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

R E S E A R C H

Open Access

A note on the

(

h

,

q

)

-zeta-type function with

weight

α

Elif Cetin

1

_{, Mehmet Acikgoz}

2

_{, Ismail Naci Cangul}

1

_{and Serkan Araci}

2*

*_{Correspondence:}

[email protected]

2_{Faculty of Arts and Science,}

Department of Mathematics, University of Gaziantep, Gaziantep, 27310, Turkey

Full list of author information is available at the end of the article

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

α

;p-adicq-integral onZp

1 Introduction

Recently, Kim has developed a new method by using theq-Volkenborn integral (orp-adic q-integral on Zp) and has added weight toq-Bernoulli numbers and polynomials and investigated their interesting properties (see []). He also showed that these polynomi-als are closely related to weightedq-Bernstein polynomials and derived novel proper-ties ofq-Bernoulli numbers with weightαby using the symmetric property of weighted q-Bernstein polynomials with the help of theq-Volkenborn integral (for more details, see []). Afterward, Araciet al.have introduced weighted (h,q)-Genocchi polynomials and defined (h,q)-zeta-type function with weightαby applying the Mellin transformation to the generating function of the (h,q)-Genocchi polynomials with weightαwhich interpo-lates for (h,q)-Genocchi polynomials with weightα at negative integers (for details, see []). In this paper, we also consider a (h,q)-zeta-type function with weightαand derive some interesting properties.

We firstly list some notations as follows.

Imagine thatpis a fixed odd prime. Throughout this work,Z,Zp,QpandCpwill denote by the ring of integers, the field ofp-adic rational numbers and the completion of the algebraic closure ofQp, respectively. Also, we denoteN∗=N∪ {} andexp(x) =ex. Let vp:Cp→Q∪ {∞}(Q is the field of rational numbers) denote thep-adic valuation of

Cp normalized so thatvp(p) = . The absolute value onCp will be denoted as| · |, and |x|p=p–vp(x)_forx∈C

p. When one speaks ofq-extensions,qis considered in many ways, e.g., as an indeterminate, a complex numberq∈C, or ap-adic numberq∈Cp. Ifq∈C, we assume that|q|< . Ifq∈Cp, we assume| –q|p<p–



p– _{so thatq}x_{=exp(xlogq) for} |x|p≤. We use the following notation:

[x]q=  –qx

 –q, [x]–q=

 – (–q)x

 +q . (.)

We want to note thatlimq→[x]q=x;cf.[–]. For a fixed positive integerd, set

X=Xd=lim_←

–

n Z /dpnZ, X∗=

<a<dp

(a,p)=

a+dpZp

and

a+dpnZp=

x∈X|x≡amoddpn,

wherea∈Zsatisfies the condition ≤a<dpn(see [–]). The followingp-adicq-Haar distribution was defined by Kim:

μq

x+pnZp

= q x

[pn_] q

for any positiven(see [, ]).

LetUD(Zp) be the set of uniformly differentiable functions onZp. We say thatf is a uniformly differentiable function at a pointa∈Zpif the difference quotient

Ff(x,y) = f(x) –f(y) x–y

has a limitf(a) as (x,y)→(a,a) and denote this byf ∈UD(Zp). In [] and [], thep-adic q-integral of the functionf∈UD(Zp) is defined by Kim as follows:

Iq(f) =

Zp

f(ξ)dμq(ξ) = lim n→∞

pn_–

ξ=

f(ξ)μq

ξ+pnZp

. (.)

The bosonic integral is considered as the bosonic limitq→,I(f) =limq→Iq(f).

Simi-larly, thep-adic fermionic integration onZpis defined by Kim [] as follows:

I–q(f) = lim q→–qIq(f) =

Zp

f(x)dμ–q(x). (.)

By using a fermionicp-adicq-integral onZp, (h,q)-Genocchi polynomials are defined by []

G(_nα_+,,h)_q(x) n+  =

Zp

q(h–)ξ_[x+ξ]n

qαdμ–q(ξ)

= lim

n→∞

 [pn_]

–q pn–

ξ=

(–)ξ_[x+ξ]n

Forx=  in (.), we haveG(nα,q,h)() :=G(nα,q,h) are called (h,q)-Genocchi numbers with weightαwhich is defined by

G(_,α_q,h)=  and qhG

(α,h)

m+()

m+  + G(_mα_+,h) m+ =

⎧ ⎨ ⎩

[]q ifm= ,  ifm= .

