# Some New Identities on theq-Genocchi Numbers and Polynomials with Weight

Authors: Seog-Hoon Rim; Joohee Jeong
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

We construct a new type ofq-Genocchi numbers and polynomials with
weight α. From these q-Genocchi numbers and polynomials
with weight α, we establish some interesting identities and relations.


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## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this paper, ℤp, ℚp, and ℂp will, respectively denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of ℚp. 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. When one talks of q-extension, q is variously considered as an indeterminate, a complex q∈ℂ, or a p-adic number q. In this paper, we assume that q∈ℂp with |1-q|p<1. As a definition of q-numbers, we use the notation of q-number of
(1.1)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q(cf. [1–11]). Note that limq→1[x]q=x. Let C(ℤp) be the space of continuous functions on ℤp. For f∈C(ℤp), the p-adic invariant integral on ℤp is defined by Kim [1, 3],
(1.2)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡N→∞1[pN]-q∑x=0pN-1f(x)(-q)x.
From (1.2), we have the well-known integral equation
(1.3)qnI-1(fn)+(-1)n-1Iq(f)=[2]q∑l=0n-1(-1)lqlf(l)(see [1, 3]), where fn(x)=f(x+n), (n∈ℕ).Forα∈ℕ, in [11], the q-Genocchi polynomials with weight α are introduced by
(1.4)t∫Zpe[x+y]qαtdμ-q(y)=∑n=0∞G̃n,q(α)(x)tnn!.
By comparing the coefficients of both sides of (1.4), we have
(1.5)G̃0,q(α)(x)=0,G̃n+1,q(α)(x)(n+1)=∫Zp[x+y]qαndμ-q(y),forn∈N.In the special case,x=0, G̃n,q(α)(0)=G̃n,q(α) are called the nth q-Genocchi numbers with weight α.


## 2.q-Genocchi Numbers and Polynomials with Weight α

In this section, we show some new identities on theq-Genocchi numbers and polynomials with weight α. And we establish the distribution relation for q-Genocchi polynomials with weight α.From (1.5), we can easily see that
(2.1)G̃n+1,q(α)(x)n+1=[2]q[α]qn(1-q)n∑l=0n(nl)(-1)lqαlx1+qαl+1.
From (1.5) and (2.1), we note that
(2.2)G̃n+1,q(α)(x)n+1=∫Zp[x+y]qαndμ-1(y)=∑l=0n(nl)[x]qαn-lqαlx∫Zp[y]qαldμ-q(y)=∑l=0n(nl)[x]qαn-lqαlxG̃l+1,q(α)l+1.
Note that (1/(l+1))(nl)=(1/(n+1))(n+1l+1) and from (2.2), we have the relation of polynomials and numbers,
(2.3)G̃n+1,q(α)(x)n+1=1(n+1)∑l=0n(n+1l+1)[x]qαn+1-lqαlxG̃l+1,q(α)=1(n+1)qαx∑l=0n+1(n+1l+1)[x]qαn+1-lqα(l+1)xG̃l+1,q(α)=1(n+1)qαx([x]qα+qαlxG̃q(α))n+1,
with the usual convention of replacing (G̃q(α))n by (G̃n,q(α)).Thus, by (2.3), we have a theorem.Theorem 2.1.
Forα∈ℕ and n∈ℤ+, one has
(2.4)qαxG̃n+1,q(α)(x)=(n+1)qαx([x]qα+qαxG̃q(α))n+1=∑l=0n+1(n+1l)[x]qαn+1-lqαxG̃l,q(α).
In (1.3), if we take n=1,
(2.5)qI-1(f1)+I-1(f)=[2]q.
We applyf(x)=e[x]qαt with (1.5), and we have the following:
(2.6)[2]q=∑n=0∞(q∫Zp[x+1]qαndμ-q(x)+∫Zp[x]qαndμ-q(x))tnn!=∑n=0∞(qG̃n+1,q(α)(1)n+1+G̃n+1,q(α)n+1)tnn!.
By comparing the coefficients on both the sides in (2.6), we get
(2.7)qG̃n+1,q(α)(1)n+1+G̃n+1,q(α)n+1={[2]qifn=0,0ifn>0.
From (2.2) and (2.7), we can derive the following:
(2.8)G̃1,q(α)(1)=1,q1-α(qαG̃q(α)+1)n+G̃n,q(α)=0ifn∈N,
with the usual convention of replacing (G̃q(α))n by G̃n,q(α).
For a fixed odd positive integerd with (p,d)=1, we set
(2.9)X=Xd=lim⁡N⟵⁡ZdpNZ,X1=Zp,X*=⋃0<a<dp,(a,p)=1(a+dpZp),a+dpNZp={x∈X∣x≡a(mod⁡dpN)},
where a∈ℤ satisfies the condition 0≤a<dpN. For the distribution relation for the q-Genocchi polynomials with weight α, we consider the following:
(2.10)∫Zp[x+y]qαndμ-q(y)=∫Zp[n+y]qαxndμ-q(y)=[d]qαn[d]-q∑a=0d-1(-1)aqa∫Zp[x+ad+y]qαdndμ-q(y).
By (1.5) and (2.10), we get a theorem.Theorem 2.2.
Forα∈ℕ and n∈ℤ+, d∈ℕ with d≡1(mod2), one has
(2.11)G̃n+1,q(α)(x)n+1=[d]qαn[d]-q∑a=0d-1(-1)aqaG̃n+1,qd(α)(x+aa).


