Some Properties on the 𝑞-Euler Numbers and Polynomials

Abstract

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

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 will denote the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, and the completion of the algebraic closure of ℚ_𝑝. The 𝑝-adic absolute value |⋅|_𝑝 is defined by |𝑝|_𝑝=1/𝑝. In this paper, we assume that 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1. As is well known, the fermionic 𝑝-adic integral on ℤ_𝑝 is defined by Kim as follows: 𝐼₋₁𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)=lim^𝑝_𝑁→∞^𝑁−1_𝑥=0(−1)^𝑥𝑓(𝑥), (1.1) where 𝑓∈𝐶(ℤ_𝑝) = the space of continuous functions on ℤ_𝑝 (see [1]).

From (1.1), we note that𝐼₋₁𝑓₁+𝐼₋₁𝑓=2𝑓(0),where𝑓₁(𝑥)=𝑓(𝑥+1). (1.2) The 𝑞-Euler polynomials are defined by2𝑞𝑒^𝑡𝑒+1^𝑥𝑡=𝑒^{𝐸𝑞(𝑥)𝑡}=^∞_𝑛=0𝐸_𝑛,𝑞(𝑥)𝑡^𝑛,𝑛! (1.3) with the usual convention about replacing 𝐸^𝑛_𝑞 by 𝐸_𝑛,𝑞 (see [2, 3]).

Let us take 𝑓(𝑦)=𝑞^𝑦𝑒^{𝑡(𝑥+𝑦)}. Then, by (1.2), we get_ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑞^𝑦𝑑𝜇₋₁𝑦=2𝑞𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_𝑛,𝑞(𝑥)𝑡^𝑛.𝑛! (1.4) By (1.3) and (1.4), we get the Witt’s formula for the 𝑞-Euler polynomials as follows:𝐸_𝑛,𝑞(𝑥)=_ℤ𝑝𝑞^𝑦𝑥+𝑦^𝑛𝑑𝜇₋₁𝑦,𝑛∈ℤ₊=ℕ∪{0}. (1.5) In the special case, 𝑥=0, 𝐸_𝑛,𝑞(0)=𝐸_𝑛,𝑞 are called the 𝑛-th 𝑞-Euler numbers.

From (1.3), we can derive the following recurrence relation for the 𝑞-Euler numbers 𝐸_𝑛,𝑞:𝐸_0,𝑞=2[2]_𝑞,𝑞𝐸_𝑞+1^𝑛+𝐸_𝑛,𝑞=2𝛿_0,𝑛, (1.6) with the usual convention about replacing 𝐸^𝑛_𝑞 by 𝐸_𝑛,𝑞 (see [4]).

By (1.5), we easily see that𝐸_𝑛,𝑞(𝑥)=^𝑛_ℓ=0𝑛ℓ𝑥^𝑛−ℓ_ℤ𝑝𝑞^𝑦𝑦^ℓ𝑑𝜇₋₁𝑦=^𝑛_ℓ=0𝑛ℓ𝑥^𝑛−ℓ𝐸_ℓ,𝑞, (1.7) where (^𝑛_ℓ)=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 ℤ_𝑝, 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 ℤ_𝑝 and (1.7).

2. Some Identities on 𝑞-Euler Polynomials

From (1.4), we note that_ℤ𝑝𝑒^{(𝑥+𝑦+𝑧)𝑡}𝑞^𝑧𝑑𝜇₋₁(𝑧)=𝑒^{(𝑥+𝑦)𝑡}2𝑞𝑒^𝑡=+1^∞_𝑛=0𝐸_𝑛,𝑞𝑡𝑥+𝑦^𝑛.𝑛! (2.1) Thus, by (1.4) and (2.1), we get𝐸_𝑛,𝑞𝑥+𝑦=_ℤ𝑝𝑥+𝑦+𝑧^𝑛𝑞^𝑧𝑑𝜇₋₁(𝑧)=^𝑛_ℓ=0𝑛ℓ𝑦^𝑛−ℓ_ℤ𝑝(𝑥+𝑧)^ℓ𝑞^𝑧𝑑𝜇₋₁(𝑧)=^𝑛_ℓ=0𝑛ℓ𝑦^𝑛−ℓ𝐸_ℓ,𝑞(𝑥),where𝑛∈ℤ₊. (2.2) By (2.2), we get𝐸_𝑛,𝑞𝑥+𝑦=^𝑛_𝑗=0𝑛𝑗𝑦^𝑛−𝑗𝐸_𝑗,𝑞(𝑥)=2[2]_𝑞𝑦^𝑛+^𝑛_𝑗=1𝑛𝑗𝑦𝑛−1𝑗−1^𝑛−𝑗𝐸_𝑗,𝑞(𝑥). (2.3) From (2.3), we can derive the following equation (2.4):^{𝑛−1𝑗=0}𝑗𝑦𝑛−1_{𝑛−1−𝑗}𝐸_𝑗+1,𝑞(𝑥)=𝐸𝑗+1^𝑛,𝑞𝑥+𝑦−2/[2]_𝑞𝑦^𝑛𝑛. (2.4) Therefore, by (2.4), we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊, one has ^𝑛_𝑗=0𝑛𝑗𝑦^𝑛−𝑗𝐸_𝑗+1,𝑞(𝑥)=1𝑗+1𝐸𝑛+1^𝑛+1,𝑞−2𝑥+𝑦[2]_𝑞𝑦^𝑛+1. (2.5)

