A New 𝑞-Analogue of Bernoulli Polynomials Associated with 𝑝-Adic 𝑞-Integrals

Abstract

We will study a new 𝑞-analogue of Bernoulli polynomials associated with 𝑝-adic 𝑞-integrals. Furthermore, we examine the Hurwitz-type 𝑞-zeta functions, replacing 𝑝-adic rational integers 𝑥 with a 𝑞-analogue [𝑥]_𝑞 for a 𝑝-adic number 𝑞 with |𝑞−1|_𝑝<1, which interpolate 𝑞-analogue of Bernoulli polynomials.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper ℤ_𝑝, ℚ_𝑝, ℂ, and ℂ_𝑝 will, respectively, represent the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, the complex number field, and the 𝑝-adic completion of the algebraic closure of ℚ_𝑝. The 𝑝-adic absolute value in ℂ_𝑝 is normalized so that |𝑝|_𝑝=1/𝑝 and 𝑞 is a 𝑝-adic number in ℂ_𝑝 with |𝑞−1|_𝑝<1. We use the notation[𝑥]_𝑞=1−𝑞^𝑥1−𝑞(1.1)(cf. [1–13]) for all 𝑥∈ℤ_𝑝. Hence, lim_𝑞→1|𝑥|_𝑞=𝑥. 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. For any 𝑛∈ℕ,𝜇_𝑞(𝑎+𝑑𝑝^𝑛ℤ_𝑝𝑞)=^𝑎[𝑑𝑝^𝑛]_𝑞(1.3)is known to be a distribution on 𝑋 (cf. [1–13]).

We say that 𝑓 is uniformly differentiable function at a point 𝑎∈ℤ_𝑝 and denote this property by 𝑓∈𝑈𝐷(ℤ_𝑝), if the difference quotients𝐹_𝑓(𝑥,𝑦)=𝑓(𝑥)−𝑓(𝑦)𝑥−𝑦(1.4)have a limit 𝑙=𝑓^′(𝑎) as (𝑥,𝑦)→(𝑎,𝑎) (cf. [2, 6, 7]). The 𝑝-adic 𝑞-integral of a function 𝑓∈𝑈𝐷(ℤ_𝑝) was defined as𝐼_𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)=lim_𝑁→∞1[𝑝^𝑁]_𝑞^𝑝𝑁−1_𝑥=0𝑓(𝑥)𝑞^𝑥.(1.5)

By using 𝑝-adic 𝑞-integrals on ℤ_𝑝, it is well known that𝑡𝑒^𝑡𝑒−1^𝑥𝑡=^∞_{𝑛=0ℤ𝑝}(𝑥+𝑠)^𝑛𝑑𝜇₁𝑡(𝑠)^𝑛,𝑛!(1.6)where 𝜇₁(𝑥+𝑝^𝑛ℤ_𝑝)=1/𝑝^𝑛. Then we note that the Bernoulli polynomials 𝐵_𝑛(𝑥) were defined as𝑡𝑒^𝑡𝑒−1^𝑥𝑡=^∞_𝑛=0𝐵_𝑛𝑡(𝑥)^𝑛.𝑛!(1.7)From (1.6) and (1.7), we have𝐵_𝑛(𝑥)=_ℤ𝑝(𝑥+𝑠)^𝑛𝑑𝜇₁(𝑠)(1.8)for all 𝑛∈ℕ∪{0}. We note that [0]_𝑞=(1−𝑞⁰)/(1−𝑞)=0.

In Section 2, we study a 𝑞-analogue of Bernoulli polynomials associated with 𝑝-adic 𝑞-integrals—simply, we say 𝑞-Bernoulli polynomials. In Section 3, we examine Hurwitz-type  𝑞-zeta functions, replacing 𝑝-adic rational integers 𝑥 with a 𝑞-analogue [𝑥]_𝑞 for a 𝑝-adic number 𝑞 with |𝑞−1|_𝑝<1, which interpolate 𝑞-analogue of Bernoulli polynomials.

2. A New 𝑞-Analogue of Bernoulli Polynomials

In this section, from the view of (1.8), we can define a new 𝑞-analogue of Bernoulli polynomials as follows:𝛽^𝑞_𝑛(𝑥)=_ℤ𝑝([𝑥]_𝑞+[𝑠]_𝑞)^𝑛𝑑𝜇_𝑞(𝑠).(2.1)We note that 𝛽^𝑞_𝑛=𝛽^𝑞_𝑛(0) are called the 𝑞-Bernoulli numbers. Then we find some properties of 𝑞-Bernoulli numbers and polynomials as follows.

