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

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.

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 qC, or a p-adic number q C p . If qC, we assume that |q|<1. If q C p , we assume |1q | p < p 1 p 1 so that q x =exp(xlogq) 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 aZ satisfies the condition 0a<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 fUD( Z p ). In [12] and [13], the p-adic q-integral of the function fUD( 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 q1, 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 ) =0and 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 qC, hN and (s)>1. It is clear that the special case h=0 and q1 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=bk+i with 0ib1. For a1(mod2) and b1(mod2). 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,ax)= [ 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,2x)= [ 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 ) (2a).

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 q1 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 .

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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.

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Correspondence to Serkan Araci.

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All authors completed the paper together. All authors read and approved the final manuscript.

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Cetin, E., Acikgoz, M., Cangul, I.N. et al. A note on the (h,q)-zeta-type function with weight α. J Inequal Appl 2013, 100 (2013). https://doi.org/10.1186/1029-242X-2013-100

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Keywords

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