# A Study on the Fermionicp-Adic q-Integral Representation on ℤp Associated with Weighted q-Bernstein and q-Genocchi Polynomials

Authors: Serkan Araci; Dilek Erdal; Jong Jin Seo
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We consider weightedq-Genocchi numbers and polynomials. We investigated some interesting properties of the weighted q-Genocchi numbers related to weighted q-Bernstein polynomials by using fermionic p-adic integrals on ℤp.


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## Body

## 1. Introduction, Definitions, and Notations

The main motivation of this paper is [1] by Kim, in which he introduced and studied properties of q-Bernoulli numbers and polynomials with weight α. Recently, many mathematicians have studied weighted special polynomials (see [1–5]).This numbers and polynomials are used in not only number theory, complex analysis, and the other branch of mathematics, but also in other parts of thep-adic analysis and mathematical physics. Kurt Hensel (1861–1941) invented the so-called p-adic numbers around the end of the nineteenth century. In spite of their being already one hundred years old, these numbers are still today enveloped in an aura of mystery within scientific community [6] although they have penetrated several mathematical fields such as number theory, algebraic geometry, algebraic topology, analysis, and mathematical physics (see, for details, [6–8]).Thep-adic q-integral (or q-Volkenborn integral) are originally constructed by Kim [9]. The q-Volkenborn integral is used in mathematical physics, for example, the functional equation of the q-zeta function, the q-Stirling numbers, and q-Mahler theory of integration with respect to the ring ℤp together with Iwasawa's p-adic q-L function.Letp be a fixed odd prime number. Throughout this paper, we use the following notations. By ℤp, we denote the ring of p-adic rational integers, ℚ denotes the field of rational numbers, ℚp denotes the field of p-adic rational numbers, and ℂp denotes the completion of algebraic closure of ℚp. Let ℕ be the set of natural numbers and ℕ*=ℕ∪{0}. The p-adic absolute value is defined by |p|p=1/p. In this paper, we assume |q-1|p<1 as an indeterminate. In [10–12], let UD (ℤp) be the space of uniformly differentiable functions on ℤp. For f∈ UD (ℤp), the fermionic p-adic q-integral on ℤp is defined by Kim(1.1)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡N→∞1[pN]-q∑x=0pN-1qxf(x)(-1)x.Forα,k,n∈ℕ* and x∈[0,1], Kim et al. defined weighted q-Bernstein polynomials as follows:(1.2)Bk,n(α)(x,q)=(nk)[x]qαk[1-x]q-αn-k,
(see [13, 14]). When we put q→1 and α=1 in (1.2), [x]qαk→xk,[1-x]q-αn-k→(1-x)n-k, and we obtain the classical Bernstein polynomials (see [13, 14]), where [x]q is a q-extension of x which is defined by(1.3)[x]q=1-qx1-q,
(see [1–4, 7, 9–12, 14–26]). Note that limq→1[x]q=x.In [3], For n∈ℕ*, S. Araci et al. defined weighted q-Genocchi polynomials as follows:(1.4)G̃n+1,q(α)(x)n+1=∫Zp[x+y]qαndμ-q(y)=[2]q(1-qα)n∑l=0n(nl)(-1)lqαlx11+qαl+1=[2]q∑m=0∞(-1)mqm[m+x]qαn.In the special case,x=0, G̃n,q(α)(0)=G̃n,q(α)  are called the q-Genocchi numbers with weight α.In [3], For α∈ℕ* and n∈ℕ, S. Araci et al. defined q-Genocchi numbers with weight α as follows:(1.5)G̃0,q(α)=0,qG̃n,q(α)(1)+G̃n,q(α)={[2]q,ifn=0,0,ifn≠0.In this paper we obtained some relations between the weightedq-Bernstein polynomials and the q-Genocchi numbers. From these relations, we derive some interesting identities on the q-Genocchi numbers and polynomials with weight α.


