On the 𝑞-Extension of Apostol-Euler Numbers and Polynomials

Abstract

Recently, Choi et al. (2008) have studied the 𝑞-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order 𝑛 and multiple Hurwitz zeta function. In this paper, we define Apostol's type 𝑞-Euler numbers 𝐸_{𝑛,𝑞,𝜉} and 𝑞-Euler polynomials 𝐸_{𝑛,𝑞,𝜉}(𝑥). We obtain the generating functions of 𝐸_{𝑛,𝑞,𝜉} and 𝐸_{𝑛,𝑞,𝜉}(𝑥), respectively. We also have the distribution relation for Apostol's type 𝑞-Euler polynomials. Finally, we obtain 𝑞-zeta function associated with Apostol's type 𝑞-Euler numbers and Hurwitz_s type 𝑞-zeta function associated with Apostol's type 𝑞-Euler polynomials for negative integers.

1. Introduction

Let 𝑝 be a fixed odd prime. Throughout this paper, ℤ_𝑝,ℚ_𝑝 and ℂ and ℂ_𝑝 will, respectively, denote the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, the complex number field, and the completion of algebraic closure of ℚ_𝑝. Let ℕ be the set of natural numbers and ℤ₊=ℕ∪{0}. Let 𝑣_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{−𝑣𝑝(𝑝)}=𝑝⁻¹. When one talks of 𝑞-extension, 𝑞 is variously considered as an indeterminate, a complex number 𝑞∈ℂ, or a 𝑝-adic number 𝑞∈ℂ_𝑝. If 𝑞∈ℂ, one normally assumes |𝑞|<1. If 𝑞∈ℂ_𝑝, then one assumes |𝑞−1|_𝑝<1. We also use the notations[𝑥]_𝑞=1−𝑞^𝑥1−𝑞,[𝑥]_−𝑞=1−(−𝑞)^𝑥1+𝑞∀𝑥∈ℤ_𝑝 (1.1)

For a fixed odd positive integer 𝑑 with (𝑝,𝑑)=1, let𝑋=𝑋_𝑑=lim_←𝑁ℤ𝑑𝑝^𝑁ℤ,𝑋₁=ℤ_𝑝,𝑋^∗=_{0<𝑎<𝑑𝑝(𝑎,𝑝)=1}(𝑎+𝑑𝑝ℤ_𝑝),𝑎+𝑑𝑝^𝑁ℤ_𝑝={𝑥∈𝑋∣𝑥≡𝑎(mod𝑑𝑝^𝑁)}, (1.2)where 𝑎∈ℤ lies in 0≤𝑎<𝑑𝑝^𝑁. The distribution is defined by𝜇_𝑞(𝑎+𝑑𝑝^𝑁ℤ_𝑝𝑞)=^𝑎[𝑑𝑝^𝑁]_𝑞. (1.3)

We say that 𝑓 is a uniformly differentiable function at a point 𝑎∈ℤ_𝑝 and denote this property by 𝑓∈UD(ℤ_𝑝), if the difference quotients 𝐹_𝑓(𝑥,𝑦)=(𝑓(𝑥)−𝑓(𝑦))/(𝑥−𝑦) have a limit 𝑙=𝑓^′(𝑎) as (𝑥,𝑦)→(𝑎,𝑎). For 𝑓∈UD(ℤ_𝑝), the 𝑝-adic invariant 𝑞-integral is defined as𝐼_𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)=lim_𝑁→∞1[𝑝^𝑁]_𝑞^𝑝𝑁−1_𝑥=0𝑓(𝑥)𝑞^𝑥. (1.4)The fermionic 𝑝-adic 𝑞-measures on ℤ_𝑝 are defined as𝜇_−𝑞(𝑎+𝑑𝑝^𝑁ℤ_𝑝)=(−𝑞)^𝑎[𝑑𝑝^𝑁]_−𝑞, (1.5)and the fermionic 𝑝-adic invariant 𝑞-integral on ℤ_𝑝 is defined as𝐼_−𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1[𝑝^𝑁]^𝑝_−𝑞^𝑁−1_𝑥=0𝑓(𝑥)(−𝑞)^𝑥 (1.6)for 𝑓∈UD(ℤ_𝑝). For details see [1–10].

