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A note on the ( h , q ) -zeta-type function with weight α

  • Elif Cetin1,
  • Mehmet Acikgoz2,
  • Ismail Naci Cangul1 and
  • Serkan Araci2Email author
Journal of Inequalities and Applications20132013:100

https://doi.org/10.1186/1029-242X-2013-100

Received: 28 January 2013

Accepted: 20 February 2013

Published: 13 March 2013

Abstract

The objective of this paper is to derive the symmetric property of an ( h , q ) -zeta function with weight α. By using this property, we give some interesting identities for ( h , q ) -Genocchi polynomials with weight α. As a result, our applications possess a number of interesting properties which we state in this paper.

MSC:11S80, 11B68.

Keywords

( h , q ) -Genocchi numbers and polynomials with weight α ( h , q ) -zeta function with weight αp-adic q-integral on Z p

1 Introduction

Recently, Kim has developed a new method by using the q-Volkenborn integral (or p-adic q-integral on Z p ) and has added weight to q-Bernoulli numbers and polynomials and investigated their interesting properties (see [1]). He also showed that these polynomials are closely related to weighted q-Bernstein polynomials and derived novel properties of q-Bernoulli numbers with weight α by using the symmetric property of weighted q-Bernstein polynomials with the help of the q-Volkenborn integral (for more details, see [2]). Afterward, Araci et al. have introduced weighted ( h , q ) -Genocchi polynomials and defined ( h , q ) -zeta-type function with weight α by applying the Mellin transformation to the generating function of the ( h , q ) -Genocchi polynomials with weight α which interpolates for ( h , q ) -Genocchi polynomials with weight α at negative integers (for details, see [3]). In this paper, we also consider a ( h , q ) -zeta-type function with weight α and derive some interesting properties.

We firstly list some notations as follows.

Imagine that p is a fixed odd prime. Throughout this work, , Z p , Q p and C p will denote by the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of Q p , respectively. Also, we denote N = N { 0 } and exp ( x ) = e x . Let v p : C p Q { } ( is the field of rational numbers) denote the p-adic valuation of C p normalized so that v p ( p ) = 1 . The absolute value on C p will be denoted as | | , and | x | p = p v p ( x ) for x C p . When one speaks of q-extensions, q is considered in many ways, e.g., as an indeterminate, a complex number q C , or a p-adic number q C p . If q C , we assume that | q | < 1 . If q C p , we assume | 1 q | p < p 1 p 1 so that q x = exp ( x log q ) for | x | p 1 . We use the following notation:
[ x ] q = 1 q x 1 q , [ x ] q = 1 ( q ) x 1 + q .
(1.1)

We want to note that lim q 1 [ x ] q = x ; cf. [123].

For a fixed positive integer d, set
X = X d = lim n Z / d p n Z , X = 0 < a < d p ( a , p ) = 1 a + d p Z p
and
a + d p n Z p = { x X x a ( mod d p n ) } ,

where a Z satisfies the condition 0 a < d p n (see [123]).

The following p-adic q-Haar distribution was defined by Kim:
μ q ( x + p n Z p ) = q x [ p n ] q

for any positive n (see [12, 13]).

