# Some Properties of Multiple Generalizedq-Genocchi Polynomials with Weight α and  Weak Weight β

Authors: J. Y. Kang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

The present paper deals with the variousq-Genocchi numbers and polynomials. We define a new type of multiple generalized q-Genocchi numbers and polynomials with weight α and weak weight β by applying the method of p-adic q-integral. We will find a link between their numbers and polynomials with weight α and weak weight β. Also we will obtain the interesting properties of their numbers and polynomials with weight α and weak weight β. Moreover, we construct a Hurwitz-type zeta function which interpolates multiple generalized q-Genocchi polynomials with weight α and weak weight β and find some combinatorial relations.


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## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this paper ℤp,ℚp,ℂ, and ℂp denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of ℚp, respectively. Let ℕ be the set of natural numbers and ℤ+=ℕ∪{0}. Let vp be the normalized exponential valuation of ℂp with |p|p=p-vp(p)=1/p (see [1–21]). When one talks of q-extension, q is variously considered as an indeterminate, a complex q∈ℂ, or a p-adic number q∈ℂp. If q∈ℂ, then one normally assumes |q|<1. If q∈ℂp, then we assume that |q-1|p<1.Throughout this paper, we use the following notation:(1.1)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q.
Hence limq→1[x]q=x for all x∈ℤp (see [1–14, 16, 18, 20, 21]).We say thatg:ℤp→ℂp is uniformly differentiable function at a point a∈ℤp and we write g∈UD(ℤp) if the difference quotients Φg:ℤp×ℤp→ℂp such that
(1.2)Φg(x,y)=g(x)-g(y)x-y
have a limit g'(a) as (x,y)→(a,a).Letd be a fixed integer, and let p be a fixed prime number. For any positive integer N, we set
(1.3)X=Xd=lim←N(ℤdpNℤ),X1=ℤp,X*=⋃0<a<dp(a,p)=1‍(a+dpℤp),a+dpNℤp={x∈X∣x≡a(moddpN)},
where a∈ℤ lies in 0≤a<dpN.For any positive integerN,
(1.4)μq(a+dpNℤp)=qa[dpN]q
is known to be a distribution on X.Forg∈UD(ℤp), Kim defined the q-deformed fermionic p-adic integral on ℤp:
(1.5)I-q(g)=∫ℤp‍g(x)dμ-q(x)=limN→∞1[pN]-q∑x=0pN-1‍g(x)(-q)x.(see [1–13]), and note that
(1.6)∫ℤp‍g(x)dμ-q(x)=∫X‍g(x)dμ-q(x).
We consider the case q∈(-1,0) corresponding to q-deformed fermionic certain and annihilation operators and the literature given there in [9, 13, 14].In [9, 12, 14, 19], we introduced multiple generalized Genocchi number and polynomials. Let χ be a primitive Dirichlet character of conductor f∈ℕ. We assume that f is odd. Then the multiple generalized Genocchi numbers, Gn,χ(r), and the multiple generalized Genocchi polynomials, Gn,χ(r)(x), associated with χ, are defined by
(1.7)Fχ(r)(t)=(2t∑a=0f-1‍χ(a)(-1)aeateft+1)r=∑n=0∞‍Gn,χ(r)tnn!,Fχ(r)(t,x)=(2t∑a=0f-1‍χ(a)(-1)aeateft+1etx)r=∑n=0∞‍Gn,χ(r)(x)tnn!.
In the special case x=0, Gn,χ(r)=Gn,χ(r)(0) are called the nth multiple generalized Genocchi numbers attached to χ.Now, having discussed the multiple generalized Genocchi numbers and polynomials, we were ready to multiple-generalize them to theirq-analogues. In generalizing the generating functions of the Genocchi numbers and polynomials to their respective q-analogues; it is more useful than defining the generating function for the Genocchi numbers and polynomials (see [12]).Our aim in this paper is to define multiple generalizedq-Genocchi numbers Gn,χ,q(α,β,r) and polynomials Gn,χ,q(α,β,r)(x) with weight α and weak weight β. We investigate some properties which are related to multiple generalized q-Genocchi numbers Gn,χ,q(α,β,r) and polynomials Gn,χ,q(α,β,r)(x) with weight α and weak weight β. We also derive the existence of a specific interpolation function which interpolate multiple generalized q-Genocchi numbers Gn,χ,q(α,β,r) and polynomials Gn,χ,q(α,β,r)(x) with weight α and weak weight β at negative integers.


