# On Multiple Generalizedw-Genocchi Polynomials and Their Applications

Authors: Lee-Chae Jang
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

We define the multiple generalizedw-Genocchi polynomials. By using fermionic p-adic invariant integrals, we derive some identities on these generalized w-Genocchi polynomials, for example, fermionic p-adic integral representation, Witt's type formula, explicit formula, multiplication formula, and recurrence formula for these w-Genocchi polynomials.


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## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this paper, ℤ, ℤp, ℚp, ℂ, and ℂp will, respectively, denote the ring of integers, the ring of p-adic integers, the field of p-adic rational numbers, the complex number field, and. the p-adic completion of the algebraic closure of ℚp. Let νp be the normalized exponential valuation of ℂp with |p|p=p-νp(p)=p-1.Theq-basic natural numbers are defined by (1.1)[n]q=1-qn1-q=1+q+q2+⋯+qn-1
for n∈ℕ, and the binomial coefficient is defined as (1.2)(nk)=n!(n-k)!k!=n(n-1)⋯(n-k+1)k!.
The binomial formulas are well known that (1.3)(1-b)n=∑i=0n(ni)(-1)ibi,1(1-b)n=∑i=0n(n+i-1i)bi(see, [1, 2]). When one  talks of q-extension, q is variously considered as an indeterminate, a complex number q∈ℂ, or a p-adic number q∈ℂp. If q∈ℂ, one normally assumes that |q|<1. If q∈ℂp, one normally assumes that |q-1|p<1. We use the notation (1.4)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q,
see [1–13] for all x∈ℤp. Note that limq→1[x]q=x for x∈ℤp in presented p-adic case.LetUD(ℤp) be denoted by the set of uniformly differentiable function on ℤp. For f∈UD(ℤp), an invariant p-adic q-integral on ℤp is defined as (1.5)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡n→∞1+q1+qpN∑x=0pN-1f(x)(-q)x.
Thus, we have the following integral relation: (1.6)lim⁡q→1qI-q(f1)+I-q(f)=(1+q)f(0),
where f1(x)=f(x+1), and the fermionic p-adic invariant integral relation: (1.7)I-1(f)=lim⁡q→1I-q(f)=∫Zpf(x)dμ-1(x),(1.8)I-1(f1)+I(f)=2f(0).Now, we recall that the definitions ofw-Euler polynomials and w-Genocchi polynomials are defined as (1.9)2extwet+1=∑n=0∞En,w(x)tnn!,2textwet+1=∑n=0∞Gn,w(x)tnn!,t∈R,w∈C,
with |1-w|p<1, respectively. In the special case x=0, En,w(0)=En,w, and Gn,w(0)=Gn,w are called w-Euler numbers and w-Genocchi numbers (see [2, 9]).In [13], Bayard and Simsek have studied multiple generalized Bernoulli polynomials as follows: (1.10)∏j=1r(ajt+log⁡(waj)(wet)aj-1)et=∑n=0∞Bn,w(r)(x;a1,…ar)tnn!,|t+log⁡(|w|)|<min⁡⁡{πa1⋯πap},
where a1,…,ar are strictly positive real numbers.The purpose of this paper is to define another construction of multiple generalizedw-Genocchi polynomials and numbers, which are different from multiple generalized Bernoulli polynomials and numbers in [13]. By using fermionic p-adic invariant integrals, we derive some identities on these generalized w-Genocchi polynomials, for example, fermionic p-adic integral representation, Witt's type formula, explicit formulas, multiplication formula, and recurrence formula for these w-Genocchi polynomials.