Let UD ( Z p ) be the set of uniformly differentiable functions on Z p . We say that f is a uniformly differentiable function at a point a Z p if the difference quotient
F f ( x , y ) = f ( x ) f ( y ) x y
has a limit f ( a ) as ( x , y ) ( a , a ) and denote this by f UD ( Z p ) . In [12] and [13], the p-adic q-integral of the function f UD ( Z p ) is defined by Kim as follows:
I q ( f ) = Z p f ( ξ ) d μ q ( ξ ) = lim n ξ = 0 p n 1 f ( ξ ) μ q ( ξ + p n Z p ) .
(1.2)
The bosonic integral is considered as the bosonic limit q 1 , I 1 ( f ) = lim q 1 I q ( f ) . Similarly, the p-adic fermionic integration on Z p is defined by Kim [8] as follows:
I q ( f ) = lim q q I q ( f ) = Z p f ( x ) d μ q ( x ) .
(1.3)
By using a fermionic p-adic q-integral on Z p , ( h , q ) -Genocchi polynomials are defined by [3]
G ˜ n + 1 , q ( α , h ) ( x ) n + 1 = Z p q ( h 1 ) ξ [ x + ξ ] q α n d μ q ( ξ ) = lim n 1 [ p n ] q ξ = 0 p n 1 ( 1 ) ξ [ x + ξ ] q α n q h ξ .
(1.4)
For x = 0 in (1.4), we have G ˜ n , q ( α , h ) ( 0 ) : = G ˜ n , q ( α , h ) are called ( h , q ) -Genocchi numbers with weight α which is defined by
G ˜ 0 , q ( α , h ) = 0 and q h G ˜ m + 1 ( α , h ) ( 1 ) m + 1 + G ˜ m + 1 ( α , h ) m + 1 = { [ 2 ] q if  m = 0 , 0 if  m 0 .
By (1.4), we have a distribution formula for ( h , q ) -Genocchi polynomials, which is shown by [3]
G ˜ n + 1 , q ( α , h ) ( x ) = [ 2 ] q [ 2 ] q a [ a ] q α n j = 0 a 1 ( 1 ) j q j h G ˜ n + 1 , q a ( α , h ) ( x + j a ) .

By applying some elementary methods, we will give symmetric properties of weighted ( h , q ) -Genocchi polynomials and a weighted ( h , q ) -zeta-type function. Consequently, our applications seem to be interesting and worthwhile for further works of many mathematicians in analytic numbers theory.

2 On the ( h , q ) -zeta-type function

In this part, we firstly recall the ( h , q ) -zeta-type function with weight α which is derived in [3] as follows:
ζ ˜ q ( α , h ) ( s , x ) = [ 2 ] q m = 0 ( 1 ) m q m h [ m + x ] q α s ,
(2.1)
where q C , h N and ( s ) > 1 . It is clear that the special case h = 0 and q 1 in (2.1) reduces to the ordinary Hurwitz-Euler zeta function. Now, we consider (2.1) in the following form:
ζ ˜ q a ( α , h ) ( s , b x + b j a ) = [ 2 ] q a m = 0 ( 1 ) m q m a h [ m + b x + b j a ] q a α s .
By applying some basic operations to the above identity, that is, for any positive integers m and b, there exist unique non-negative integers k and i such that m = b k + i with 0 i b 1 . For a 1 ( mod 2 ) and b 1 ( mod 2 ) . Thus, we can compute as follows:
(2.2)
From this, we can easily discover the following:
j = 0 a 1 ( 1 ) j q j b h ζ ˜ q a ( α , h ) ( s , b x + b j a ) = [ a ] q α s [ 2 ] q a j = 0 a 1 ( 1 ) j q j b h i = 0 b 1 ( 1 ) i q i a h m = 0 ( 1 ) m q m b a h [ a b ( m + x ) + a i + b j ] q α s .
(2.3)
Replacing a by b and j by i in (2.2), we have the following:
ζ ˜ q b ( α , h ) ( s , a x + a i b ) = [ b ] q α s [ 2 ] q b j = 0 a 1 ( 1 ) j q j b h m = 0 ( 1 ) m q m b a h [ a b ( m + x ) + a i + b j ] q α s .

By considering the above identity in (2.3), we can easily state the following theorem.

Theorem 1 The following identity is true:
[ 2 ] q b [ a ] q α s i = 0 a 1 ( 1 ) i q i b h ζ ˜ q a ( α , h ) ( s , b x + b i a ) = [ 2 ] q a [ b ] q α s i = 0 b 1 ( 1 ) i q i a h ζ ˜ q b ( α , h ) ( s , a x + a i b ) .
Now, setting b = 1 in Theorem 1, we have the following distribution formula:
ζ ˜ q ( α , h ) ( s , a x ) = [ 2 ] q [ 2 ] q a [ a ] q α s i = 0 a 1 ( 1 ) i q i h ζ ˜ q a ( α , h ) ( s , x + i a ) .
(2.4)

Putting a = 2 in (2.4) leads to the following corollary.