Theorem 2.1. For 𝑛∈ℕ∪{0}, one has𝛽^𝑞_𝑛=1(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙1[𝑙+1]_𝑞.(2.2)

Proof . From (1.5) with 𝑥=0, we can find the following:𝛽^𝑞_𝑛=_ℤ𝑝[𝑠]^𝑛_𝑞𝑑𝜇_𝑞(𝑠)=lim^𝑝_𝑁→∞^𝑁−1_𝑗=0[𝑗]^𝑛_𝑞𝑞^𝑗[𝑝^𝑁]_𝑞=^𝑛_𝑙=0𝑛𝑙(−1)^𝑙1(1−𝑞)^𝑛−1lim^𝑝_𝑁→∞^𝑁−1_𝑗=0𝑞^𝑗(𝑙+1)11−𝑞^𝑝𝑁=1(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙1[𝑙+1]_𝑞.(2.3)

Theorem 2.2. For 𝑛∈ℕ∪{0} and 𝑑 being an odd positive integer with (𝑝,𝑑)=1, one has𝛽^𝑞_𝑛(𝑥)=[𝑑]_𝑞^𝑛𝑛−1_𝑙=0𝑛𝑙𝛽^𝑞𝑑_𝑙^𝑑−1_𝑖=0𝑞^𝑖(𝑙+1)𝑥([𝑑]_𝑞𝑑𝑖+[𝑑]_𝑞𝑑)^𝑛−𝑙.(2.4)

Proof . From (1.5), we can derive (2.4) as follows: 𝛽^𝑞_𝑛(𝑥)=_ℤ𝑝([𝑥]_𝑞+[𝑠]_𝑞)^𝑛𝑑𝜇_𝑞(𝑠)=lim_𝑁→∞1[𝑑𝑝^𝑁]_𝑞^{𝑑𝑝𝑁−1}_𝑎=0([𝑥]_𝑞+[𝑎]_𝑞)^𝑛𝑞^𝑎=lim_𝑁→∞1−𝑞1−𝑞^{𝑑𝑝𝑁𝑑−1𝑝}_𝑖=0^𝑁−1_𝑘=0([𝑥]_𝑞+[𝑖+𝑑𝑘]_𝑞)^𝑛𝑞^{𝑖+𝑑𝑘}=lim_𝑁→∞1[𝑑]_𝑞1[𝑝^𝑁]_𝑞𝑑^𝑑−1_𝑖=0𝑞^{𝑖𝑝𝑁−1}_𝑘=0[𝑑]^𝑛_𝑞𝑥([𝑑]_𝑞𝑑𝑖+[𝑑]_𝑞𝑑+𝑞^𝑖[𝑘]_𝑞𝑑)^𝑛(𝑞^𝑑)^𝑘=lim_𝑁→∞1[𝑑]_𝑞1[𝑝^𝑁]_𝑞𝑑^𝑑−1_𝑖=0𝑞^{𝑖𝑝𝑁−1𝑛}_𝑘=0𝑙=0𝑛𝑙𝑥([𝑑]_𝑞𝑑𝑖+[𝑑]_𝑞𝑑)^𝑛−𝑙(𝑞^𝑖[𝑘]_𝑞𝑑)^𝑙(𝑞^𝑑)^𝑘=[𝑑]_𝑞^{𝑛−1𝑑−1𝑛}_𝑖=0𝑙=0𝑛𝑙𝑞^𝑖(𝑙+1)𝑥([𝑑]_𝑞𝑑𝑖+[𝑑]_𝑞𝑑)^𝑛−𝑙lim_𝑁→∞1[𝑝^𝑁]_𝑞𝑑^𝑝𝑁−1_𝑘=0[𝑘]^𝑙_𝑞𝑑(𝑞^𝑑)^𝑘=[𝑑]_𝑞^{𝑛−1𝑑−1𝑛}_𝑖=0𝑙=0𝑛𝑙𝑞^𝑖(𝑙+1)𝑥([𝑑]_𝑞𝑑𝑖+[𝑑]_𝑞𝑑)^𝑛−𝑙_ℤ𝑝[𝑠]^𝑙_𝑞𝑑𝑑𝜇_𝑞𝑑(𝑠)=[𝑑]_𝑞^𝑛𝑛−1_𝑙=0𝑛𝑙𝛽^𝑞𝑑_𝑙^𝑑−1_𝑖=0𝑞^𝑖(𝑙+1)𝑥([𝑑]_𝑞𝑑𝑖+[𝑑]_𝑞𝑑)^𝑛−𝑙,(2.5)since 𝑎=𝑖+𝑑𝑘 and([𝑥]_𝑞+[𝑖+𝑑𝑘]_𝑞)^𝑛=[𝑑]^𝑛_𝑞𝑥([𝑑]_𝑞𝑑𝑖+[𝑑]_𝑞𝑑+𝑞^𝑖[𝑘]_𝑞𝑑)^𝑛(2.6)for 𝑎=0,1,…,𝑑𝑝^𝑁−1, 𝑖=0,1,…,𝑑−1, and 𝑘=0,1,…,𝑝^𝑁−1.
Let 𝐺^𝑞(𝑥,𝑡) be the generating function of 𝑞-Bernoulli polynomials as follows:𝐺^𝑞(𝑥,𝑡)=^∞_𝑛=0𝛽^𝑞_𝑛𝑡(𝑥)^𝑛.𝑛!(2.7)From (2.2) and (2.7), we can obtain the following theorem.

