# q-Genocchi Numbers and Polynomials Associated with Fermionic p-Adic Invariant Integrals on ℤp

Authors: Leechae Jang; Taekyun Kim
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

The main purpose of this paper is to present a systemic study of some
families of multiple Genocchi numbers and polynomials. In particular, by using the
fermionicp-adic invariant integral on ℤp, we construct p-adic Genocchi numbers and
polynomials of higher order. Finally, we derive the following interesting formula: Gn+k,q(k)(x)=2kk!(n+kk)∑l=0∞∑d0+d1+⋯+dk=k−1,di∈ℕ(−1)l(l+x)n, where Gn+k,q(k)(x) are the q-Genocchi polynomials of order k.


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## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this
paper, ℤp,ℚp,ℂ, and ℂp will, respectively, denote the ring of p-adic rational 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=pνp(p)=1/p. 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 |q|<1.
If q∈ℂp,
then we assume that |1−q|p<1,
see [1–6].Inℂ,
the ordinary Euler polynomials are defined as (1.1)2et+1ext=∑n=0∞En(x)tnn!,(|t|<π). In the case x=0,En(0)=En are called Euler numbers, see [1–13]. Let δ0,n be the Kronecker symbol. From (1.1) we derive
the following relation: (1.2)E0=1,(E+1)n+En=2δ0,n,n∈ℕ, (cf. [7–13]). Here, we use the technique
method notation by replacing En by En(n≥0),
symbolically. The first few are 1,−1/2,0,1/4,…, and E2k=0 for k=1,2,….
A sequence consisting of the Genocchi numbers Gn satisfies the following
relations: (1.3)2tet+1=∑n=0∞Gntnn!,(|t|<π), see [11, 12]. It satisfies G1=1,G3=G5=G7=⋯=G2k+1=0,k=1,2,3,…, and even coefficients are given by (1.4)G2n=2(1−22n)B2n=2nE2n−1(0), where Bn is Bernoulli numbers. The first few Genocchi
numbers for even integers are −1,1,−3,17,−155,2073,….
The first few prime Genocchi numbers are −3 and 17,
which occur at n=6 and 8.
There are no others with n<105. We now define the Genocchi polynomials as follows: (1.5)2tet+1ext=∑n=0∞Gn(x)tnn!,(|t|<π). Thus, we note
that (1.6)Gn(x)=∑l=0n(nl)Glxn−l.In this paper, we use the following notations:[x]q=(1−qx)/(1−q) and [x]−q=(1+(−q)x)/(1+q).
Let UD(ℤp) be the space of uniformly differentiable
functions on ℤp.
For f∈UD(ℤp),
the p-adic q-deformed fermionic integral on ℤp is defined as (1.7)I−q(f)=∫ℤpf(x)dμ−q(x)=lim⁡N→∞1[pN]q∑x=0pN−1f(x)(−q)x, see [1–4]. The fermionic p-adic invariant integral on ℤp can be obtained as q→1.
That is, (1.8)I−1(f)=lim⁡q→1I−q(f)=∫ℤpf(x)dμ−1(x). From (1.8), we easily derive the
following integral equation related to fermionic invariant p-adic integral on ℤp: (1.9)I−1(f1)+I−1(f)=2f(0), where f1(x)=f(x+1), see [5].The purpose of this paper is to present a systemic
study of some families of multiple Genocchi numbers and polynomials by using
the fermionic multivariatep-adic invariant integral on ℤp.
In addition, we will investigate some interesting identities related to
Genocchi numbers and polynomials.


