AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation79530410.1155/2012/795304795304Research ArticleOn the q-Euler Numbers and Polynomials with Weight 0KimT.ChoiJ.SadekIbrahimDivision of General Education-MathematicsKwangwoon UniversitySeoul 139-701Republic of Koreakw.ac.kr20121512012201213102011291120112012Copyright © 2012 T. Kim and J. Choi.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The purpose of this paper is to investigate some properties of q-Euler numbers and polynomials with weight 0. From those q-Euler numbers with weight 0, we derive some identities on the q-Euler numbers and polynomials with weight 0.

1. Introduction

Let p be a fixed odd prime number. Throughout this paper p, p, and p will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of p. The p-adic absolute value is defined by |x|p=1/pr where x=prs/t for s,t with (p,t)=(p,s)=1 and r. In this paper, we assume that α and qp with |1-q|p<1. As well-known definition, the Euler polynomials are defined by2et+1ext=eE(x)t=n=0En(x)tnn!, with the usual convention about replacing En(x) by En(x) (see ).

In this special case, x=0, En(0)=En are called the nth Euler numbers (see ). Recently, the q-Euler numbers with weight α are defined byẼ0,q(α)=1,q(qαẼq(α)+1)n+Ẽn,q(α)=0if  n>0, with the usual convention about replacing (Ẽq(α))n by Ẽn,q(α) (see [3, 12]). The q-number of x is defined by [x]q=(1-qx)/(1-q) (see ). Note that limq1[x]q=x. Let us define the notation of q-Euler numbers with weight 0 as Ẽn,q(0)=Ẽn,q. The purpose of this paper is to investigate some interesting identities on the q-Euler numbers with weight 0.

2. On the Extended <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M40"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Euler Numbers of Higher-Order with Weight 0

Let C(p) be the space of continuous functions on p. For fC(p), the fermionic p-adic q-integral on p is defined by Kim as follows:Iq(f)=Zpf(x)dμ-q(x)=limNq1+qpNx=0pN-1f(x)(-q)x, (see ). By (2.1), we getqnIq(fn)+(-1)n-1Iq(f)=ql=0n-1(-1)n-1-lf(l)ql, where fn(x)=f(x+n) and n (see [4, 5]).

By (1.2), (2.1), and (2.2), we see thatZp[x]qαndμ-q(x)=Ẽn,q(α)=q(1-q)n[α]qnl=0n(nl)(-1)l11+qαl+1.

In the special case, n=1, we getZpextdμ-q(x)=qqet+1=1+q-1et+q-1=n=0Hn(-q-1)tnn!, where Hn(-q-1) are the nth Frobenius-Euler numbers. From (2.4), we note that the q-Euler numbers with weight 0 are given byẼn,q=Zpxndμ-q(x)=Hn(-q-1),for  nZ+.

Therefore, by (2.5), we obtain the following theorem.

Theorem 2.1.

For n+, one has Ẽn,q=Hn(-q-1), where Hn(-q-1) are called the nth Frobenius-Euler numbers.

Let us define the generating function of the q-Euler numbers with weight 0 as follows:F̃q(t)=n=0Ẽn,qtnn!.

Then, by (2.3) and (2.7), we getF̃q(t)=qm=0(-1)mqmemt=1+qqet+1.

Now we define the q-Euler polynomials with weight 0 as follows:n=0Ẽn,q(x)tnn!=1+qqet+1ext.

Thus, (2.4) and (2.9), we getZpe(x+y)tdμ-q(y)=1+qqet+1ext=n=0Ẽn,q(x)tnn!.

From (2.10), we haven=0Ẽn,q(x)tnn!=(1+q-1et+q-1)ext=n=0Hn(-q-1,x)tnn!, where Hn(-q-1,x) are called the nth Frobenius-Euler polynomials (see ).

Therefore, by (2.11), we obtain the following theorem.

Theorem 2.2.

For n+, one has Ẽn,q(x)=Zp(x+y)ndμ-q(x)=Hn(-q-1,x), where Hn(-q-1,x) are called the nth Frobenius-Euler polynomials.

From (2.2) and Theorem 2.2, we note thatqnHm(-q-1,n)+Hm(-q-1)=ql=0n-1(-1)llmql, where n with n1 (mod 2).

Therefore, by (2.13), we obtain the following corollary.

Corollary 2.3.

For n, with n1 (mod 2) and m+, one has qnHm(-q-1,n)+Hm(-q-1)=ql=0n-1(-1)llmql.

In particular, q=1, we get Em(n)+Em=2l=0n-1(-1)llm, where Em and Em(n) are called the mth Euler numbers and polynomials which are defined by2et+1=m=0Emtmm!,2et+1ext=m=0Em(x)tmm!.