Corollary 1 The following identity holds true:
ζ ˜ q ( α , h ) ( s , 2 x ) = [ 2 ] q [ 2 ] q 2 [ 2 ] q α s ( ζ ˜ q 2 ( α , h ) ( s , x ) q h ζ ˜ q 2 ( α , h ) ( s , x + 1 2 ) ) .

Taking s = m into Theorem 1, we have the symmetric property of ( h , q ) -Genocchi polynomials by the following theorem.

Theorem 2 The following identity is true:
[ 2 ] q b [ a ] q α m 1 j = 0 a 1 ( 1 ) i q i b h G ˜ m , q a ( α , h ) ( b x + b i a ) = [ 2 ] q a [ b ] q α m 1 i = 0 b 1 ( 1 ) i q i a h G ˜ m , q b ( α , h ) ( a x + a i b ) .
Now also, setting b = 1 and replacing x by x a in the above theorem, we can rewrite the following ( h , q ) -Genocchi polynomials with weight α:
G ˜ n , q ( α , h ) ( x ) = [ 2 ] q [ 2 ] q a [ a ] q α n 1 i = 0 a 1 ( 1 ) i q i h G ˜ n , q a ( α , h ) ( x + i a ) ( 2 a ) .
Due to Araci et al. [3], we develop as follows:
n = 0 G ˜ n , q ( α , h ) ( x + y ) t n n ! = [ 2 ] q t m = 0 ( 1 ) m q m h e t [ x + y + m ] q α = [ 2 ] q t m = 0 ( 1 ) m q m h e t [ y ] q α e ( q α y t ) [ x + m ] q α = ( n = 0 [ y ] q α n t n n ! ) ( n = 0 q α ( n 1 ) y G ˜ n , q ( α , h ) ( x ) t n n ! ) .
By using the Cauchy product, we see that
n = 0 ( j = 0 n ( n j ) q α ( j 1 ) y G ˜ j , q ( α , h ) ( x ) [ y ] q α n j ) t n n ! .

Thus, by comparing the coefficients of t n n ! , we state the following corollary.

Corollary 2 The following equality holds true:
G ˜ n , q ( α , h ) ( x + y ) = j = 0 n ( n j ) q α ( j 1 ) y G ˜ j , q ( α , h ) ( x ) [ y ] q α n j .
(2.5)

By using Theorem 2 and (2.5), we readily derive the following symmetric relation after some applications.

Theorem 3 The following equality holds true:
[ 2 ] q b i = 0 m ( m i ) [ a ] q α i 1 [ b ] q α m i G ˜ i , q a ( α , h ) ( b x ) S ˜ m i : q b , h + i 1 ( α ) ( a ) = [ 2 ] q a i = 0 m ( m i ) [ b ] q α i 1 [ a ] q α m i G ˜ i , q b ( α , h ) ( a x ) S ˜ m i : q a , h + i 1 ( α ) ( b ) ,

where S ˜ m : q , i ( α ) ( a ) = j = 0 a 1 ( 1 ) j q j i [ j ] q α m .

When q 1 into Theorem 3, it leads to the following corollary.

Corollary 3 The following identity holds true:
i = 0 m ( m i ) a i 1 b m i G i ( b x ) S m i ( a ) = i = 0 m ( m i ) b i 1 a m i G i ( a x ) S m i ( b ) ,
where S m ( a ) = j = 0 a 1 ( 1 ) j j m and G n ( x ) are called the ordinary Genocchi polynomials which are defined via the following generating function:
n = 0 G n ( x ) t n n ! = 2 t e t + 1 e x t .

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

We would like to thank the referees for their valuable suggestions and comments on the present paper. The third author is supported by Uludag University Research Fund, Project Number F-2012/16 and F-2012/19.

Authors’ Affiliations

(1)
Faculty of Arts and Science, Department of Mathematics, Uludag University, Bursa, Turkey
(2)
Faculty of Arts and Science, Department of Mathematics, University of Gaziantep, Gaziantep, Turkey

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© Cetin et al.; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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