## 2. On the Weightedq-Genocchi Numbers and Polynomials

By the definition ofq-Genocchi polynomials with weight α, we easily get (2.1)G̃n+1,q(α)(x)n+1=∫Zp[x+y]qαndμ-q(y)=∫Zp([x]qα+qαx[y]qα)ndμ-q(y)=∑k=0n(nk)[x]qαn-kqαkx∫Zp[y]qαkdμ-q(y)=∑k=0n(nk)[x]qαn-kqαkxG̃k+1,q(α)k+1.Therefore, we obtain the following theorem.Theorem 2.1.
Forn,α∈ℕ*, one has
(2.2)G̃n,q(α)(x)=q-αx∑k=0n(nk)qαkxG̃k,q(α)[x]qαn-k,
with usual convention about replacing (G̃q(α))n by G̃n,q(α).By Theorem2.1, we have(2.3)G̃n,q(α)(x)=q-αx(qαxG̃q(α)+[x]qα)n.By (1.4), we get (2.4)G̃n+1,q-1(α)(1-x)n+1=∫Zp[1-x+y]q-αndμ-q(y)=[2]q-1(1-q-α)n∑l=0n(nl)q-αl(1-x)(-1)l11+q-αl-1=(-1)nqαn[2]q(1-qα)n∑l=0n(nl)(-1)lqαlx11+qαl+1=(-1)nqαnG̃n+1,q(α)(x)n+1.Therefore, we obtain the following theorem.Theorem 2.2.
Forn,α∈ℕ*, one has
(2.5)G̃n,q-1(α)(1-x)=(-1)n-1qα(n-1)G̃n,q(α)(x).From (1.5) and Theorem 2.1, we have the following theorem.Theorem 2.3.
Forn,α∈ℕ*, one has
(2.6)G̃0,q(α)=0,q1-α(qαG̃q(α)+1)n+G̃n,q(α)={[2]q,ifn=0,0,ifn≠0,
with usual convention about replacing (Gq(α))n by Gn,q(α).Forn,α∈ℕ, by Theorem 2.3, we note that (2.7)q2αG̃n,q(α)(2)=(qα(qαG̃q(α)+1)+1)n=∑k=0n(nk)qkα(qαG̃q+1)k=nq2α-1([2]q-G̃1,q(α))-qα-1∑k=2n(nk)qkαG̃k,q(α)=qα-1[2]q-qα-1∑k=1n(nk)qαkG̃k,q(α)=qα-1[2]q+q2α-2G̃n,q(α)ifn>1.Therefore, we have the following theorem.Theorem 2.4.
Forn∈ℕ, one has
(2.8)G̃n,q(α)(2)=[2]qqα+1+1q2G̃n,q(α).From Theorem2.2 and (2.5), we see that(2.9)(n+1)∫Zp[1-x]q-αndμ-q(x)=(n+1)(-1)nqnα∫Zp[x-1]qαndμ-q(x)=(-1)nqnαG̃n+1,q(α)(-1)=G̃n+1,q-1(α)(2).Therefore, we get the following theorem.Theorem 2.5.
Forn,α∈ℕ*, one has
(2.10)(n+1)∫Zp[1-x]q-αndμ-q(x)=G̃n+1,q-1(α)(2).Letn,α∈ℕ. By Theorems 2.4 and 2.5, we get(2.11)(n+1)∫Zp[1-x]q-αndμ-q(x)=qα[2]q+q2G̃n+1,q-1(α).From (2.11), we get the following corollary.Corollary 2.6.
Forn,α∈ℕ*, one has
(2.12)∫Zp[1-x]q-αndμ-q(x)=qαn+1[2]q+q2G̃n+1,q-1(α)n+1.