Classical Euler numbers are defined by the generating function2𝑒^𝑡=+1^∞_𝑛=0𝐸_𝑛𝑡^𝑛,𝑛! (1.7)and these numbers are interpolated by the Euler zeta function which is defined as𝜁_𝐸(𝑠)=^∞_𝑛=0(−1)^𝑛𝑛^𝑠,𝑠∈ℂ. (1.8)

After Carlitz [11] gave 𝑞-extensions of the classical Bernoulli numbers and polynomials, the 𝑞-extensions of Bernoulli and Euler numbers and polynomials have been studied by several authors (cf. [1–16, 18–26, 34–39]).

By using 𝑝-adic 𝑞-integral, the 𝑞-Euler numbers 𝐸_𝑛,𝑞 are defined as𝐸_𝑛,𝑞=_ℤ𝑝[𝑡]^𝑛_𝑞𝑑𝜇_−𝑞(𝑡),for𝑛∈ℕ. (1.9)The 𝑞-Euler numbers 𝐸_𝑛,𝑞 are defined by means of the generating function𝐹_𝑞(𝑡)=[2]_𝑞^∞_𝑛=0(−1)^𝑛𝑞^𝑛𝑒^{[𝑛]𝑞𝑡} (1.10)(cf. [8, 26]). Kim [22] gave a new construction of the 𝑞-Euler numbers 𝐸_𝑛,𝑞 which can be uniquely determined by𝐸_0,𝑞=[2]_𝑞2,(𝑞𝐸+1)^𝑛+𝐸_𝑛,𝑞=[2]_𝑞,if𝑛=0,0,if𝑛≠0, (1.11)with the usual convention of replacing 𝐸^𝑛 by 𝐸_𝑛,𝑞.

The twisted 𝑞-Euler numbers and 𝑞-Euler polynomials are very important in several fields of mathematics and physics, and so they have been studied by many authors. Simsek [37, 38] constructed generating functions of 𝑞-generalized Euler numbers and polynomials and twisted 𝑞-generalized Euler numbers and polynomials. Recently, Y. H. Kim et al. [27] gave the twisted 𝑞-Euler zeta function associated with twisted 𝑞-Euler numbers and obtained 𝑞-Euler's identity. They also have a 𝑞-extension of the Euler zeta function for negative integers and the 𝑞-analog of twisted Euler zeta function. Kim [24] defined twisted 𝑞-Euler numbers and polynomials of higher order and studied multiple twisted 𝑞-Euler zeta functions.

The Apostol-Bernoulli and the Apostol-Euler polynomials and numbers have been studied by several authors (cf. [15, 17, 32, 33, 40, 41]). Recently, 𝑞-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials and numbers have been studied by many authors with great interest. In [15], Cenkci and Can introduced and investigated 𝑞-extensions of the Bernoulli polynomials. Choi et al. [16] have studied some 𝑞-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order 𝑛 and multiple Hurwitz zeta function.

In this paper, we define Apostol's type 𝑞-Euler numbers and 𝑞-Euler polynomials. Then, we have the generating functions of Apostol's type 𝑞-Euler numbers and 𝑞-Euler polynomials and the distribution relation for Apostol's type 𝑞-Euler polynomials. In Section 2, we define Apostol's type 𝑞-Euler numbers 𝐸_{𝑛,𝑞,𝜉} and 𝑞-Euler polynomials 𝐸_{𝑛,𝑞,𝜉}(𝑥). Then, we obtain the generating functions of 𝐸_{𝑛,𝑞,𝜉} and 𝐸_{𝑛,𝑞,𝜉}(𝑥), respectively. We also have the distribution relation for Apostol's type 𝑞-Euler polynomials. In Section 3, we obtain 𝑞-zeta function associated with Apostol's type 𝑞-Euler numbers and Hurwitz's type 𝑞-zeta function associated with Apostol's type 𝑞-Euler polynomials for negative integers.