Theorem 2.3. Let 𝐺^𝑞(𝑥,𝑡) be as in the above generating function. Then, one has𝐺^𝑞(𝑥,𝑡)=(1−𝑞)^∞_𝑚=0𝑞^𝑚𝑒^{([𝑥]𝑞+[𝑚]𝑞)𝑡}.(2.8)

Proof . By using (2.2) and (2.7), we can derive (2.8) as follows:𝐺^𝑞(𝑥,𝑡)=𝑒^{([𝑥]𝑞+𝛽𝑞)𝑡}=𝑒^{[𝑥]𝑞𝑡}𝑒^𝛽𝑞𝑡=𝑒^{[𝑥]𝑞𝑡∞}_𝑛=0𝛽^𝑞_𝑛𝑡^𝑛𝑛!=𝑒^{[𝑥]𝑞𝑡∞}_𝑛=01(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙1[𝑙+1]_𝑞𝑡^𝑛𝑛!=𝑒^{[𝑥]𝑞𝑡∞}_𝑛=01(1−𝑞)^𝑛𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙11−𝑞^𝑙+1𝑡^𝑛𝑛!=𝑒^{[𝑥]𝑞𝑡∞}_𝑛=01(1−𝑞)^𝑛𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙∞_𝑚=0𝑞^(𝑙+1)𝑚𝑡^𝑛𝑛!=𝑒^{[𝑥]𝑞𝑡}(1−𝑞)^∞_𝑚=0𝑞^𝑚∞_𝑛=01(1−𝑞)^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑚𝑡^𝑛𝑛!=𝑒^{[𝑥]𝑞𝑡}(1−𝑞)^∞_𝑚=0𝑞^𝑚∞_𝑛=0[𝑚]^𝑛_𝑞𝑡^𝑛𝑛!=(1−𝑞)^∞_𝑚=0𝑞^𝑚𝑒^{([𝑥]𝑞+[𝑚]𝑞)𝑡}.(2.9)

3. A New Formula for Hurwitz-Type  𝑞-Zeta Functions

In this section, we consider the generating functions 𝐹(𝑡,𝑥) which interpolate the 𝑞-Bernoulli polynomials 𝛽_𝑛^∗𝑞(𝑥) as follows:𝐹(𝑡,𝑥)=∑^∞_𝑚=0𝑞^𝑚𝑒^{([𝑥]𝑞+[𝑚]𝑞)𝑡}=∑^∞_𝑚=0𝛽_𝑛^∗𝑞𝑡(𝑥)^𝑚.𝑚!(3.1)From (3.1), we directly obtain the following theorem.

Theorem 3.1. For each 𝑘∈ℕ∪{0}, one has𝛽_𝑘^∗𝑞(𝑥)=^∞_𝑚=0𝑞^𝑚([𝑥]_𝑞+[𝑚]_𝑞)^𝑘.(3.2)

Proof . By the 𝑘th differentiation on both sides of (3.1), we can derive (3.2) as follows:𝛽_𝑘^∗𝑞𝑑(𝑥)=^𝑘𝑑𝑡^𝑘𝐹(𝑥,𝑡)|_𝑡=0=∑^∞_𝑚=0𝑞^𝑚([𝑥]_𝑞+[𝑚]_𝑞)^𝑘.(3.3)
We remark that−𝛽_𝑘^∗𝑞(𝑥)𝑘=1𝑘^∞_𝑚=0𝑞^𝑚([𝑥]_𝑞+[𝑚]_𝑞)^𝑘(3.4)for 𝑘∈ℕ. From (3.2), we derive a 𝑞-extension of Hurwitz-type zeta function as follows: for 𝑠∈ℂ with ℜ(𝑠)>1 and ℜ(𝑥)>0, we define𝜁^𝑞1(𝑠,𝑥)=1−𝑠^∞_𝑚=0𝑞^𝑚([𝑥]_𝑞+[𝑚]_𝑞)^𝑠.(3.5)Note that the functions 𝜁^𝑞(𝑠,𝑥) are analytic on ℜ(𝑠)>1 and they have simple pole at 𝑠=1. From (3.2), (3.4), and (3.5), we can see that Hurwitz-type  𝑞-zeta functions interpolate 𝑞-Bernoulli polynomials as follows.

Theorem 3.2. For each 𝑘∈ℕ, one has𝜁^𝑞𝛽(1−𝑘,𝑥)=−^𝑘∗𝑞(𝑥)𝑘.(3.6)

Acknowledgment

This paper was supported by Konkuk University in 2008.