By (2.2), we easily see thatqZpf(x+1)dμ-q(x)+Zpf(x)dμ-q(x)=qf(0).

Thus, by (2.16), we getq=qZpe(x+1)tdμ-q(x)+Zpextdμ-q(x)=n=0(qZp(x+1)ndμ-q(x)+Zpxndμ-q(x))tnn!=n=0(qHn(-q-1,1)+Hn(-q-1))tnn!.

Therefore, by (2.16), we obtain the following theorem.

Theorem 2.4.

For n+, one has qHn(-q-1,1)+Hn(-q-1)={1+q,if  n=0,0,if  n>0,

where Hn(-q-1,x) are called the nth Frobenius-Euler polynomials and Hn(-q-1) are called the nth Frobenius-Euler numbers. In particular, q=1, we haveEn(1)+En={2,if  n=0,0,if  n>0, where En are called the nth Euler numbers.

From (2.5) and Theorem 2.2, we note thatẼn,q(x)=Zp(x+y)ndμ-q(y)=l=0n(nl)Zpyldμ-q(y)xn-l=l=0n(nl)Ẽn,qxn-l=(x+Ẽq)n,   where the usual convention about replacing (Ẽq)l by Ẽl,q. By Theorems 2.2 and 2.4, we getqẼn,q(1)+Ẽn,q={q,if  n=0,0,if  n>0.

From (2.20) and (2.21), we haveq(Ẽq+1)n+Ẽn,q={q,if  n=0,0,if  n>0.

For n, by (2.20) and (2.22), we haveq2Ẽn,q(2)=q2(Ẽq+1+1)n=q2l=1n(nl)(Ẽq+1)l+q(1+q-Ẽ0,q)=q+q2-ql=0n(nl)Ẽl,q=q+q2-q(Ẽq+1)n=q+q2+Ẽn,q-qqδ0,n.

Therefore, by (2.23), we obtain the following theorem.

Theorem 2.5.

For n, one has q2Ẽn,q(2)=q+q2+Ẽn,q.

For n+, we haveẼn,q-1(1-x)=Zp(1-x+x1)ndμ-q-1(x1)=(-1)nZp(x1+x)ndμ-q(x1)=(-1)nẼn,q(x).

Therefore, by (2.25), we obtain the following theorem.

Theorem 2.6.

For n+, one has Ẽn,q-1(1-x)=(-1)nẼn,q(x).

From (2.20), we haveZp(1-x)ndμ-q(x)=(-1)nZp(x-1)ndμ-q(x)=(-1)nẼn,q(-1).

By Theorem 2.6 and (2.27), we getZp(1-x)ndμ-q(x)=Ẽn,q-1(2)=1+q+q2Ẽn,q-1if  n>0.

Therefore, by (2.28), we obtain the following theorem.

Theorem 2.7.

For n, one has Zp(1-x)ndμ-q(x)=1+q+q2Ẽn,q-1.

Let C(p) be the space of continuous functions on p. For fC(p), p-adic analogue of Bernstein operator of order n for f is given byBn(fx)=k=0nBk,n(x)f(kn)=k=0nf(kn)(nk)xk(1-x)n-k, where n,k+ (see [1, 6, 7]).

For n,k+, p-adic Bernstein polynomial of degree n is defined byBk,n(x)=(nk)xk(1-x)n-k,xZp (see [1, 6, 7]).

Let us take the fermionic p-adic q-integral on p for one Bernstein polynomials in (2.31) as follows:ZpBk,n(x)dμ-q(x)=(nk)Zpxk(1-x)n-kdμ-q(x)=(nk)l=0n-k(n-kl)(-1)lZpxk+ldμ-q(x)=(nk)l=0n-k(n-kl)(-1)lẼk+l,q.

By simple calculation, we easily getZpBk,n(x)dμ-q(x)=ZpBn-k,n(1-x)dμ-q(x)=(nk)l=0k(kl)(-1)k+lZp(1-x)n-ldμ-q(x)=(nk)l=0k(kl)(-1)k+l(1+q+q2Ẽn-l,q-1)=(nk)l=0k(kl)(-1)k+lq2Ẽn-l,q-1+q(nk)(-1)kδ0,kif  n>k.

Therefore, by (2.32) and (2.33), we obtain the following theorem.

Theorem 2.8.

For n+ with n>k>0, one has l=0n-k(n-kl)(-1)lẼk+l,q=l=0k(kl)(-1)k+lq2Ẽn-l,q-1.

In particular, k=0, we getl=0n(nl)(-1)lẼl,q=q2Ẽn,q-1+q.

By Theorems 2.1 and 2.2, we getl=0n-k(n-kl)(-1)lHk+l(-q-1)=l=0k(kl)(-1)k+lq2Hn-l(-q), where n,k+ with n>k>0.