2. on The 𝑞-Extensions of The Apostol-Euler Numbers and Polynomials

In this section, we will assume 𝑞∈ℂ_𝑝 with |𝑞−1|_𝑝<1. For 𝑛∈ℤ₊, let 𝐶_𝑝𝑛={𝜉∣𝜉^𝑝𝑛=1} be the cyclic group of order 𝑝^𝑛, and let 𝑇_𝑝 be the space of locally constant space, that is,𝑇_𝑝=lim_𝑛→∞𝐶_𝑝𝑛=_𝑛≥0𝐶_𝑝𝑛. (2.1)

Let 𝜉∈𝑇_𝑝. We define Apostol's type 𝑞-Euler numbers by𝐸_{𝑛,𝑞,𝜉}=_ℤ𝑝𝑞^−𝑥𝜉^𝑥[𝑥]^𝑛_𝑞𝑑𝜇_−𝑞(𝑥). (2.2)Then, we have𝐸_{𝑛,𝑞,𝜉}=[2]_𝑞(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙11+𝑞^𝑙𝜉, (2.3)where (^𝑛_𝑙) are the binomial coefficients.

Apostol's type 𝑞-Euler polynomials are defined as𝐸_{𝑛,𝑞,𝜉}(𝑥)=_ℤ𝑝𝑞^−𝑦𝜉^𝑦[𝑥+𝑦]^𝑛_𝑞𝑑𝜇_−𝑞(𝑦). (2.4)Since[𝑥+𝑦]^𝑛_𝑞=([𝑥]_𝑞+𝑞^𝑥[𝑦]_𝑞)^𝑛=^𝑛_𝑙=0𝑛𝑙[𝑥]_𝑞^𝑛−𝑙𝑞^𝑙𝑥[𝑦]^𝑙_𝑞, (2.5)we have from (2.4) that𝐸_{𝑛,𝑞,𝜉}(𝑥)=^𝑛_𝑙=0𝑛𝑙[𝑥]_𝑞^𝑛−𝑙𝑞^𝑙𝑥_ℤ𝑝𝑞^−𝑦𝜉^𝑦[𝑦]^𝑙_𝑞𝑑𝜇_−𝑞(𝑦). (2.6)By (2.2) and (2.6), we have𝐸_{𝑛,𝑞,𝜉}(𝑥)=^𝑛_𝑙=0𝑛𝑙[𝑥]_𝑞^𝑛−𝑙𝑞^𝑙𝑥𝐸_{𝑙,𝑞,𝜉}. (2.7)Since[𝑥+𝑦]^𝑛_𝑞=1(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^{(𝑥+𝑦)𝑙}=1(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥𝑞^𝑙𝑦, (2.8)we have_ℤ𝑝𝑞^−𝑦𝜉^𝑦[𝑥+𝑦]^𝑛_𝑞𝑑𝜇_−𝑞1(𝑦)=(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥_ℤ𝑝𝑞^{(𝑙−1)𝑦}𝜉^𝑦𝑑𝜇_−𝑞(𝑦). (2.9)Therefore, we also have𝐸_{𝑛,𝑞,𝜉}(𝑥)=[2]_𝑞1(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙𝑞^𝑙𝑥(−1)^𝑙11+𝑞^𝑙𝜉. (2.10)

Note that (2.7) and (2.10) are two representations for 𝐸_{𝑛,𝑞,𝜉}(𝑥). Hence, we have the following result.

Theorem 2.1. For 𝑛∈ℤ₊ and 𝜉∈𝑇_𝑝, one has𝐸_{𝑛,𝑞,𝜉}=[2]_𝑞(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙11+𝑞^𝑙𝜉,𝐸_{𝑛,𝑞,𝜉}(𝑥)=[2]_𝑞(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥1+𝑞^𝑙𝜉=^𝑛_𝑙=0𝑛𝑙[𝑥]_𝑞^𝑛−𝑙𝑞^𝑙𝑥𝐸_{𝑙,𝑞,𝜉}. (2.11)