## 3. Novel Identities on the Weightedq-Genocchi Numbers

In this section, we derive concerning the some interesting properties ofq-Genocchi numbers via the p-adic q-integral on ℤp, in the sense of fermionic and weighted q-Bernstein polynomials.(3.1)Bk,n(α)(x,q)=(nk)[x]qαk[1-x]q-αn-k,wheren,k,α∈N*.By (3.1), Kim et al. get the symmetry of q-Bernstein polynomials weighted α as follows:(3.2)Bk,n(α)(x,q)=Bn-k,n(α)(1-x,q-1),
(see [4]). Thus, from Corollary 2.6, (3.1), and (3.2), we see that(3.3)∫ZpBk,n(α)(x,q)dμ-q(x)=∫ZpBn-k,n(α)(1-x,q-1)dμ-q(x)=(nk)∑l=0k(kl)(-1)k+l∫Zp[1-x]q-αn-ldμ-q(x)=(nk)∑l=0k(kl)(-1)k+l(qα[2]qn-l+1+q2G̃n-l+1,q-1(α)n-l+1).Forn,k∈ℕ* and α∈ℕ with n>k, we obtain(3.4)∫ZpBk,n(x)dμ-q(x)=(nk)∑l=0k(kl)(-1)k+l(qα[2]qn-l+1+q2G̃n-l+1,q-1(α)n-l+1)={qα[2]qn+1+q2G̃n+1,q-1(α)n+1,ifk=0,∑l=0k(kl)(-1)k+l(qα[2]qn-l+1+q2G̃n-l+1,q-1(α)n-l+1),ifk≠0.Let us take the fermionicp-adic q-integral on ℤp for the weighted q-Bernstein polynomials of degree n as follows:(3.5)∫ZpBk,n(α)(x,q)dμ-q(x)=(nk)∫Zp[x]qαk[1-x]q-αn-kdμ-q(x)=(nk)∑l=0n-k(n-kl)(-1)lG̃l+k+1,q(α)l+k+1.Therefore, by (3.4) and (3.5), we obtain the following theorem.Theorem 3.1.
Forn,k∈ℕ* and α∈ℕ with n>k, one has
(3.6)∑l=0n-k(n-kl)(-1)lG̃l+k+1,q(α)l+k+1={q2[2]qn+1+q2G̃n+1,q-1(α)n+1,ifk=0,∑l=0k(kl)(-1)k+l(qα[2]qn-l+1+q2G̃n-l+1,q-1(α)n-l+1),ifk≠0.Letn1,n2,k∈ℕ* and α∈ℕ with n1+n2>2k.   Then, we get(3.7)∫ZpBk,n1(α)(x,q)Bk,n2(α)(x,q)dμ-q(x)=(n1k)(n2k)∑l=02k(2kl)(-1)2k+l∫Zp[1-x]q-αn1+n2-ldμ-q(x)=((n1k)(n2k)∑l=02k(2kl)(-1)2k+l(qα[2]qn1+n2-l+1+q2G̃n1+n2-l+1,q-1(α)n1+n2-l+1))={qα[2]qn1+n2+1+q2G̃n1+n2+1,q-1(α)n1+n2+1,ifk=0,(n1k)(n2k)∑l=02k(2kl)(-1)2k+l(qα[2]qn1+n2-l+1+q2G̃n1+n2-l+1,q-1(α)n1+n2-l+1),ifk≠0.Therefore, we obtain the following theorem.Theorem 3.2.
Forn1,n2,k∈ℕ* and α∈ℕ with n1+n2>2k, one has