By (.), we have a distribution formula for (h,q)-Genocchi polynomials, which is shown by []

G(_nα_+,,h)_q(x) = []q []qa[a]

n qα

a–

j=

(–)jqjhG(_nα_+,,h)_qa

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 mathemati-cians 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 weightαwhich is derived in [] as follows:

ζ_q(α,h)(s,x) = []q

∞

m=

(–)m_qmh [m+x]s

qα

, (.)

whereq∈C,h∈Nand(s) > . It is clear that the special caseh=  andq→ in (.) re-duces to the ordinary Hurwitz-Euler zeta function. Now, we consider (.) in the following form:

ζ_q(αa,h)

s,bx+bj a

= []qa ∞

m=

(–)m_qmah [m+bx+bj_a]s_qaα

.

By applying some basic operations to the above identity, that is, for any positive integers mandb, there exist unique non-negative integerskandisuch thatm=bk+iwith ≤i≤ b– . Fora≡(mod) andb≡(mod). Thus, we can compute as follows:

ζ_q(αa,h)

s,bx+bj a

= [a]s_qα[]qa ∞

m=

(–)m_qmah [ma+abx+bj]s_qaα

= [a]s_qα[]qa ∞

m=

b–

i=

(–)i+mb_q(i+mb)ah [(i+mb)a+abx+bj]s_qaα

= [a]s_qα[]qa

b–

i=

(–)iqiah

∞

m=

(–)m_qmbah [ab(m+x) +ai+bj]s

qα

From this, we can easily discover the following:

a–

j=

(–)jqjbhζ_q(αa,h)

s,bx+bj a

= [a]s qα[]qa

a–

j=

(–)j_qjbh b–

i=

(–)i_qiah

∞

m=

(–)m_qmbah [ab(m+x) +ai+bj]s_qα

. (.)

Replacingabybandjbyiin (.), we have the following:

ζ(α,h) qb

s,ax+ai b

= [b]s_qα[]_qb

a–

j=

(–)jqjbh

∞

m=

(–)m_qmbah [ab(m+x) +ai+bj]s

qα .

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

Theorem  The following identity is true:

[]_qb

[a]s_qα a–

i=

(–)iqibhζ_q(αa,h)

s,bx+bi a

=[]qa [b]s_qα

b–

i=

(–)iqiahζ(α,h) qb

s,ax+ai b

.

Now, settingb=  in Theorem , we have the following distribution formula:

ζ_q(α,h)(s,ax) = []q []qa[a]s

qα a–

i=

(–)iqihζ_q(αa,h)

s,x+ i a

. (.)

Puttinga=  in (.) leads to the following corollary.

Corollary  The following identity holds true:

ζ_q(α,h)(s, x) = []q []q[]s_qα

ζ_q(α,h)(s,x) –qhζ

(α,h)

q

s,x+ 

.

Takings= –minto Theorem , we have the symmetric property of (h,q)-Genocchi poly-nomials by the following theorem.

Theorem  The following identity is true:

[]qb[a]m_qα– a–

j=

(–)iqibhG(_mα_,,_qha)

bx+bi a

= []qa[b]m–

qα b–

i=

(–)iqiahG(_mα_,,_qhb)

ax+ai b

.

Now also, settingb=  and replacingxby _ax in the above theorem, we can rewrite the following (h,q)-Genocchi polynomials with weightα:

G(_nα_,q,h)(x) = []q []qa[a]

n–

qα a–

i=

(–)iqihG(_nα_,q,ha)

x+i a

Due to Araciet al.[], we develop as follows:

∞

n=

G(α,h)

n,q (x+y) tn n! = []qt

∞

m=

(–)mqmhet[x+y+m]qα

= []qt

∞

m=

(–)mqmhet[y]qα

e(qαyt)[x+m]qα

=

_∞

n=

[y]n_qα tn n!

_∞

n=

qα(n–)yG_n(α_,q,h)(x)t n

n!

.

By using the Cauchy product, we see that

∞

n=

_n

j=

n j

qα(j–)yG_j(_,α_q,h)(x)[y]n_q–αj

tn n!.

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

Corollary  The following equality holds true:

G(_nα_,q,h)(x+y) = n

j=

n j

qα(j–)yG_j(_,α_q,h)(x)[y]n_qα–j. (.)

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

Theorem  The following equality holds true:

[]_qb

m

i=

m i

[a]i_q–α[b]m_qα–iG(_iα_,q,ah)(bx)S(

α)

m–i:qb_,h+i–(a)

= []qa

m

i=

m i

[b]i_q–α[a]_qmα–iG(α,h) i,qb (ax)S

(α)

m–i:qa_,h+i–(b),

whereS(_mα_:)_q,i(a) =_ja_=–(–)jqji[j]m_qα.

Whenq→ into Theorem , it leads to the following corollary.