## 2. Multiple Generalizedw-Genocchi Polynomials and Numbers

Letr∈ℕ and a1,…,ar be strictly positive real numbers. The multiple generalized w-Genocchi polynomials Gn,w(r)(x;a1,…,ar) are defined as (2.1)∏j=1r(2t)r(wet)aj+1ext=∑n=0∞Gn,w(r)(x;a1,…ar)tnn!,fort∈R,w∈C,
where |logw+t|≤min1≤j≤r{π/aj}. The values of Gn,w(r)(x;a1,…,ar) at x=0 are called the multiple generalized w-Genocchi numbers: when r=1, w=1, and aj=0  (j=1,…r), the polynomials or numbers are called the ordinary Genocchi polynomials or numbers.It is known that(2.2)t∫Zpwzet(z+x)dμ-1(z)=2twet+1=∑n=0∞Gn,w(x)tnn!,tr∫Zp⋯∫Zpwz1+z2+⋯+zret(z1+⋯+zr+x)dμ-1(z1)⋯dμ-1(zr)=(2twet+1)rext=∑n=0∞Gn,w(r)(x)tnn!.In fact, let us taket∈ℝ,w∈ℂ, and we apply the above difference integral formula (1.8) for f(z)=wazetaz, then we obtain (2.3)2(wet)a+1etx=∫Zpwazet(az+x)dμ-1(z).
By (2.3), we easily see that (2.4)∏j=1r(2t)r(wet)aj+1ext=tr∫Zp⋯∫Zpwa1z1+⋯+arzret(a1z1+⋯+arzr+x)dμ-1(z1)⋯dμ-1(zr)=∑n=0∞∫Zp⋯∫Zpwa1z1+⋯+arzr(a1z1+⋯+arzr+x)n×dμ-1(z1)⋯dμ-1(zr)tn+rn!,(2.5)G0,w(r)(x;a1,…,ar)=⋯=Gr-1,w(r)(x;a1,…,ar))=0.By (2.4) and (2.5), we obtain the following fermionic p-adic integral representation formula for these numbers.Theorem 2.1 (p-adic integral representation).
Letr∈ℕ and a1,…,ar be strictly positive real numbers. Then one has a fermionic p-adic invariant integral representation for the multiple generalized w-Genocchi polynomials Gn,w(r)(x;a1,…,ar) as follows:
(2.6)Gn+r,w(r)(x;a1,…,ar)r!(n+rr)=∫Zp⋯∫Zpwa1z1+⋯+arzr(a1z1+⋯+arzr+x)ndμ-1(z1)⋯dμ-1(zr)
for n≥r and
(2.7)G0,w(r)(x;a1,…,ar)=⋯=Gr-1,w(r)(x;a1…,ar))=0.We remark that if we setr=1 and a1=1, then we have the following equation: (2.8)Gn+r,w(1)(x;1)1!(n+11)=Gn+1,w(r)(x)n+1=En,w(x).
The generalized w-Genocchi polynomials are given by (2.9)2t(wet)a+1ext=∑n=0∞Gn,w(x;a)tnn!,∫Zpwazet(az+x)dμ-1(z)=∑n=0∞∫Zpwaz(az+x)ndμ-1(z)tn.
By comparing the coefficients on both sides in (2.9), we obtain the following identity on the generalized w-Genocchi polynomials (2.10)Gn,w(x;a)n!=∫Zpwaz(az+x)ndμ-1(z).
Similarly, from (2.4), we can obtain the following Witt's type formula for the multiple generalized w-Genocchi polynomials.Theorem 2.2 (Witt's type formula).
Letr∈ℕ and a1,…,ar be strictly positive real numbers. Then one has
(2.11)Gn,w(r)(x;a1,…,ar)n!=∫Zp⋯∫Zpwa1z1+⋯+arzr(a1z1+⋯+arzr+x)ndμ-1(z1)⋯dμ-1(zr).From (2.4), we can directly calculate the following: (2.12)Gn,w(r)(x;a1,…,ar)=∫Zp⋯∫Zpwa1z1+⋯+arzr(a1z1+⋯+arzr+x)n×dμ-1(z1)⋯dμ-1(zr)n!=∑i=0n(ni)xn-i∫Zp⋯∫Zpwa1z1+⋯+arzr×(a1z1+⋯+arzr)idμ-1(z1)⋯dμ-1(zr)n!=∑i=0n(ni)2(n-i)!xn-iGi,w(r)(a1,…,ar).