## 2.  The Generating Functions of Multiple Generalizedq-Genocchi Numbers and Polynomials with Weight α and Weak Weight β

Many mathematicians constructed various kinds of generating functions of theq-Gnocchi numbers and polynomials by using p-adic q-Vokenborn integral. First we introduce multiple generalized q-Genocchi numbers and polynomials with weight α and weak weight β.Let us define the generalizedq-Genocchi numbers Gn,χ,q(α,β) and polynomials Gn,χ,q(α,β)(x) with weight α and weak weight β, respectively,
(2.1)Fχ,q(α,β)(t)=∑n=0∞‍Gn,χ,q(α,β)tnn!=∫X‍tχ(x)e[x]qαtdμ-qβ(x),Fχ,q(α,β)(t,x)=∑n=0∞‍Gn,χ,q(α,β)(x)tnn!=∫X‍tχ(y)e[x+y]qαtdμ-qβ(y).
By using the Taylor expansion of e[x]qαt, we have
(2.2)∑n=0∞‍∫X‍χ(x)[x]qαndμ-qβ(x)tnn!=∑n=0∞‍Gn,χ,q(α,β)tn-1n!=G0,χ,q(α,β)+∑n=0∞‍Gn+1,χ,q(α,β)n+1tnn!.
By comparing the coefficient of both sides of tn/n! in (2.2), we get
(2.3)Gn+1,χ,q(α,β)n+1=[2]qβ(1-qα)n∑a=0f-1‍(-1)aqβaχ(a)∑l=0n‍(nl)(-1)lqαal11+qf(αl+β).
From (2.2) and (2.3), we can easily obtain that
(2.4)∑n=0∞‍Gn,χ,q(α,β)tnn!=∑n=0∞‍(t∫X‍χ(x)[x]qαndμ-qβ(x))tnn!=[2]qβt∑l=0∞‍(-1)lqβlχ(l)e[l]qαt.
Therefore, we obtain
(2.5)Fχ,q(α,β)(t)=[2]qβt∑l=0∞‍(-1)lqβlχ(l)e[l]qαt=∑n=0∞‍Gn,χ,q(α,β)tnn!.Similarly, we find the generating function of generalizedq-Genocchi polynomials with weight α and weak weight β:
(2.6)G0,χ,q(α,β)(x)=0,Gn+1,χ,q(α,β)(x)n+1=∫X‍χ(y)[x+y]qαndμ-qβ(y)=[2]qβ∑l=0∞‍(-1)lqβlχ(l)[x+l]qαn.
From (2.6), we have
(2.7)Fχ,q(α,β)(t,x)=[2]qβt∑l=0∞‍(-1)lqβlχ(l)e[x+l]qαt=∑n=0∞‍Gn,χ,q(α,β)(x)tnn!.
Observe that Fχ,q(α,β)(t)=Fχ,q(α,β)(t,0). Hence we have Gn,χ,q(α,β)=Gn,χ,q(α,β)(0). If q→1 into (2.7), then we easily obtain Fχ(t,x).First, we define the multiple generalizedq-Genocchi numbers Gn,χ,q(α,β,r) with weight α and weak weight β:
(2.8)Fχ,q(α,β,r)(t)=[2]qβrtr∑k1,…,kr=0∞‍(-1)∑i=1r‍kiqβ∑i=1r‍ki(∏i=1r‍χ(ki))e[∑i=1r‍ki]qαt=tr∫X‍⋯∫X‍︸r-timesχ(x1)⋯χ(xr)e[x1+⋯+xr]qαtdμ-qβ(x1)⋯dμ-qβ(xr)=∑n=0∞‍Gn,χ,q(α,β,r)tnn!.
Then we have
(2.9)∑n=0∞‍∫X‍⋯∫X‍︸r-timesχ(x1)⋯χ(xr)[x1+⋯+xr]qαndμ-qβ(x1)⋯dμ-qβ(xr)tnn!=∑n=0∞‍Gn,χ,q(α,β,r)tn-rn!=∑n=0r-1‍Gn,χ,q(α,β,r)tn-rn!+∑n=0∞‍Gn+r,χ,q(α,β,r)(n+rr)r!tnn!,
where (n+rr)=(n+r)!/n!r!.By comparing the coefficients on the both sides of (2.9), we obtain the following theorem.Theorem 2.1.
