# On the Barnes' Type Related to Multiple Genocchi Polynomials onℤp

Authors: J. Y. Kang; H. Y. Lee; N. S. Jung; C. S. Ryoo
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Using fermionicp-adic invariant integral on ℤp, we construct the Barnes' type multiple Genocchi numbers and polynomials. From those numbers and polynomials,
we derive the twisted Barnes' type multiple Genocchi numbers and polynomials. Moreover,
we will find the Barnes' type multiple Genocchi zeta function.


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## Body

## 1. Introduction

Recently, theoretical physicists have devised ultrametric structures similar to tree-like structures arising in the study of physical systems because the fact is that the physical space may no longer be Archimedean seems plausible to some mathematical physicists at a very small distance. They have looked for construct-related models usingp-adic numbers and p-adic analysis. So, p-adic numbers are used in mathematical physics (in particular string theory, field theory) as well as in other areas of natural sciences which complicated fractal behavior and hierarchical structures. Also, q-Volkenborn integral which is made by Kim is used in the functional equation of the q-zeta function, the q-Stirling numbers, and Mahler theory of integration with respect to the ring ℤp together with Iwasawas p-adic q-L function. So we study the q-Gnocchi numbers and polynomials in q-type of special generating functions (see [1, 2]).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[1–15]).Letd be a fixed integer and let p be a fixed prime number. For any positive integer N, we set
(1.1)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.We say thatf:ℤp→ℂp is uniformly differentiable function at a point a∈ℤp, and we write f∈UD(ℤp) if the difference quotients Φf:ℤp×ℤp→ℂp such that
(1.2)Φf(x,y)=f(x)-f(y)x-y
have a limit f'(a) as (x,y)→(a,a).As well-known definition, the Genocchi polynomials are defined by(1.3)F(t)=2tet+1=eGt=∑n=0∞‍Gntnn!,F(t,x)=2tet+1ext=eG(x)t=∑n=0∞‍Gn(x)tnn!,
with the usual convention of replacing Gn(x) by Gn(x). In the special case, x=0, Gn(0)=Gn are called the nth Genocchi numbers.Forf∈UD(ℤp), Kim defined the q-deformed fermionic p-adic integral on Zp,
(1.4)I-q(f)=∫ℤp‍f(x)dμ-q(x)=limN→∞1[pN]-q∑x=0pN-1‍f(x)(-q)x.
Note that
(1.5)I-1(f)=limq→1I-q(f).
If we take f(x)=tetx, by (1.4), we see that
(1.6)∫ℤp‍tetxdμ-1(x)=2t1+et=∑n=0∞‍Gntnn!.
By the same method, we note that
(1.7)∫ℤp‍tet(x+y)dμ-1(x)=2t1+etetx=∑n=0∞‍Gn(x)tnn!.
And note that
(1.8)∫ℤp‍f(x)dμ-1(x)=∫X‍f(x)dμ-1(x).
Barnes' type multiple zeta function ζN(s,w∣a1,…,aN) depends on the parameters a1,…,aN that will be assumed to be positive. It is defined by
(1.9)ζN(s,w∣a1,…,aN)=∑m1,…,mN=0∞‍(w+m1a1+⋯+mNaN)-s,
for Re(s)>N, Re(w)>0. Barnes showed that ζN had a meromorphic continuation in s (with simple pole only at s=1,…,N) and defined his multiple gamma function ΓN(w) in terms of the s-derivative at s=0, which may be recalled here as follows:
(1.10)Ψn(w∣a1,…,aN)=∂sζN(s,w∣a1…,aN)∣s=0.


