[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.
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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