From (2.12), we get the following explicit formula.Theorem 2.3 (explicit formula).
Letr∈ℕ and a1,…,ar be strictly positive real numbers. Then one has
(2.13)Gn,w(r)(x;a1,…,ar)=∑i=0n(ni)2(n-i)!xn-iGi,w(r)(a1,…,ar).Next we discuss the multiplication formula for the multiple generalizedw-Genocchi polynomials as follows: (2.14)Gn,w(r)(x;a1,…,ar)=∫Zp⋯∫Zpwa1z1+⋯+arzr(a1z1+⋯+arzr+x)ndμ-1(z1)⋯dμ-1(zr)n!=lim⁡N→∞∑z1,…,zr=0mpN-1wa1z1+⋯+arzr(a1z1+⋯+arzr+x)n(-1)z1+⋯+zr=mn∑t1,…,tr=0m-1wa1t1+⋯+artr(-1)t1+⋯+trlim⁡N→∞∑y1,…,yr=0pN-1(-1)m(y1+⋯+yr)×(wm)a1y1+⋯+aryr(x+a1t1+⋯+artrm+a1y1+⋯+aryr)nn!=mn∑t1,…,tr=0m-1wa1t1+⋯+artr(-1)t1+⋯+trn!∫Zp⋯∫Zp(wm)a1y1+⋯+aryr×(x+a1t1+⋯+artrm+a1y1+⋯+aryr)ndμ-1(y1)⋯dμ-1(yr)=mn∑t1,…,tr=0m-1wa1t1+⋯+artr(-1)t1+⋯+tr×Gn,wn(r)(x+a1t1+⋯+artrm;a1,…,ar).
Thus, we obtain the following multiplication formula for the multiple generalized w-Genocchi polynomials.Theorem 2.4 (multiplication formula).
Letr∈ℕ and a1,…,ar be strictly positive real numbers. For any m∈ℕ, one has
(2.15)Gn,w(r)(mx;a1,…,ar)=mn∑t1,…,tr=0m-1wa1t1+⋯+artr(-1)t1+⋯+tr×Gn,wn(r)(x+a1t1+⋯+artrm;a1,…,ar).Corollary 2.5.
(1) If one setsw=a1=⋯=ar=1 and r,n∈ℕ, then one obtains Raabe type formula for multiple Genocchi polynomials Gn(r)(x) as follows:
(2.16)Gn(r)(mx)=mn∑t1,…,tr=0m-1Gn(r)(x+∑i=1ntim),
where (2t/(et+1))rext=∑n=0∞Gn(r)(x)(tn/n!).
(2) If one setsw=1 and r,n∈ℕ, then one obtains Carlitz's multiplication formula for the multiple generalized Genocchi polynomials Gn(r)(x;a1,…,ar) as follows:
(2.17)Gn(r)(mx;a1,…,ar)=mn∑t1,…,tr=0m-1Gn(r)(x+∑i=1naitim;a1,…,ar),
where ((2t)r/(∏j=1r(eajt+1)))ext=∑n=0∞Gn(r)(mx;a1,…,ar)(tn/n!).Finally, we discuss the recurrence formula for the multiple generalizedw-Genocchi polynomials as follows. Let r∈ℕ and a1,…,ar be strictly positive real numbers. For any k=1,…,r, we can directly derive the following equation: (2.18)∑n=0∞(∑j=0n(nj)Gj,w(k)(x∣a1,…,ak)Gn-j,w(r-k)(ak+1,…,ar))tnn!=∑n=0∞(∑j=0nGj,w(k)(x∣a1,…,ak)tjj!Gn-j,w(r-k)(ak+1,…,ar)tn-j(n-j)!)=∑n=0∞(∑m+l=n,m,l≥0Gm,w(k)(x∣a1,…,ak)tmm!Gl,w(r-k)(ak+1,…,ar)tll!)=(∑m=0∞Gm,w(k)(x∣a1,…,ak)tmm!)(∑l=0∞Gl,w(r-k)(ak+1,…,ar)tll!)=(∏j=1k(2t)k(wet)ja+1ext)(∏j=k+1r(2t)r-k(wet)ja+1)=∏j=1r((2t)r(wet)ja+1ext)=∑n=0∞Gn(r)(mx∣a1,…,ar)tnn!.
By comparing the coefficients on both sides in (2.18), we obtain the recurrence formula for the multiple generalized w-Genocchi polynomials.Theorem 2.6 (recurrence formula).
Letr∈ℕ and a1,…,ar be strictly positive real numbers. For any k=1,…,r, one has
(2.19)Gn(r)(mx∣a1,…,ar)=∑j=0n(nj)Gj,w(k)(x∣a1,…,ak)Gn-j,w(r-k)(ak+1,…,ar).



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