## 2. Barnes' Type Multiple Genocchi Polynomials

In this section, we assume thatw1,…,wr∈ℤp. For x∈ℚ+, r∈ℕ, we define Barnes' type multiple Genocchi polynomials as follows:
(2.1)∫ℤp‍⋯∫ℤp‍︸r-timestre(x+w1y1+⋯+wryr)tdμ-1(y1)⋯dμ-1(yr)=2rtr(1+etw1)(1+etw2)⋯(1+etwr)etx=∑n=0∞‍Gn(r)(x∣w1,…,wr)tnn!.
In the special case, x=0, Gn(r)(w1,…,wr)=Gn(r)(0∣w1,…,wr) are called the nth Barnes' type multiple Genocchi numbers.Thus, we have(2.2)G0(r)(x∣w1,…,wr)=G1(r)(x∣w1,…,wr)=⋯=Gr-1(r)(x∣w1,…,wr)=0,Gn+r(r)(x∣w1,…,wr)(n+rr)r!=∫ℤp‍⋯∫ℤp‍︸r-times(x+w1y1+⋯+wryr)ndμ-1(y1)⋯dμ-1(yr).Forr≥1, n>0, we can readily prove the following (2.3) from (2.2):
(2.3)Gn+r(r)(x∣w1,…,wr)=∑l=0n‍(n+rl+r)xn-lGl+r(r)(w1,…,wr),
where (n+rl+r) is a binomial coefficient.Theorem 2.1 (Property of distribution ofGn(r)(x∣w1,…,wr)).
Forr≥1, n>0 and m∈ℕ with m≡1(mod2),
(2.4)Gn(r)(x∣w1,…,wr)=mnmr∑n1,…,nr=0m-1‍(-1)∑i=1rniGn(r)(x+∑i=1rwinim∣w1,…,wr).
Letχ be Dirichlet's character with conductor d∈ℕ with d≡1(mod2):
(2.5)F(r)(t,x∣w1,…,wr)=2rtr∏i=1r(1+etwi)etx=∑n=0∞‍Gn(r)(x∣w1,…,wr)tnn!.
By (2.1) and (2.5), we see that (2.6)∫ℤp‍⋯∫ℤp‍︸r-timestre(x+w1y1+⋯+wryr)tdμ-1(y1)⋯dμ-1(yr)=2rtr∑m1,…,mr=0∞‍(-1)m1+⋯+mre(x+w1m1+⋯+wrmr)t.
From (2.2) and (2.6), we have
(2.7)Gn+r(r)(x∣w1,…,wr)(n+rr)r!=∫ℤp‍⋯∫ℤp‍︸r-times(x+w1y1+⋯+wryr)ndμ-1(y1)⋯dμ-1(yr)=2r∑m1,…,mr=0∞‍(-1)m1+⋯+mr(x+w1m1+⋯+wrmr)n.
By (2.5), we see that
(2.8)Fχ(r)(t,x∣w1,…,wr)=2rtr(∑a=0d-1(-1)aχ(a)ew1at)⋯(∑a=0d-1(-1)aχ(a)ewrat)(1+ew1dt)⋯(1+ewrdt)etx=2rtr∑m1,…,mr=0∞‍(-1)m1+⋯+mr(∏i=1r‍χ(mi))e(x+w1m1+⋯+wrmr)t.
We define the generalized Barnes' type multiple Genocchi polynomials attached toχ as follows: for m≥0 and wi∈ℤp(i=1,2,…,n),
(2.9)G0,χ(r)(x∣w1,…,wr)=G1,χ(r)(x∣w1,…,wr)=⋯=Gr-1,χ(r)(x∣w1,…,wr)=0,Gn+r(r)(x∣w1,…,wr)(n+rr)r!=∫X‍⋯∫X‍︸r-times(∏i=1r‍χ(xi))(x+∑i=1r‍wiyi)ndμ-1(y1)⋯dμ-1(yr).For simple calculation of fermionicp-adic invariant on ℤp, we note the following theorem.Theorem 2.2 (Property of distribution ofGn,χ(r)(x∣w1,…,wr)).
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(2.11)Gn+r,χ(r)(x∣w1,…,wr)(n+rr)r!=2r∑m1,…,mr=0∞‍(-1)∑i=1rmi(∏i=1r‍χ(mi))(x+∑i=1r‍wimi)n.