Let us replace 𝑦 by −𝑦 in Theorem 2.1. Then we get ^𝑛_𝑗=0𝑛𝑗(−1)^𝑛−𝑗𝑦^𝑛−𝑗𝐸_𝑗+1,𝑞(𝑥)=1𝑗+1𝐸𝑛+1^𝑛+1,𝑞−()𝑥−𝑦−1^𝑛+12[2]_𝑞𝑦^𝑛+1. (2.6) Thus, we have^𝑛_𝑗=0𝑛𝑗(−1)^𝑗𝑦^𝑛−𝑗𝐸_𝑗+1,𝑞(𝑥)=1𝑗+1()𝑛+1−1^𝑛𝐸_𝑛+1,𝑞+2𝑥−𝑦[2]_𝑞𝑦^𝑛+1. (2.7) Therefore, by Theorem 2.1 and (2.7), we obtain the following corollary.

Corollary 2.2. For 𝑛∈ℤ₊, one has ^[𝑛/2]_𝑗=1𝑛𝑦2𝑗^𝑛−2𝑗𝐸_2𝑗+1,𝑞(𝑥)=𝐸2𝑗+1^𝑛+1,𝑞𝑥+𝑦+(−1)^𝑛𝐸_𝑛+1,𝑞𝑥−𝑦.2𝑛+2 (2.8)

From (2.2), we have^𝑛_𝑗=11𝑗𝑦𝑛−1𝑗−1^𝑛−𝑗(−1)^𝑗𝐸_𝑗,𝑞(𝑥)=(−1)^𝑛𝐸_𝑛,𝑞𝑥−𝑦−𝑦^𝑛2/[2]_𝑞𝑛. (2.9) Therefore, by (2.3) and (2.9), we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℕ, one has ^[(𝑛+1)/2]_𝑗=1𝑛𝑦2𝑗−1^{𝑛+1−2𝑗}𝐸_2𝑗,𝑞(𝑥)𝑗=𝐸_𝑛+1,𝑞𝑥+𝑦+(−1)^𝑛+1𝐸_𝑛+1,𝑞𝑥−𝑦−4/[2]_𝑞𝑦^𝑛+1.4𝑛+4 (2.10)

Letting 𝑦=1 in Theorem 2.1, we see that𝑞^𝑛_𝑗=0𝑛𝑗𝐸_𝑗+1,𝑞(𝑥)=𝑗+1𝑞𝐸_𝑛+1,𝑞(𝑥+1)−2𝑞/[2]_𝑞,𝑛+1𝑞𝐸_𝑛+1,𝑞(𝑥+1)=^𝑛+1_ℓ=0ℓ𝐸𝑛+1_𝑞+1^ℓ𝑥^𝑛+1−ℓ=2−𝐸_0,𝑞𝑥^𝑛+1−^𝑛+1_ℓ=1ℓ𝐸𝑛+1_ℓ,𝑞𝑥^𝑛+1−ℓ=2𝑥^𝑛+1−^𝑛+1_ℓ=0ℓ𝐸𝑛+1_ℓ,𝑞𝑥^𝑛+1−ℓ=2𝑥^𝑛+1−𝐸_𝑛+1,𝑞(𝑥). (2.11) Therefore, by (2.11), we obtain the following theorem.