Now, we will find the generating function of 𝐸_{𝑛,𝑞,𝜉} and 𝐸_{𝑛,𝑞,𝜉}(𝑥), respectively. Let 𝐹(𝑡) be the generating function of 𝐸_{𝑛,𝑞,𝜉}. Then, we have𝐹(𝑡)=^∞_𝑛=0𝐸_{𝑛,𝑞,𝜉}𝑡^𝑛=𝑛!_∞𝑛=0([2]_𝑞(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙11+𝑞^𝑙𝜉)𝑡^𝑛𝑛!=[2]_𝑞^∞_𝑛=01(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙(^∞_𝑚=0𝑞^𝑙𝑚𝜉^𝑚(−1)^𝑚)𝑡^𝑛𝑛!=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝜉^𝑚∞_𝑛=01(1−𝑞)^𝑛(^𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑚)𝑡^𝑛𝑛!=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝜉^𝑚∞_𝑛=01(1−𝑞)^𝑛(1−𝑞^𝑚)^𝑛𝑡^𝑛𝑛!=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝜉^𝑚∞_𝑛=0[𝑚]^𝑛_𝑞𝑡^𝑛𝑛!=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝜉^𝑚𝑒^{[𝑚]𝑞𝑡}. (2.12)Therefore, the generating function 𝐹(𝑡) of 𝐸_{𝑛,𝑞,𝜉} equals𝐹(𝑡)=^∞_𝑛=0𝐸_{𝑛,𝑞,𝜉}𝑡^𝑛𝑛!=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝜉^𝑚𝑒^{[𝑚]𝑞𝑡}. (2.13)Note that_ℤ𝑝𝑞^−𝑥𝜉^𝑥𝑒^{[𝑥]𝑞𝑡}𝑑𝜇_−𝑞(𝑥)=^∞_{𝑛=0ℤ𝑝}𝑞^−𝑥𝜉^𝑥[𝑥]^𝑛_𝑞𝑑𝜇_−𝑞𝑡(𝑥)^𝑛=𝑛!_∞𝑛=0𝐸_{𝑛,𝑞,𝜉}𝑡^𝑛𝑛!=𝐹(𝑡). (2.14)For the generating function of 𝐸_{𝑛,𝑞,𝜉}(𝑥), we have_ℤ𝑝𝑞^−𝑦𝜉^𝑦𝑒^{[𝑥+𝑦]𝑞𝑡}𝑑𝜇_−𝑞(𝑦)=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝜉^𝑚𝑒^{[𝑚+𝑥]𝑞𝑡}. (2.15)Hence, we obtain the following theorem.

Theorem 2.2. For 𝜉∈𝑇_𝑝, one has_ℤ𝑝𝑞^−𝑥𝜉^𝑥𝑒^{[𝑥]𝑞𝑡}𝑑𝜇_−𝑞(𝑥)=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝜉^𝑚𝑒^{[𝑚]𝑞𝑡},_ℤ𝑝𝑞^−𝑦𝜉^𝑦𝑒^{[𝑥+𝑦]𝑞𝑡}𝑑𝜇_−𝑞(𝑦)=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝜉^𝑚𝑒^{[𝑚+𝑥]𝑞𝑡}. (2.16)(2.17)

Since (2.16) equals to the generating functions (2.17) equals to the generating functions ∑^∞_𝑛=0𝐸_{𝑛,𝑞,𝜉}(𝑥)(𝑡^𝑛/𝑛!), we have the following result.

Corollary 2.3. For 𝑛∈ℤ₊ and 𝜉∈𝑇_𝑝, one has𝐸_{𝑛,𝑞,𝜉}=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝜉^𝑚[𝑚]^𝑛_𝑞,𝐸_{𝑛,𝑞,𝜉}(𝑥)=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝜉^𝑚[𝑚+𝑥]^𝑛_𝑞. (2.18)

