# Some Properties on theq-Euler Numbers and Polynomials

Authors: T. Kim; S.-H. Lee
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We give some new identities onq-Euler numbers and polynomials by using the fermionic p-adic integral on ℤp.


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## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this paper ℤp, ℚp, and ℂp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of ℚp. The p-adic absolute value |·|p is defined by |p|p=1/p. In this paper, we assume that q∈ℂp with |1-q|p<1. As is well known, the fermionic p-adic integral on ℤp is defined by Kim as follows: (1.1)I-1(f)=∫Zpf(x)dμ-1(x)=lim⁡N→∞⁡∑x=0pN-1(-1)xf(x),
where f∈C(ℤp) = the space of continuous functions on ℤp (see [1]).From (1.1), we note that(1.2)I-1(f1)+I-1(f)=2f(0),wheref1(x)=f(x+1).
The q-Euler polynomials are defined by(1.3)2qet+1ext=eEq(x)t=∑n=0∞En,q(x)tnn!,
with the usual convention about replacing Eqn by En,q (see [2, 3]).Let us takef(y)=qyet(x+y). Then, by (1.2), we get(1.4)∫Zpe(x+y)tqydμ-1(y)=2qet+1ext=∑n=0∞En,q(x)tnn!.
By (1.3) and (1.4), we get the Witt’s formula for the q-Euler polynomials as follows:(1.5)En,q(x)=∫Zpqy(x+y)ndμ-1(y),n∈Z+=N∪{0}.
In the special case, x=0, En,q(0)=En,q are called the n-th q-Euler numbers.From (1.3), we can derive the following recurrence relation for the q-Euler numbers En,q:(1.6)E0,q=2[2]q,q(Eq+1)n+En,q=2δ0,n,
with the usual convention about replacing Eqn by En,q (see [4]).By (1.5), we easily see that(1.7)En,q(x)=∑l=0n(nl)xn-l∫Zpqyyldμ-1(y)=∑l=0n(nl)xn-lEl,q,
where (nℓ)=n!/ℓ!(n-ℓ)!=n(n-1)⋯(n-ℓ+1)/ℓ! (see [1, 2, 4–13]). Cohen introduced many interesting and valuable identities related to Euler and Bernoulli numbers and polynomials in his book (see [14]). In [13], Ryoo has introduced the q-Euler numbers and polynomials with weight α, and Simsek et al. have studied q-Euler numbers and polynomials, and they introduced many interesting identities and properties (see [3, 15, 16]). In this paper, we consider the q-Euler numbers and polynomials with weight α=1. By applying the fermionic p-adic integral on ℤp, we derive many not only new but also some interesting identities on the q-extension of Euler numbers and polynomials. In particular, we consider that Theorems 2.5, 2.6, 2.7, and 2.9 are important identities because these identities are closely related to Frobenius-Euler numbers and polynomials. As is well known, Frobenius-Euler numbers and polynomials are important to study p-adic l-functions in the number theory and mathematical physics related to fermionic distributions. In [17], Bayad and Kim have studied some interesting identities and properties on the q-Euler numbers and polynomials associated with Bernstein polynomials. Recently, several authors have studied some properties of q-Euler numbers and polynomials (see [1–19]). The purpose of this paper is to give some interesting new identities for the q-Euler numbers and polynomials by using the fermionic p-adic integral on ℤp and (1.7).