Letq∈ℂp with |1-q|p<1 and n∈ℤ+. Then one has
(2.10)G0,χ,q(α,β,r)=G1,χ,q(α,β,r)=⋯=Gr-1,χ,q(α,β,r)=0,Gn+r,χ,q(α,β,r)(n+rr)r!=∫X‍⋯∫X‍︸r-timesχ(x1)⋯χ(xr)[x1+⋯+xr]qαndμ-qβ(x1)⋯dμ-qβ(xr)=[2]qβr(1-qα)n∑a1,…,ar=0f-1‍∑l=0n‍(nl)(∏i=1r‍χ(ai))(-1)l+∑i=1r‍aiq(αl+β)∑i=1r‍ai(1+qf(αl+β))r=[2]qβr∑m=0∞‍∑a1,…,ar=0f-1‍(m+r-1m)(-1)∑i=1r‍ai+mqβ(∑i=1r‍ai+fm)×(∏i=1r‍χ(ai))[∑i=1r‍ai+fm]qαn.From now on, we define the multiple generalizedq-Genocchi polynomials Gn,χ,q(α,β,r)(x) with weight α and weak weight β. 						(2.11)Fχ,q(α,β,r)(t,x)=[2]qβrtr∑k1,…,kr=0∞‍(-1)∑i=1r‍kiqβ∑i=1r‍ki(∏i=1r‍χ(ki))e[∑i=1r‍ki+x]qαt=tr∫X‍⋯∫X‍︸r-timesχ(y1)⋯χ(yr)e[x+y1+⋯+yr]qαtdμ-qβ(y1)⋯dμ-qβ(yr)=∑n=0∞‍Gn,χ,q(α,β,r)(x)tnn!.
Then we have
(2.12)∑n=0∞‍∫X‍⋯∫X‍︸r-timesχ(y1)⋯χ(yr)[x+y1+⋯+yr]qαndμ-qβ(y1)⋯dμ-qβ(yr)tnn!=∑n=0∞‍Gn,χ,q(α,β,r)(x)tn-rn!=∑n=0r-1‍Gn,χ,q(α,β,r)(x)tn-rn!+∑n=0∞‍Gn+r,χ,q(α,β,r)(x)(n+rr)r!tnn!,
where (n+rr)=(n+r)!/n!r!.By comparing the coefficients on the both sides of (2.12), we have the following theorem.Theorem 2.2.
Letq∈ℂp with |1-q|p<1 and n∈ℤ+. Then one has
(2.13)G0,χ,q(α,β,r)(x)=G1,χ,q(α,β,r)(x)=⋯=Gr-1,χ,q(α,β,r)(x)=0,Gn+r,χ,q(α,β,r)(x)(n+rr)r!=∫X‍⋯∫X‍︸r-timesχ(y1)⋯χ(yr)[x+y1+⋯+yr]qαndμ-qβ(y1)⋯dμ-qβ(yr)=[2]qβr(1-qα)n∑a1,…,ar=0f-1‍∑l=0n‍(nl)(∏i=1r‍χ(ai))(-1)l+∑i=1r‍aiqαlx+(αl+β)∑i=1r‍ai(1+qf(αl+β))r=[2]qβr∑m=0∞‍∑a1,…,ar=0f-1‍(m+r-1m)(-1)∑i=1r‍ai+mqβ(∑i=1r‍ai+fm)×(∏i=1r‍χ(ai))[∑i=1r‍ai+fm+x]qαn.In (2.11), we simply identify that
(2.14)limq→1Fχ,q(α,β,r)(t,x)=2rtr∑k1,…,kr=0∞‍(-1)∑i=1r‍ki(∏i=1r‍χ(ki))e(∑i=1r‍ki+x)t=(2t∑a=0f-1‍(-1)aχ(a)eat1+eft)retx=Fχ(r)(t,x).So far, we have studied the generating functions of the multiple generalizedq-Genocchi numbers Gn,χ,q(α,β,r) and polynomials Gn,χ,q(α,β,r)(x) with weight α and weak weight β.