## 3. Higher-Orderq-Genocchi Numbers and Polynomials with Weight α

In this section, we define higher-orderq-Genocchi polynomials G̃n+1,q(α)(h,k∣x) and numbers G̃n+1,q(α)(h,k) with weight α. We find an integral equation for higher-order q-Genocchi numbers with weight α. And we establish a combination property.Letα∈ℤ and h,k∈ℤ+, for n∈ℤ+, then we define higher-order q-Genocchi polynomials with weight α as follows:
(3.1)G̃n+1,q(α)(h,k∣x)(n+1)=∫Zp⋯∫Zp︸k-times[x1+x2+⋯+xk+x]qαnq(h-1)x1+⋯+(h-k)xkdμ-q(x1)⋯dμ-q(xk)=[2]qk[α]qn(1-q)n∑l=0n(nl)(-1)lqαlx(1+qαl+h)⋯(1+qαl+h-k+1)=[2]qk[α]qn(1-q)m∑l=0n(nl)(-1)lqαlx(-qαl+h:q-1)k,
where (x:q)h=∏i=0h-1(1-xqi).In the special case,x=0, G̃n+1,q(α)(h,k|0)=G̃n+1,q(α)(h,k) are called the (n+1)th (h,k)-Genocchi numbers with weight α.In (3.1), apply the following identity:
(3.2)[x1+x2+⋯+xk]qα(1-qα)+qα(x1+x2+⋯+xk)=1,
and we have a theorem.Theorem 3.1.
Forα∈ℕ and h,k∈ℤ+, one has
(3.3)G̃n+1,q(α)(h,k)n+1=(1-qα)G̃n+2,q(α)(h,k)n+2+G̃n+1,q(α)(h+α,k)n+1.We consider, forα∈ℕ and h,k∈ℤ+,
(3.4)∑j=0i(ij)(qα-1)jG̃n+j-i+1(α)(h-α,k)n+j-i+1=∑j=0i(ij)(qα-1)j∫Zp⋯∫Zp︸k-times[∑l=1kxl]qαn-i-jq∑l=1k(h-α-l)xldμ-q(x1)⋯dμ-q(xk)=∑j=0i(ij)(qα-1)j∫Zp⋯∫Zp︸k-times[∑l=1kxl]qαn-i+jq∑l=1k(h-l)xldμ-q(x1)⋯dμ-q(xk)=∑j=0i(ij)(qα-1)jG̃n+j-i+1(α)(h,k)n+j-i+1.
Therefore, we obtain the following combinatorial property.Theorem 3.2.
Forα∈ℕ and h,k∈ℤ+, one has
(3.5)∑j=0i(ij)(qα-1)jG̃n+j-i+1(α)(h-α,k)n+j-i+1=∑j=0i-1(i-1j)(qα-1)jG̃n+j-i+1(α)(h,k)n+j-i+1.



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*Source: 271784-2011-12-22.xml*