Now, we will find the distribution relation for 𝐸_{𝑛,𝑞,𝜉}(𝑥). By (2.4), we have𝐸_{𝑛,𝑞,𝜉}(𝑥)=_𝑋𝑞^−𝑦𝜉^𝑦[𝑥+𝑦]^𝑛_𝑞𝑑𝜇_−𝑞(𝑦)=lim_𝑁→∞1[𝑑𝑝^𝑁]_−𝑞^{𝑑𝑝𝑁−1}_𝑦=0𝜉^𝑦(−1)^𝑦[𝑥+𝑦]^𝑛_𝑞=lim_𝑁→∞1[𝑑𝑝^𝑁]_−𝑞^𝑑−1𝑝_𝑎=0^𝑁−1_𝑦=0𝜉^{𝑎+𝑑𝑦}(−1)^{𝑎+𝑑𝑦}[𝑥+𝑎+𝑑𝑦]^𝑛_𝑞. (2.19)Note that for odd numbers 𝑑 and 𝑝,[𝑑𝑝^𝑁]_−𝑞=[𝑑]_−𝑞[𝑝^𝑁]_−𝑞𝑑,[𝑥+𝑎+𝑑𝑦]_𝑞=[𝑑]_𝑞[𝑥+𝑎𝑑+𝑦]_𝑞𝑑. (2.20)By (2.19), we have𝐸_{𝑛,𝑞,𝜉}1(𝑥)=[𝑑]_−𝑞^𝑑−1_𝑎=0𝜉^𝑎(−1)^𝑎lim_𝑁→∞1[𝑝^𝑁]_−𝑞𝑑^𝑝𝑁−1_𝑦=0(𝜉^𝑑)^𝑦(−1)^𝑦[𝑑]^𝑛_𝑞[𝑥+𝑎𝑑+𝑦]^𝑛_𝑞𝑑=[𝑑]^𝑛_𝑞[𝑑]_−𝑞^𝑑−1_𝑎=0𝜉^𝑎(−1)^𝑎_ℤ𝑝(𝜉^𝑑)^𝑦(𝑞^𝑑)^−𝑦[𝑥+𝑎𝑑+𝑦]^𝑛_𝑞𝑑𝑑𝜇_−𝑞𝑑(𝑦). (2.21)Therefore, we obtain the distribution relation for 𝐸_{𝑛,𝑞,𝜉}(𝑥) as follows.

Theorem 2.4. For 𝑛∈ℤ₊,𝜉∈𝑇_𝑝, and 𝑑∈ℤ₊ with 𝑑≡1(mod2), one has 𝐸_{𝑛,𝑞,𝜉}(𝑥)=[𝑑]^𝑛_𝑞[𝑑]_−𝑞^𝑑−1_𝑎=0𝜉^𝑎(−1)^𝑎𝐸_{𝑛,𝑞𝑑,𝜉}^𝑑(𝑥+𝑎𝑑). (2.22)

3. Further Remark on The Basic 𝑞-Zeta Functions Associated with Apostol's Type 𝑞-Euler Numbers and Polynomials

In this section, we assume that 𝑞∈ℂ with |𝑞|<1. Let 𝜉∈𝑇_𝑝. For 𝑠∈ℂ, 𝑞-zeta function associated with Apostol's type 𝑞-Euler numbers is defined as𝜁_𝑞,𝜉(𝑠)=[2]_𝑞^∞_𝑛=1𝜉^𝑛(−1)^𝑛[𝑛]^𝑠_𝑞, (3.1)which is analytic in whole complex 𝑠-plane. Substituting 𝑠=−𝑘 with 𝑘∈ℤ₊ into 𝜁_𝑞,𝜉(𝑠) and using Corollary 2.3, then we arrive at𝜁_𝑞,𝜉(−𝑘)=[2]_𝑞^∞_𝑛=1𝜉^𝑛(−1)^𝑛[𝑛]^𝑘_𝑞=𝐸_{𝑘,𝑞,𝜉}. (3.2)

Now, we also consider Hurwitz's type 𝑞-zeta function associated with the Apostol's type 𝑞-Euler polynomials as follows:𝜁_𝑞,𝜉(𝑠,𝑥)=[2]_𝑞^∞_𝑛=0𝜉^𝑛(−1)^𝑛[𝑛+𝑥]^𝑠_𝑞. (3.3)Substituting 𝑠=−𝑘 with 𝑘∈ℤ₊ into 𝜁_𝑞,𝜉(𝑠,𝑥) and using Corollary 2.3, then we arrive at𝜁_𝑞,𝜉(−𝑘,𝑥)=[2]_𝑞^∞_𝑛=0𝜉^𝑛(−1)^𝑛[𝑛+𝑥]^𝑘_𝑞=𝐸_{𝑘,𝑞,𝜉}(𝑥). (3.4)Hence, we obtain 𝑞-zeta function associated with Apostol's type 𝑞-Euler numbers and Hurwitz's type 𝑞-zeta function associated with Apostol's type 𝑞-Euler polynomials for negative integers.