## 3.  Modified Multiple Generalizedq-Genocchi Polynomials with Weight α and Weak Weight β

In this section, we will investigate about modified multiple generalizedq-Genocchi numbers and polynomials with weight α and weak weight β. Also, we will find their relations in multiple generalized q-Genocchi numbers and polynomials with weight α and weak weight β.Firstly, we modify generating functions ofGn,χ,q(α,β,r) and Gn,χ,q(α,β,r)(x). We access some relations connected to these numbers and polynomials with weight α and weak weight β. For this reason, we assign generating function of modified multiple generalized q-Genocchi numbers and polynomials with weight α and weak weight β which are implied by Gn,χ,q(α,β,r) and Gn,χ,q(α,β,r)(x). We give relations between these numbers and polynomials with weight α and weak weight β.We modify (2.11) as follows:
(3.1)𝔉χ,q(α,β,r)(t,x)=Fχ,q(α,β,r)(q-αxt,x),
where Fχ,q(α,β,r)(t,x) is defined in (2.11).From the above we know that(3.2)𝔉χ,q(α,β,r)(t,x)=∑n=0∞‍q-(n+r)αxGn,χ,q(α,β,r)(x)tnn!.
After some elementary calculations, we attain
(3.3)𝔉χ,q(α,β,r)(t,x)=q-αrxe(q-αx[x]qαt)Fχ,q(α,β,r)(t),
where Fχ,q(α,β,r)(t) is defined in (2.8).From the above, we can assign the modified multiple generalizedq-Genocchi polynomials εn,χ,q(α,β,r)(x) with weight α and weak weight β as follows:
(3.4)𝔉χ,q(α,β,r)(t,x)=∑n=0∞‍εn,χ,q(α,β,r)(x)tnn!.
Then we have
(3.5)εn,χ,q(α,β,r)(x)=q-(n+r)αxGn,χ,q(α,β,r)(x).Theorem 3.1.
Forr∈ℕ and n∈ℤ+, one has
(3.6)εn,χ,q(α,β,r)(x)=q-(n+r)αx∑i=0n‍(ni)qαix[x]qαn-iGi,χ,q(α,β,r).Corollary 3.2.
Forr∈ℕ and n∈ℤ+, by using (3.7), one easily obtains
(3.7)εn,χ,q(α,β,r)(x)=q-(n+r)αx∑m=0∞‍∑j=0n‍∑l=0n-j‍(nj,l,n-j-l)(n-j+m-1m)(-1)lqα{(j+l)x+m}Gj,χ,q(α,β,r).Secandly, by using generating function of the multiple generalizedq-Genocchi polynomials with weight α and weak weight β, which is defined by (2.11), we obtain the following identities.By using (2.13), we find that
(3.8)Gn+r,χ,q(α,β,r)(x)(n+rr)r!=[2]qβr∑m=0∞‍∑a1,…,ar=0f-1‍(m+r-1m)(-1)∑i=1r‍ai+m×qβ(∑i=1r‍ai+fm)(∏i=1r‍χ(ai))[∑i=1r‍ai+fm+x]qαn=[2]qβr∑a1,…,ar=0f-1‍∑l=0n‍∑a=0l‍(na,l-a,n-l)(-1)a+∑i=1r‍aiq{α(a+n-l)+β}∑i=1r‍ai×(∏i=1r‍χ(ai))[x]qαn-l(1-qα)l(1+qf{α(a+n-l)+β})r.
Thus we have the following theorem.Theorem 3.3.
Letq∈ℂp with |1-q|p<1 and r∈ℕ. Then one has
(3.9)Gn+r,χ,q(α,β,r)(x)(n+rr)r!=[2]qβr∑a1,…,ar=0f-1‍∑l=0n‍∑a=0l‍(na,l-a,n-l)(-1)a+∑i=1r‍aiq{α(a+n-l)+β}∑i=1r‍ai×(∏i=1r‍χ(ai))[x]qαn-l(1-qα)l(1+qf{α(a+n-l)+β})r.By using (2.13), we have
(3.10)Fχ,q(α,β,r)(t,x)=[2]qβrtr∑n=0∞‍∑l=0n‍(nl)(-1)lqαlx(1-q)n∑a1,…,ar=0f-1‍(-1)∑i=1r‍ai×q(αl+β)(∑i=1r‍ai)(∏i=1r‍χ(ai))∑m=0∞‍(m+r-1m)(-qf(αl+β))mtnn!.