## 3. Twisted Barnes' Type Multiple Genocchi Polynomials

LetTp=limN→∞CpN=∪N≥0CpN, where CpN={ζ∣ζpN=1} is the cyclic group of order pN. For ε∈Tp, we denote by ϕε:ℤp→ℂp the locally constant function x↦ϕε(x)=εx. If we take f(x)=ϕε(x)etx, then we get
(3.1)∫ℤp‍tϕε(x)etxdμ-1(x)=2t1+εet.
By (3.1), we easily see that
(3.2)∫X‍tϕε(x)χ(x)etxdμ-1(x)=2t∑a=0d-1(-1)aεaχ(a)eat1+εdedt.
From (3.1) and (3.2), we define the twisted Genocchi numbers and the generalized twisted Genocchi numbers attached to χ as follows:
(3.3)2t1+εet=∑n=0∞‍Gn,ϵtnn!,2t∑a=0d-1(-1)aεaχ(a)eat1+εdedt=∑n=0∞‍Gn,χ,εtnn!.
In (3.1), (3.2), and (3.3), we get
(3.4)∫ℤp‍ϕε(x)xndμ-1(x)=Gn+1,εn+1,∫X‍ϕε(x)χ(x)xndμ-1(x)=Gn+1,χ,εn+1.
By using fermionic multivariate p-adic invariant integral on ℤp, we define the twisted Barnes' type multiple Genocchi polynomials as follows:
(3.5)∫ℤp‍⋯∫ℤp‍︸r-timestr(∏i=1r‍ϕε(yi))e(x+w1y1+⋯+wryr)tdμ-1(y1)⋯dμ-1(yr)=2rtr(1+εew1t)⋯(1+εewrt)etx=∑n=0∞‍Gn,ε(r)(x∣w1,…,wr)tnn!,ε∈Tp.Gn,ε(r)(x∣w1,…,wr) are called the twisted Barnes' type multiple Genocchi polynomials. In the special case, x=0, Gn,ε(r)(w1,…,wr)=Gn,ε(r)(0∣w1,…,wr) are called the twisted Barnes' type multiple Genocchi numbers.From (3.5), we can derive the following:
(3.6)∫ℤp‍⋯∫ℤp‍︸r-timestr(∏i=1r‍ϕε(yi))(x+w1y1+⋯+wryr)ndμ-1(y1)⋯dμ-1(yr)=Gn+r,ε(r)(x∣w1,…,wr)(n+rr)r!.Letχ be the primitive Drichlet character with conductor d∈ℕ with d≡1(mod2). Then we define the generalized twisted Barnes' type multiple Genocchi numbers attached to χ:
(3.7)∫ℤp‍⋯∫ℤp‍︸r-timestr(∏i=1r‍ϕε(xi))(∏i=1r‍χ(xi))e(w1x1+⋯+wrxr)tdμ-1(x1)⋯dμ-1(xr)=2rtr(∑l=0d-1(-1)lεlχ(l)ew1lt)⋯(∑l=0d-1(-1)lεlχ(l)ewrlt)(1+εdew1dt)⋯(1+εdewrdt)=∑n=0∞‍Gn,ε,χ(r)(w1,…,wr)tnn!=Fε,χ(r)(t∣w1,…,wr).
From (3.7), we see that
(3.8)Fε,χ(r)(t∣w1,…,wr)=2rtr∑m1,…,mr=0∞‍(-1)∑i=1rmiε∑i=1rmi(∏i=1r‍χ(mi))et(∑i=1rwimi).
By using (3.7), we easily see that
(3.9)Gn,ε,χ(r)(w1,…,wr)(n+rr)r!=∫X‍⋯∫X‍︸r-times(∏i=1r‍ϕε(xi))(∏i=1r‍χ(xi))(w1x1+⋯+wrxr)ndμ-1(x1)⋯dμ-1(xr).Theorem 3.1 (Property of distribution ofGn,ε,χ(r)(x∣w1,…,wr)).
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## 4. Remark

Letw1,…,wr be taken nonnegative in complex plane. We consider the Barnes' type multiple Genocchi zeta function ζ(s,x∣w1,…,wr) as follows:
(4.1)ζ(s,x∣w1,…,wr)=2r∑m1,…,mr=0∞‍(-1)m1+⋯+mr(x+w1m1+⋯+wrmr)s,
where s∈ℂ, and Re(x)>0.By (4.1) and (2.7), we get
(4.2)ζ(-n,x∣w1,…,wr)=Gn+r(r)(x∣w1,…,wr)(n+rr)r!,
where n∈ℤ+.Define(4.3)Lχ(s,x∣w1,…,wr)=2r∑m1,…,mr=0∞‍(-1)m1+⋯+mr(∏i=1rχ(mi))(x+w1m1+⋯+wrmr)s,
where s∈ℂ, and Re(x)>0.From (4.3), we note that
(4.4)Lχ(-n,x∣w1,…,wr)=Gn+r,χ(r)(x∣w1,…,wr)(n+rr)r!,n∈ℤ+.



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*Source: 679495-2012-10-09.xml*