(3.8)∫ZpBk,n1(α)(x,q)Bk,n2(α)(x,q)dμ-q(x)={qα[2]qn1+n2+1+q2G̃n1+n2+1,q-1(α)n1+n2+1,ifk=0,(n1k)(n2k)∑l=02k(2kl)(-1)2k+l(qα[2]qn1+n2-l+1+q2G̃n1+n2-l+1,q-1(α)n1+n2-l+1),ifk≠0.From the binomial theorem, we can derive(3.9)∫ZpBk,n1(α)(x,q)Bk,n2(α)(x,q)dμ-q(x)=∏i=12(nik)∑l=0n1+n2-2k(n1+n2-2kl)(-1)l∫Zp[x]qα2k+ldμ-q(x)=∏i=12(nik)∑l=0n1+n2-2k(n1+n2-2kl)(-1)lG̃l+2k+1,q(α)l+2k+1.Thus, for Theorem3.4 and (3.13), we can obtain the following corollary.Corollary 3.3.
Forn1,n2,k∈ℕ* and α∈ℕ with n1+n2>2k, one has
(3.10)∑l=0n1+n2-2k(n1+n2-2kl)(-1)lG̃l+2k+1,q(α)l+2k+1={qα[2]qn1+n2+1+q2G̃n1+n2+1,q-1(α)n1+n2+1,ifk=0,∑l=02k(2kl)(-1)2k+l(qα[2]qn1+n2-l+1+q2G̃n1+n2-l+1,q-1(α)n1+n2-l+1),ifk≠0.Forx∈ℤp and s∈ℕ with s≥2, let n1,n2,…,ns,k∈ℕ* and α∈ℕ with ∑l=1snl>sk. Then, we take the fermionic p-adic q-integral on ℤp for the weighted q-Bernstein polynomials of degree n as follows:(3.11)∫ZpBk,n1(α)(x,q)Bk,n2(α)(x,q)⋯Bk,ns(α)(x,q)︸s-timesdμ-q(x)=∏i=1s(nik)∫Zp[x]qαsk[1-x]q-αn1+n2+⋯+ns-skdμ-q(x)=∏i=1s(nik)∑l=0sk(skl)(-1)l+sk∫Zp[1-x]q-αn1+n2+⋯+ns-skdμ-q(x)={qα[2]qn1+n2+⋯+ns+1+q2G̃n1+n2+⋯+ns+1,q-1(α)n1+n2+⋯+ns+1,ifk=0,∏i=1s(nik)∑l=0sk(skl)(-1)sk+l×(qα[2]qn1+n2+⋯+ns-l+1+q2G̃n1+n2+⋯+ns-l+1,q-1(α)n1+n2+⋯+ns-l+1),ifk≠0.Therefore, we obtain the following theorem.Theorem 3.4.
Fors∈ℕ with s≥2, let n1,n2,…,ns,k∈ℕ* and α∈ℕ with ∑l=1snl>sk. Then, one has
(3.12)∫Zp∏i=1sBk,ni(α)(x)dμ-q(x)={qα[2]qn1+n2+⋯+ns+1q2G̃n1+n2+⋯+ns+1,q-1(α)n1+n2+⋯+ns-l+1,ifk=0,∏i=1s(nik)∑l=0sk(skl)(-1)sk+l(qα[2]qn1+n2-l+1+q2G̃n1+n2-l+1,q-1(α)n1+n2-l+1),ifk≠0.From the definition of weightedq-Bernstein polynomials and the binomial theorem, we easily get(3.13)∫ZpBk,n1(α)(x,q)Bk,n2(α)(x,q)⋯Bk,ns(α)(x,q)︸s-timesdμ-q(x)=∏i=1s(nik)∑l=0n1+⋯+ns-sk(∑d=1s(nd-k)l)(-1)l∫Zp[x]qαsk+ldμ-q(x)=∏i=1s(nik)∑l=1n1+⋯+ns-sk(∑d=1s(nd-k)l)(-1)lG̃l+sk+1,q(α)l+sk+1.Therefore, from (3.13) and Theorem 3.4, we have the following corollary.Corollary 3.5.
Fors∈ℕ with s≥2, let n1,n2,…,ns,k∈ℕ* and α∈ℕ with ∑l=1snl>sk. One has
(3.14)∑l=0n1+⋯+ns-sk(∑d=1s(nd-k)l)(-1)lG̃l+sk+1,q(α)l+sk+1={qα[2]qn1+n2+⋯+ns+1+q2G̃n1+n2+⋯+ns+1,q-1(α)n1+n2+⋯+ns+1,ifk=0,∑l=0sk(skl)(-1)sk+l(qα[2]qn1+n2+⋯+ns-l+1+q2G̃n1+n2+⋯+ns-l+1,q-1(α)n1+n2+⋯+ns-l+1),ifk≠0.



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