## 2. Genocchi Numbers Associated with Fermionicp-Adic Invariant Integral on ℤp

From (1.9) we can derive (2.1)t∫ℤpextdμ−1(x)=2tet+1=∑n=0∞Gntnn!,t∫ℤpe(x+y)tdμ−1(y)=2tet+1ext=∑n=0∞Gn(x)tnn!, where Gn(x) are Genocchi polynomials. It is easy to check
that (2.2)t∫ℤpe(2x+1)tdμ−1(x)=2tet+e−t=tsecht=12∑n=0∞(∑l=0n(nl)2lGl)tnn!. By comparing the coefficient on
both sides in (2.1), we easily see that (2.3)∫ℤp(x+y)ndμ−1(x)=Gn+1(x)n+1. Therefore, we obtain the
following proposition.Proposition 2.1.
Fork∈ℤ+,
(i)
∫ℤp(x+y)ndμ−1(x)=Gn+1(x)/(n+1) (Witt's formula for Genocchi polynomials);(ii)
∫ℤpe(2x+1)tdμ−1(x)=(1/(n+1))((1/2)∑l=0n+1(n+1l)2lGl),
where (nl)=(n(n−1)⋯(n−l+1))/l!.Letϑℂp={x∈ℂp∣|x|p≤1} be the integer ring of ℂp.
We note that i=(−1)1/2∈ϑℂp.
By using Taylor expansion, we see that (2.4)eix=∑n=0∞inxnn!=∑n=0∞(−1)nx2n(2n)!+i∑n=0∞(−1)nx2n+1(2n+1)!. In the p-adic number field, sinx and cos⁡x are defined as (2.5)sin⁡x=x1!−x33!+x55!−x77!+⋯,cos⁡x=1−x22!+x44!−x66!+⋯. From (2.4) and (2.5), we
derive (2.6)eix=cos⁡x+isinx. This is equivalent
to (2.7)cos⁡x=eix+e−ix2,sinx=eix−e−ix2i. By (2.7), we easily see
that (2.8)sec⁡t=2eit+e−it=∫ℤpe(2x+1)itdμ−1(x)=∑n=0∞∫ℤp(2x+1)ndμ−1(x)intnn!. It is not difficult to show that ∫ℤp(2x+1)2n+1dμ−1(x)=0 for n∈ℤ+=N∪{0}. From (2.8), we note that (2.9)sec⁡t=∑n=0∞in∫ℤp(2x+1)ndμ−1(x)tnn!=∑n=0∞(−1)n∫ℤp(2x+1)2ndμ−1(x)t2n(2n)!. Thus, we have (2.10)tsec⁡t=12∑n=0∞(−1)n(∑l=02n+1(2n+1l)2lGl)t2n+1(2n+1)!.Now we consider the fermionic multivariatep-adic invariant integral on ℤp as follows: (2.11)tk∫ℤp⋯∫ℤp︸k‐timese(x1+x2+⋯+xk)tdμ−1(x1)⋯dμ−1(xk)=2ktk(et+1)⋯(et+1)︸k‐times=∑n=0∞Gn(k)tnn!, where Gn(k) are the nth Genocchi number of order k.
By comparing the coefficient on both sides in (2.11), we see that G0(k)=G1(k)=⋯=Gn(k)=0,
and (2.12)∫ℤp⋯∫ℤp︸k‐times(x1+x2+⋯+xk)ndμ−1(x1)⋯dμ−1(xk)(n+k)k=Gn+k(k), where (n+k)k is the Jordan factor which is defined by (n+k)k=(n+k)⋯(n+1).
Thus, we note that (2.13)∫ℤp⋯∫ℤp︸k‐times(x1+x2+⋯+xk)ndμ−1(x1)⋯dμ−1(xk)=Gn+k(k)(n+kn)n!, for k∈N,n∈ℤ+.Theorem 2.2.
Forn∈ℤ+,k∈ℕ, (2.14)∫ℤp⋯∫ℤp︸k‐times(x1+x2+⋯+xk)ndμ−1(x1)⋯dμ−1(xk)=Gn+k(k)(n+kn)n!.The multinomial coefficient is well known
as(2.15)(x1+x2+⋯+xk)n=∑l1+⋯+lk=nl1,…,lk>0(nl1,…,lk)x1l1x2l2⋯xklk. Therefore, we obtain the