## 2. Some Identities onq-Euler Polynomials

From (1.4), we note that(2.1)∫Zpe(x+y+z)tqzdμ-1(z)=e(x+y)t2qet+1=∑n=0∞En,q(x+y)tnn!.
Thus, by (1.4) and (2.1), we get(2.2)En,q(x+y)=∫Zp(x+y+z)nqzdμ-1(z)=∑l=0n(nl)yn-l∫Zp(x+z)lqzdμ-1(z)=∑l=0n(nl)yn-lEl,q(x),wheren∈Z+.
By (2.2), we get(2.3)En,q(x+y)=∑j=0n(nj)yn-jEj,q(x)=2[2]qyn+∑j=1nnj(n-1j-1)yn-jEj,q(x).
From (2.3), we can derive the following equation (2.4):(2.4)∑j=0n-1(n-1j)yn-1-jEj+1,q(x)j+1=En,q(x+y)-(2/[2]q)ynn.
Therefore, by (2.4), we obtain the following theorem.Theorem 2.1.
Forn∈ℤ+, one has
(2.5)∑j=0n(nj)yn-jEj+1,q(x)j+1=1n+1(En+1,q(x+y)-2[2]qyn+1).Let us replacey by -y in Theorem 2.1. Then we get
(2.6)∑j=0n(nj)(-1)n-jyn-jEj+1,q(x)j+1=1n+1(En+1,q(x-y)-(-1)n+12[2]qyn+1).
Thus, we have(2.7)∑j=0n(nj)(-1)jyn-jEj+1,q(x)j+1=1n+1((-1)nEn+1,q(x-y)+2[2]qyn+1).
Therefore, by Theorem 2.1 and (2.7), we obtain the following corollary.Corollary 2.2.
Forn∈ℤ+, one has
(2.8)∑j=1[n/2](n2j)yn-2jE2j+1,q(x)2j+1=En+1,q(x+y)+(-1)nEn+1,q(x-y)2n+2.From (2.2), we have(2.9)∑j=1n1j(n-1j-1)yn-j(-1)jEj,q(x)=(-1)nEn,q(x-y)-yn(2/[2]q)n.
Therefore, by (2.3) and (2.9), we obtain the following theorem.Theorem 2.3.
Forn∈ℕ, one has
(2.10)∑j=1[(n+1)/2](n2j-1)yn+1-2jE2j,q(x)j=En+1,q(x+y)+(-1)n+1En+1,q(x-y)-(4/[2]q)yn+14n+4.Lettingy=1 in Theorem 2.1, we see that(2.11)q∑j=0n(nj)Ej+1,q(x)j+1=qEn+1,q(x+1)-2q/[2]qn+1,qEn+1,q(x+1)=∑l=0n+1(n+1l)(Eq+1)lxn+1-l=(2-E0,q)xn+1-∑l=1n+1(n+1l)El,qxn+1-l=2xn+1-∑l=0n+1(n+1l)El,qxn+1-l=2xn+1-En+1,q(x).
Therefore, by (2.11), we obtain the following theorem.Theorem 2.4.
Forn∈ℤ+, one has
(2.12)∑j=0n(nj)qEj+1,q(x)j+1=-En+1,qn+1+2xn+1n+1-2[2]q1n+1.Replacingy by 1 and n by 2n in Corollary 2.2, we have(2.13)∑j=0n(2n2j)E2j+1,q(x)2j+1=E2n+1,q(x+1)+E2n+1,q(x-1)4n+2=(1/q)(qE2n+1,q(x+1)+E2n+1,q(x))+(qE2n+1,q(x)+E2n+1,q(x-1))4n+2-E2n+1,q(x)q(4n+2)-qE2n+1,q(x)4n+2=2x2n+1q(4n+2)+2(x-1)2n+14n+2-E2n+1,q(x)q(4n+2)-qE2n+1,q(x)4n+2.
Therefore, by (2.13), we obtain the following theorem.Theorem 2.5.
Forn∈ℤ+, one has