Thus we have
(3.11)∑n=0∞‍Gn,χ,q(α,β,r)(x)tnn!=∑n=0∞‍[2]qβrtr∑l=0n‍(nl)(-1)lqαlx(1-qα)-n∑a1,…,ar=0f-1‍(-1)∑i=1r‍ai×q(αl+β)(∑i=1r‍ai)(∏i=1r‍χ(ai))(1+qf(αl+β))-rtnn!.
By comparing the coefficients of both sides of (n+r)!/tn+r in the above, we arrive at the following theorem.Theorem 3.4.
Letq∈ℂp with |1-q|p<1, r∈ℕ. Then one has
(3.12)Gn+r,χ,q(α,β,r)(x)(n+rr)r!=[2]qβr∑l=0n‍(nl)(-1)lqαlx(1-qα)-n∑a1,…,ar=0f-1‍(-1)∑i=1r‍ai×q(αl+β)(∑i=1r‍ai)(∏i=1r‍χ(ai))(1+qf(αl+β))-r.From (2.12), we easily know that
(3.13)∑n=0∞‍Gn,χ,q(α,β,r)(x)tnn!=∑n=0∞‍[2]qβr(n+rr)r!∑k1,…,kr=0∞‍(-1)∑i=1r‍kiqβ∑i=1r‍ki(∏i=1r‍χ(ki))×[x+∑i=1r‍ki]qαntn+r(n+r)!.
From the above, we get the following theorem.Theorem 3.5.
Letr∈ℕ, k∈ℤ+. Then one has
(3.14)G0,χ,q(α,β,r)(x)=G1,χ,q(α,β,r)(x)=⋯=Gr-1,χ,q(α,β,r)(x)=0,Gl+r,χ,q(α,β,r)(x)=[2]qβr(l+rr)r!∑k1,…,kr=0∞‍(-1)∑i=1r‍kiqβ∑i=1r‍ki(∏i=1r‍χ(ki))[x+∑i=1r‍ki]qαl.From (2.13), we have
(3.15)∑n=0∞‍Gn,χ,q(α,β,r)(x)tnn!∑n=0∞‍Gn,χ,q(α,β,s)(x)tnn!=[2]qβr+str+s∑a1,…,ar=0f-1‍∑m=0∞‍(m+r-1m)(-1)∑i=1r‍ai+mqβ(∑i=1r‍ai+fm)×(∏i=1r‍χ(ai))e[∑i=1r‍ai+fm+x]qαt∑b1,…,bs=0f-1‍∑k=0∞‍(k+s-1k)(-1)∑i=1s‍bi+k×qβ(∑i=1s‍bi+fk)(∏i=1s‍χ(bi))e[∑i=1r‍bi+fk+x]qαt.
By using Cauchy product in (3.15), we obtain
(3.16)∑n=0∞‍∑j=0n‍(nj)Gj,χ,q(α,β,r)(x)Gn-j,χ,q(α,β,s)(x)tnn!=[2]qβr+str+s∑n=0∞‍∑j=0n‍∑a1,…,ar=0f-1‍∑b1,…,bs=0f-1‍(j+r-1j)(n-j+s-1n-j)×(-1)∑i=1r‍ai+∑i=1s‍bi+nqβ(∑i=1r‍ai+∑i=1s‍bi+fn)(∏i=1r‍χ(ai))(∏i=1s‍χ(bi))×e[∑i=1r‍ai+fj+x]qαte[∑i=1s‍bi+f(n-j)+x]qαt.
From (3.16), we have
(3.17)∑m=0∞‍(∑j=0m‍(mj)Gj,χ,q(α,β,r)(x)Gm-j,χ,q(α,β,s)(x))tmm!=∑m=0∞‍[2]qβr+str+s∑n=0∞‍∑j=0n‍∑a1,…,ar=0f-1‍∑b1,…,bs=0f-1‍(j+r-1j)(n-j+s-1n-j)×(-1)∑i=1r‍ai+∑i=1s‍bi+nqβ(∑i=1r‍ai+∑i=1s‍bi+fn)(∏i=1r‍χ(ai))(∏i=1s‍χ(bi))×([∑i=1r‍ai+fj+x]qα+[∑i=1s‍bi+f(n-j)+x]qα)mtmm!.