Theorem 2.4. For 𝑛∈ℤ₊, one has ^𝑛_𝑗=0𝑛𝑗𝑞𝐸_𝑗+1,𝑞(𝑥)𝐸𝑗+1=−^𝑛+1,𝑞+𝑛+12𝑥^𝑛+1−2𝑛+1[2]_𝑞1.𝑛+1 (2.12)

Replacing 𝑦 by 1 and 𝑛 by 2𝑛 in Corollary 2.2, we have2𝑛𝐸^𝑛_𝑗=02𝑗^2𝑗+1,𝑞(𝑥)=𝐸2𝑗+1_2𝑛+1,𝑞(𝑥+1)+𝐸_2𝑛+1,𝑞(𝑥−1)=4𝑛+21/𝑞𝑞𝐸_2𝑛+1,𝑞(𝑥+1)+𝐸_2𝑛+1,𝑞(𝑥)+𝑞𝐸_2𝑛+1,𝑞(𝑥)+𝐸_2𝑛+1,𝑞(𝑥−1)−𝐸4𝑛+2_2𝑛+1,𝑞(𝑥)𝑞()𝐸4𝑛+2−𝑞^2𝑛+1,𝑞(𝑥)=4𝑛+22𝑥^2𝑛+1𝑞()+2()4𝑛+2𝑥−1^2𝑛+1−𝐸4𝑛+2^2𝑛+1,𝑞(𝑥)𝑞()𝐸4𝑛+2−𝑞^2𝑛+1,𝑞(𝑥).4𝑛+2 (2.13) Therefore, by (2.13), we obtain the following theorem.

Theorem 2.5. For 𝑛∈ℤ₊, one has 2𝑛𝐸^𝑛_𝑗=02𝑗^2𝑗+1,𝑞(𝑥)=𝑥2𝑗+1𝑞(^2𝑛+1)+()2𝑛+1𝑥−1^2𝑛+1−𝐸2𝑛+1^2𝑛+1,𝑞(𝑥)𝑞()𝐸4𝑛+2−𝑞^2𝑛+1,𝑞(𝑥).4𝑛+2 (2.14)

Replacing 𝑦 by 1 and 𝑛 by 2𝑛 in Theorem 2.3, we have2𝑛𝐸^𝑛_𝑗=12𝑗−1^2𝑗,𝑞(𝑥)𝑗=𝐸_2𝑛+1,𝑞(𝑥+1)−𝐸_2𝑛+1,𝑞(𝑥−1)−18𝑛+4(2𝑛+1)[2]_𝑞=2𝑥^2𝑛+1𝑞()−2()8𝑛+4𝑥−1^2𝑛+1−18𝑛+4(2𝑛+1)[2]_𝑞−𝐸_2𝑛+1,𝑞(𝑥)𝑞()+8𝑛+4𝑞𝐸_2𝑛+1,𝑞(𝑥)=𝑥8𝑛+4_2𝑛+1()−()2𝑞2𝑛+1𝑥−1^2𝑛+12()−12𝑛+1(2𝑛+1)[2]_𝑞−𝐸_2𝑛+1,𝑞(𝑥)𝑞()+8𝑛+4𝑞𝐸_2𝑛+1,𝑞(𝑥).8𝑛+4 (2.15) Therefore, by (2.15), we obtain the following theorem.

Theorem 2.6. For 𝑛∈ℤ₊, one has 2𝑛𝐸^𝑛_ℓ=12𝑗−1^2𝑗,𝑞(𝑥)𝑗=𝑥^2𝑛+1()−2()2𝑞2𝑛+1𝑥−1^2𝑛+12()−12𝑛+1(2𝑛+1)[2]_𝑞−𝐸_2𝑛+1,𝑞(𝑥)𝑞()𝐸8𝑛+4+𝑞^2𝑛+1,𝑞(𝑥).8𝑛+4 (2.16)

Replacing 𝑦 by 1/2 and 𝑛 by 2𝑛 in Theorem 2.3, we get2𝑛1^𝑛_𝑗=12𝑗−12^{2𝑛+1−2𝑗}𝐸_2𝑗(𝑥)𝑗=𝐸_2𝑛+1,𝑞(𝑥+1/2)−𝐸_2𝑛+1(𝑥−1/2)−4/[2]_𝑞(1/2)^2𝑛+1.8𝑛+4 (2.17) Thus, by (2.17), we get^𝑛_𝑗=122𝑛2𝑗−1^2𝑗𝐸_2𝑗(𝑥)𝑗=2^2𝑛𝐸_2𝑛+1,𝑞(𝑥+1/2)−𝐸_2𝑛+1(𝑥−1/2)−14𝑛+2(2𝑛+1)[2]_𝑞. (2.18) Note that𝑞𝐸_2𝑛+1,𝑞1𝑥+2=𝑞𝐸_2𝑛+1,𝑞1𝑥−2+1=𝑞^2𝑛+1_ℓ=0ℓ2𝑛+11𝑥−2^{2𝑛+1−ℓ}𝐸_𝑞+1^ℓ=1𝑥−2^2𝑛+12−𝐸_0,𝑞−^2𝑛+1_ℓ=1ℓ2𝑛+11𝑥−2^{2𝑛+1−ℓ}𝐸_ℓ,𝑞1=2𝑥−2^2𝑛+1−𝐸_2𝑛+1,𝑞1𝑥−2. (2.19) Therefore, by (2.18) and (2.19), we obtain the following theorem.