following corollary.Corollary 2.3.
Forn∈ℤ+, k∈ℕ, (2.16)∑l1+⋯+lk=nl1,…,lk>0(nl1,…,lk)(Gl1+1l1+1)(Gl2+1l2+1)⋯(Glk+1lk+1)=Gn+k(k)(n+kn)n!.Forq∈ℂp with |1−q|<1,
it is not difficult to show that (2.17)t∫ℤpqxextdμ−1(x)=2tqet+1. Now, we define the q-extension of the Genocchi numbers as
follows: (2.18)2tqet+1=∑n=0∞Gn,qtnn!. By (2.17) and (2.18), we easily see
that (2.19)Gn+1,qn+1=∫ℤpqxxndμ−1(x). With the same motivation to
construct the Genocchi polynomials of higher order, we can consider the q-extension of higher-order Genocchi numbers as
follows: (2.20)tk∫ℤp⋯∫ℤp︸k‐timesqx2+2x3+⋯+(k−1)xke(x+x1+x2+⋯+xk)tdμ−1(x1)⋯dμ−1(xk)=tk2k(et+1)(qet+1)⋯(qk−1et+1)︸k‐timesext=∑n=0∞Gn,q(k)tnn!, where Gn,q(k) are the q-Genocchi polynomials of order k.
The basic q-natural numbers are defined
as (2.21)[n]q=1−qn1−q=1+q+q2+⋯+qn−1,n∈N. The q-factorial of n is defined as (2.22)[n]q!=[n]q[n−1]q⋯[2]q[1]q=(1+q+⋯+qn−1)⋯(1+q)⋅1.The q-binomial coefficient is also defined
as (2.23)(nk)q=[n]q![n−k]q![k]q!=[n]q[n−1]q⋯[n−k+1]q[k]q!. Note that lim⁡q→1(nk)q=(nk)=(n(n−1)⋯(n−k+1))/k!. The q-binomial coefficient satisfies the following
recurrsion formula: (2.24)(n+1k)q=(nk−1)q+qk(nk)q=qn−k(nk−1)q+(nk)q. From this recurrsion formula, we
can derive (2.25)(nk)q=∑d0+d1+⋯+dk=k−1di∈Nq0⋅d0+1⋅d1+⋯+k⋅dk. The q-binomial expansion is given
by (2.26)∏i=1n(a+bqi−1)=∑k=0n(nk)qq(k2)an−kbk,∏i=1n(1−bqi−1)−1=∑k=0∞(n+k−1k)qbk.
By (2.20) and (2.26), we see
that
(2.27)tk∫ℤp⋯∫ℤp︸k‐timesqx2+2x3+⋯+(k−1)xke(x+x1+x2+⋯+xk)tdμ−1(x1)⋯dμ−1(xk)=tk2k∏i=1k(1−(−qi−1)et)−1ext=tk2k∑l=0∞(k+l−1l)q(−1)le(l+x)t=tk∑n=0∞(2k∑l=0∞(k+l−1l)q(−1)l(l+x)n)tnn!. Therefore, we obtain the
following theorem.Theorem 2.4.
Forn∈ℤ+,k∈N,
we have (2.28)∫ℤp⋯∫ℤp︸k‐timesqx2+2x3+⋯+(k−1)xk(x1+x2+⋯+xk)ndμ−1(x1)⋯dμ−1(xk)=2k∑l=0∞(k+l−1l)q(−1)l(l+x)n.By (2.20), it is not difficult to show
that (2.29)Gn+k,q(k)(x)=k!(n+kk)∫ℤp⋯∫ℤp︸k‐timesqx2+2x3+⋯+(k−1)xk×(x+x1+x2+⋯+xk)k×dμ−1(x1)⋯dμ−1(xk)(n+k)k,G0,q(k)=G1,q(k)=⋯=Gn,q(k)=0, where n=0,1,2,….
Therefore, we obtain the following corollary.Corollary 2.5.
Forn∈ℤ+,k∈ℕ, (2.30)Gn+k,q(k)(x)k!(n+kk)=2k∑l=0∞(k+l−1l)q(−1)l(l+x)n.Corollary 2.6.
Forn∈ℤ+,k∈ℕ, (2.31)Gn+k,q(k)(x)=2kk!(n+kk)∑l=0∞∑d0+d1+⋯+dk=k−1di∈Nq0⋅d0+1⋅d1+⋯+k⋅dk(−1)l(l+x)n.



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