeta values, Poly-Bernoulli numbers and related zeta functions, Nagoya Math. J., 153 (1999), 189-209.

[23] Q.M.Luo, B. N.Guo, F.Qi, and L.Debnath,Generalization of Bernoulli numbers and polynomials, IJMMS, Vol. 2003, Issue 59, 2003, 3769-3776.

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[25] L.C.Jang, T.Kim, D.H. Lee and D.W. Park, An application of polylogarithms in the analogs of Genocchi numbers, Notes on Number Theory and Discrete Mathematics, Vol.7, No. 3, pp. 65-69, 2001.

[26] Y.Simsek, q-Hardy-Berndt type sums associated with q-Genocchi type zeta and l-functions, http://arxiv. org/abs/0710.5681v1.

[27] Hassan Jolany and Hamed Faramarzi, Generalizations on poly-Eulerian numbers and polynomials, Scientia Magna, Vol. 6 (2010), No. 1, 9-18.

Hassan Jolany, Hossein Mohebbi

(Department of Mathematics, statistics, and computer Science, University of

Tehran, Iran)

R.Eizadi Alikelaye

(Islamic Azad University of QAZVIN, Iran)

E-mail: jolany@ut.ac.ir, re.eizadi@gmail.com

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