(2.14)∑j=0n(2n2j)E2j+1,q(x)2j+1=x2n+1q(2n+1)+(x-1)2n+12n+1-E2n+1,q(x)q(4n+2)-qE2n+1,q(x)4n+2.Replacingy by 1 and n by 2n in Theorem 2.3, we have(2.15)∑j=1n(2n2j-1)E2j,q(x)j=E2n+1,q(x+1)-E2n+1,q(x-1)8n+4-1(2n+1)[2]q=2x2n+1q(8n+4)-2(x-1)2n+18n+4-1(2n+1)[2]q-E2n+1,q(x)q(8n+4)+qE2n+1,q(x)8n+4=x2n+12q(2n+1)-(x-1)2n+12(2n+1)-1(2n+1)[2]q-E2n+1,q(x)q(8n+4)+qE2n+1,q(x)8n+4.
Therefore, by (2.15), we obtain the following theorem.Theorem 2.6.
Forn∈ℤ+, one has
(2.16)∑l=1n(2n2j-1)E2j,q(x)j=x2n+12q(2n+1)-2(x-1)2n+12(2n+1)-1(2n+1)[2]q-E2n+1,q(x)q(8n+4)+qE2n+1,q(x)8n+4.Replacingy by 1/2 and n by 2n in Theorem 2.3, we get(2.17)∑j=1n(2n2j-1)(12)2n+1-2jE2j(x)j=E2n+1,q(x+1/2)-E2n+1(x-1/2)-(4/[2]q)(1/2)2n+18n+4.
Thus, by (2.17), we get(2.18)∑j=1n(2n2j-1)22jE2j(x)j=22n(E2n+1,q(x+1/2)-E2n+1(x-1/2))4n+2-1(2n+1)[2]q.
Note that(2.19)qE2n+1,q(x+12)=qE2n+1,q(x-12+1)=q∑l=02n+1(2n+1l)(x-12)2n+1-l(Eq+1)l=(x-12)2n+1(2-E0,q)-∑l=12n+1(2n+1l)(x-12)2n+1-lEl,q=2(x-12)2n+1-E2n+1,q(x-12).
Therefore, by (2.18) and (2.19), we obtain the following theorem.Theorem 2.7.
Forn∈ℕ, one has
(2.20)∑j=1n(2n2j-1)22jE2j(x)j=22n(x-1/2)2n+12n+1-[2]q2n4n+2E2n+1,q(x-12)-1(2n+1)[2]q.Replacingy by 1 and n by 2n+1 in Corollary 2.2, we see that(2.21)∑j=0n(2n+12j)E2j+1,q(x)2j+1=E2n+2,q(x+1)-E2n+2,q(x-1)4n+4=((1/q)(qE2n+2,q(x+1)+E2n+2,q(x))4n+4)-(qE2n+2,q(x)+E2n+2,q(x-1)4n+4)-(E2n+2,q(x)q(4n+4)-qE2n+2,q(x)4n+4)=2x2n+2q(4n+4)-2(x-1)2n+24n+4-(E2n+2,q(x)q(4n+4)-qE2n+2,q(x)4n+4).
Therefore, by (2.21), we obtain the following theorem.Theorem 2.8.
Forn∈ℤ+, one has
(2.22)∑j=0n(2n+12j)E2j+1,q(x)2j+1=x2n+2q(2n+2)-(x-1)2n+22n+2-(1-q)[2]qE2n+2,q(x)q(4n+4).Replacingn by 2n+1 and y by 1 in Theorem 2.3, we get(2.23)∑j=1n+1(2n+12j-1)E2j,q(x)j=E2n+2,q(x+1)+E2n+2,q(x-1)8n+8-1(2n+2)[2]q=qE2n+2,q(x+1)+E2n+2,q(x)q(8n+8)+qE2n+2,q(x)+E2n+2,q(x)8n+8-E2n+2,q(x)q(8n+8)-qE2n+2,q(x)8n+8-1(2n+2)[2]q=2x2n+2q(8n+8)+q2(x-1)2n+28n+8-E2n+2,q(x)q(8n+8)-qE2n+2,q(x)8n+8-1(2n+2)[2]q.
Therefore, by (2.23), we obtain the following theorem.Theorem 2.9.
Forn∈ℤ+, one has
(2.24)∑j=1n+1(2n+12j-1)E2j,q(x)j=x2n+2q(4n+4)+q(x-1)2n+24n+4-(1+q2)E2n+2,q(x)8n+8-1(2n+2)[2]q.



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*Source: 284826-2012-05-31.xml*