By comparing the coefficients of both sides of tm+r+s/(m+r+s)! in (3.17), we have the following theorem.Theorem 3.6.
Letr∈ℕands∈ℤ+. Then one has
(3.18)∑j=0l+r+s‍(l+r+sj)Gj,χ,q(α,β,r)(x)Gl+r+s-j,χ,q(α,β,s)(x)(l+r+sl)(r+s)!=[2]qβr+s∑n=0∞‍∑j=0n‍∑a1,…,ar=0f-1‍∑b1,…,bs=0f-1‍(j+r-1j)(n-j+s-1n-j)×(-1)∑i=1r‍ai+∑i=1s‍bi+nqβ(∑i=1r‍ai+∑i=1s‍bi+fn)(∏i=1r‍χ(ai))(∏i=1s‍χ(bi))×([∑i=1r‍ai+fj+x]qα+[∑i=1s‍bi+f(n-j)+x]qα)l.Corollary 3.7.
In (3.18) setting s=1, one has
(3.19)∑j=0l+r+1‍(l+r+1j)Gj,χ,q(α,β,r)(x)Gl+r+1-j,χ,q(α,β,1)(x)(l+r+1l)(r+1)!=[2]qβr+1∑n=0∞‍∑j=0n‍∑a1,…,ar=0f-1‍∑b1=0f-1‍(j+r-1j)(-1)∑i=1r‍ai+b1+nqβ(∑i=1r‍ai+b1+fn)×(χ(b1)∏i=1r‍χ(ai))([∑i=1r‍ai+fj+x]qα+[b1+f(n-j)+x]qα)l.By using (2.13) we have the following theorem.Theorem 3.8.
Distribution theorem is as follows:(3.20)Gn+r,χ,q(α,β,r)=[f]qαn[f]-qβr∑a1,…,ar=0f-1‍(-1)∑i=1r‍aiqβ∑i=1r‍ai(∏i=1r‍χ(ai))Gn+r,qf(α,β,r)(a1+⋯+arf),Gn+r,χ,q(α,β,r)(x)=[f]qαn[f]-qβr∑a1,…,ar=0f-1‍(-1)∑i=1r‍aiqβ∑i=1r‍ai(∏i=1r‍χ(ai))Gn+r,qf(α,β,r)(x+a1+⋯+arf).


## 4. Interpolation Function of Multiple Generalizedq-Genocchi Polynomials with Weight α and Weak Weight β

In this section, we see interpolation function of multiple generalizedq-Genocchi polynomials with weak weight α and find some relations.Let us define interpolation function of theGk+r,q(α,β,r)(x) as follows.Definition 4.1.
Letq,s∈ℂ with |q|<1 and 0<x≤1. Then one defines
(4.1)ζχ,q(α,β,r)(s,x)=[2]qβr∑k1,…,kr=0∞‍(-1)∑i=1r‍kiqβ∑i=1r‍ki(∏i=1r‍χ(ki))[x+∑i=1r‍ki]qαs.
We call ζq(α,β,r)(s,x) the multiple generalized Hurwitz type q-zeta funtion.
In (4.1), setting r=1, we have
(4.2)ζχ,q(α,β,1)(s,x)=[2]qβ∑l=0∞‍(-1)lqβlχ(l)[x+l]qαs=ζχ,q(α,β)(s,x).Remark 4.2.
It holds that(4.3)limq→1ζχ,q(α,β,r)(s,x)=2r∑k1,…,kr=0∞‍(-1)∑i=1r‍ki(∏i=1r‍χ(ki))(x+∑i=1r‍ki)s.
Substituting s=-n,n∈ℤ+ into (4.1), then we have,
(4.4)ζχ,q(α,β,r)(-n,x)=[2]qβr∑k1,…,kr=0∞‍(-1)∑i=1r‍kiqβ∑i=1r‍ki(∏i=1r‍χ(ki))[x+∑i=1r‍ki]qαn.Setting (3.14) into the above, we easily get the following theorem.Theorem 4.3.
Letr∈ℕ, n∈ℤ+. Then one has
(4.5)ζχ,q(α,β,r)(-n,x)=Gn+r,χ,q(α,β,r)(x)(n+rr)r!.



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*Source: 179385-2012-09-29.xml*