Corollary  The following identity holds true:

m

i=

m i

ai–bm–iGi(bx)Sm–i(a)

= m

i=

m i

where Sm(a) =a_j=–(–)j_jm_{and Gn(x)are called the ordinary Genocchi polynomials which} are defined via the following generating function:

∞

n=

Gn(x)t n

n!= t et_{+ }e

xt_.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

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, 27310, Turkey.

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.

Received: 28 January 2013 Accepted: 20 February 2013 Published: 13 March 2013 References

1. Kim, T: On the weightedq-Bernoulli numbers and polynomials. Adv. Stud. Contemp. Math.21(2), 207-215 (2011) 2. Kim, T, Bayad, A, Kim, YH: A study on thep-adicq-integrals representation onZpassociated with the weighted

q-Bernstein andq-Bernoulli polynomials. J. Inequal. Appl.2011, Article ID 513821 (2011)

3. Araci, S, Seo, JJ, Erdal, D: New construction weighted (h,q)-Genocchi numbers and polynomials related to zeta type function. Discrete Dyn. Nat. Soc.2011, Article ID 487490 (2011). doi:10.1155/2011/487490

4. Bayad, A, Kim, T: Identities involving values of Bernstein,q-Bernoulli, andq-Euler polynomials. Russ. J. Math. Phys. 18(2), 133-143 (2011)

5. Ryoo, CS: Some relations between twistedq-Euler numbers and Bernstein polynomials. Adv. Stud. Contemp. Math. 21(2), 217-223 (2011)

6. Ryoo, CS, Kim, T: An analogue of the zeta function and its applications. Appl. Math. Lett.19, 1068-1072 (2006) 7. Kim, T:q-generalized Euler numbers and polynomials. Russ. J. Math. Phys.13(3), 293-308 (2006)

8. Kim, T: Some identities on theq-Euler polynomials of higher order andq-Stirling numbers by the fermionicp-adic integral on_Zp. Russ. J. Math. Phys.16, 484-491 (2009)

9. Kim, T: On theq-extension of Euler and Genocchi numbers. J. Math. Anal. Appl.326, 1458-1465 (2007)

10. Kim, T: On the analogs of Euler numbers and polynomials associated withp-adicq-integral onZpatq= 1. J. Math. Anal. Appl.331, 779-792 (2007)

11. Kim, T, Lee, SH, Han, HH, Ryoo, CS: On the values of the weightedq-zeta andL-functions. Discrete Dyn. Nat. Soc.2011, Article ID 476381 (2011)

12. Kim, T:q-Volkenborn integration. Russ. J. Math. Phys.9, 288-299 (2002)

13. Kim, T: On aq-analogue of thep-adic log gamma functions and related integrals. J. Number Theory76, 320-329 (1999)

14. Kim, T: Note on the Eulerq-zeta functions. J. Number Theory129(7), 1798-1804 (2009)

15. Kim, T: Some formulae for theq-Bernstein polynomials andq-deformed binomial distributions. J. Comput. Anal. Appl. 14(5), 917-933 (2012)

16. Kim, T: An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionicp-adic invariantq-integrals onZp. Rocky Mt. J. Math.41(1), 239-247 (2011)

17. Kim, T, Lee, SH: Some properties on theq-Euler numbers and polynomials. Abstr. Appl. Anal.2012, Article ID 284826 (2012)

18. Dolgy, DV, Kang, DJ, Kim, T, Lee, B: Some new identities on the twisted (h,q)-Euler numbers andq-Bernstein polynomials. J. Comput. Anal. Appl.14(5), 974-984 (2012)

19. Ozden, H, Cangul, IN, Simsek, Y: Multivariate interpolation functions of higher orderq-Euler numbers and their applications. Abstr. Appl. Anal.2008, Article ID 390857 (2008)

20. Araci, S, Acikgoz, M, Park, KH, Jolany, H: On the unification of two families of multiple twisted type polynomials by usingp-adicq-integral onZpatq= –1. Bull. Malays. Math. Sci. Soc. (to appear)

21. Araci, S, Acikgoz, M, Park, KH: A note on theq-analogue of Kim’sp-adic log gamma type functions associated with

q-extension of Genocchi and Euler numbers with weightα. Bull. Korean Math. Soc. (accepted)

22. Araci, S, Erdal, D, Seo, JJ: A study on the fermionicp-adicq-integral representation onZpassociated with weighted

q-Bernstein andq-Genocchi polynomials. Abstr. Appl. Anal.2011, Article ID 649248 (2011)

23. Jolany, H, Araci, S, Acikgoz, M, Seo, JJ: A note on the generalizedq-Genocchi measure with weightα. Bol. Soc. Parana. Mat.31(1), 17-27 (2013)

doi:10.1186/1029-242X-2013-100