Theorem 2.7. For 𝑛∈ℕ, one has ^𝑛_𝑗=122𝑛2𝑗−1^2𝑗𝐸_2𝑗(𝑥)𝑗=2^2𝑛(𝑥−1/2)^2𝑛+1−[2]2𝑛+1^𝑞2^𝑛𝐸4𝑛+2^2𝑛+1,𝑞1𝑥−2−1(2𝑛+1)[2]_𝑞. (2.20)

Replacing 𝑦 by 1 and 𝑛 by 2𝑛+1 in Corollary 2.2, we see that2𝑛+1𝐸^𝑛_𝑗=02𝑗^2𝑗+1,𝑞(𝑥)=𝐸2𝑗+1_2𝑛+2,𝑞(𝑥+1)−𝐸_2𝑛+2,𝑞(𝑥−1)=4𝑛+41/𝑞𝑞𝐸_2𝑛+2,𝑞(𝑥+1)+𝐸_2𝑛+2,𝑞(𝑥)−4𝑛+4𝑞𝐸_2𝑛+2,𝑞(𝑥)+𝐸_2𝑛+2,𝑞(𝑥−1)−𝐸4𝑛+4_2𝑛+2,𝑞(𝑥)𝑞()𝐸4𝑛+4−𝑞^2𝑛+2,𝑞(𝑥)=4𝑛+42𝑥^2𝑛+2𝑞()−2()4𝑛+4𝑥−1^2𝑛+2−𝐸4𝑛+4^2𝑛+2,𝑞(𝑥)𝑞()𝐸4𝑛+4−𝑞^2𝑛+2,𝑞(𝑥).4𝑛+4 (2.21) Therefore, by (2.21), we obtain the following theorem.

Theorem 2.8. For 𝑛∈ℤ₊, one has 2𝑛+1𝐸^𝑛_𝑗=02𝑗^2𝑗+1,𝑞(𝑥)=𝑥2𝑗+1𝑞(^2𝑛+2)−()2𝑛+2𝑥−1^2𝑛+2−[2]2𝑛+21−𝑞^𝑞𝐸_2𝑛+2,𝑞(𝑥)𝑞().4𝑛+4 (2.22)

Replacing 𝑛 by 2𝑛+1 and 𝑦 by 1 in Theorem 2.3, we get2𝑛+1𝐸^𝑛+1_𝑗=12𝑗−1^2𝑗,𝑞(𝑥)𝑗=𝐸_2𝑛+2,𝑞(𝑥+1)+𝐸_2𝑛+2,𝑞(𝑥−1)−18𝑛+8(2𝑛+2)[2]_𝑞=𝑞𝐸_2𝑛+2,𝑞(𝑥+1)+𝐸_2𝑛+2,𝑞(𝑥)𝑞()+8𝑛+8𝑞𝐸_2𝑛+2,𝑞(𝑥)+𝐸_2𝑛+2,𝑞(𝑥)−𝐸8𝑛+8_2𝑛+2,𝑞(𝑥)𝑞()−8𝑛+8𝑞𝐸_2𝑛+2,𝑞(𝑥)−18𝑛+8(2𝑛+2)[2]_𝑞=2𝑥^2𝑛+2𝑞()+()8𝑛+8𝑞2𝑥−1^2𝑛+2−𝐸8𝑛+8^2𝑛+2,𝑞(𝑥)𝑞()𝐸8𝑛+8−𝑞^2𝑛+2,𝑞(𝑥)−18𝑛+8(2𝑛+2)[2]_𝑞. (2.23) Therefore, by (2.23), we obtain the following theorem.

Theorem 2.9. For 𝑛∈ℤ₊, one has 2𝑛+1𝐸^𝑛+1_𝑗=12𝑗−1^2𝑗,𝑞(𝑥)𝑗=𝑥^2𝑛+2𝑞()()4𝑛+4+𝑞𝑥−1^2𝑛+2−4𝑛+41+𝑞²𝐸_2𝑛+2,𝑞(𝑥)−18𝑛+8(2𝑛+2)[2]_𝑞. (2.24)

Acknowledgments

This paper was supported by the research grant of Kwangwoon University in 2012. The authors would like to express their sincere gratitude to referees for their valuable comments and suggestions.