On the Symmetries of the 𝑞-Bernoulli Polynomials

Abstract

Kupershmidt and Tuenter have introduced reflection symmetries for the 𝑞-Bernoulli numbers and the Bernoulli polynomials in (2005), (2001), respectively. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the classical Bernoulli numbers. In this paper, we study the new symmetries of the 𝑞-Bernoulli numbers and polynomials, which are different from Kupershmidt's and Tuenter's results. By using our symmetries for the 𝑞-Bernoulli polynomials, we can obtain some interesting relationships between 𝑞-Bernoulli numbers and polynomials.

1. Introduction

Let 𝑝 be a fixed prime. Throughout this paper, ℤ_𝑝, ℚ_𝑝, ℂ, and ℂ_𝑝 will, respectively, denote the ring of 𝑝-adic rational integer, the field of 𝑝-adic rational numbers, the complex number field, and the completion of algebraic closure of ℚ_𝑝. The 𝑝-adic absolute value in ℂ_𝑝 is normalized so that |𝑝|_𝑝<𝑝⁻¹. Let 𝑞 be variously considered as an indeterminate, a complex number 𝑞∈ℂ, or a 𝑝-adic number 𝑞∈ℂ_𝑝. If 𝑞∈ℂ_𝑝, we assume that |1−𝑞|_𝑝<1. We say that 𝑓 is uniformly differentiable function at a point 𝑎∈ℤ_𝑝, and we denote this property by 𝑓∈UD(ℤ_𝑝) if the difference quotients, 𝐹_𝑓∶ℤ_𝑝×ℤ_𝑝→ℤ_𝑝by𝐹_𝑓(𝑥,𝑦)=𝑓(𝑥)−𝑓(𝑦)𝑥−𝑦,(1.1) have a limit 𝑙=𝑓^(𝑎) as (𝑥,𝑦)→(𝑎,𝑎). The 𝑝-adic invariant integral on ℤ_𝑝 is defined as 𝐼(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝑥=lim_𝑁→∞1𝑝^{𝑁𝑝𝑁−1}_𝑥=0𝑓(𝑥)(1.2) [1–22]. From this integral, we derive several further interesting properties of symmetry for the 𝑞-Bernoulli numbers and polynomials in this paper. Kupershmidt [14] and Tuenter [20] have introduced reflection symmetries for the 𝑞-Bernoulli numbers and the Bernoulli polynomials. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the classical Bernoulli numbers. In this paper, we study the new symmetries of the 𝑞-Bernoulli numbers and polynomials, which are different from Kupershmidt's and Tuenter's results. By using our symmetries for the 𝑞-Bernoulli polynomials, we can obtain some interesting relationships between 𝑞-Bernoulli numbers and polynomials.

2. On the Symmetries of the 𝑞-Bernoulli Polynomials

For 𝑓∈UD(ℤ_𝑝), the 𝑝-adic invariant integral on ℤ_𝑝 is defined as 𝐼(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝑥=lim_𝑁→∞1𝑝^{𝑁𝑝𝑁−1}_𝑥=0𝑓(𝑥).(2.1) Let 𝑓₁(𝑥) be a translation with 𝑓₁(𝑥)=𝑓(𝑥+1). Then, we have 𝐼(𝑓₁)=𝐼(𝑓)+𝑓^(0).(2.2) From (2.2), we can also derive 𝐼(𝑓_𝑛)=𝐼(𝑓)+^𝑛−1_𝑖=0𝑓^(𝑖),𝑓^(𝑖)=𝑑𝑓(𝑖)𝑑𝑥.(2.3) Let 𝑓(𝑥)=𝑞^𝑥𝑒^𝑡𝑥, then we have _ℤ𝑝𝑞^𝑥𝑒^𝑡𝑥𝑑𝑥=𝑡+log𝑞𝑞𝑒^𝑡−1.(2.4) It is known that the 𝑞-Bernoulli polynomials are defined as 𝑡+log𝑞𝑞𝑒^𝑡−1𝑒^𝑥𝑡=^∞_𝑛=0𝐵_𝑛,𝑞(𝑥)𝑡^𝑛𝑛!(2.5) [17, 19]. Now we define an integral representation for the 𝑞-extension of Bernoulli numbers as follows: _ℤ𝑝𝑞^𝑥𝑒^𝑡𝑥𝑑𝑥=log𝑞+𝑡𝑞𝑒^𝑡−1=^∞_𝑛=0𝐵_𝑛,𝑞𝑡^𝑛𝑛!.(2.6)

From (2.3), (2.4), and (2.6), we can derive _ℤ𝑝𝑞^𝑦(𝑥+𝑦)^𝑛𝑑𝑦=𝐵_𝑛,𝑞(𝑥),_ℤ𝑝𝑞^𝑥𝑥^𝑛𝑑𝑥=𝐵_𝑛,𝑞.(2.7) By (2.3), we easily see that 1log𝑞+𝑡(_ℤ𝑝𝑞^𝑛+𝑥𝑒^{(𝑛+𝑥)𝑡}𝑑𝑥−_ℤ𝑝𝑞^𝑥𝑒^𝑥𝑡𝑑𝑥)=𝑞^𝑛𝑒^𝑛𝑡−1𝑡+log𝑞_ℤ𝑝𝑞^𝑥𝑒^𝑥𝑡𝑑𝑥=𝑞^𝑛𝑒^𝑛𝑡−1=𝑞𝑒^𝑡−1_𝑛−1_𝑖=0𝑞^𝑖𝑒^𝑖𝑡=^∞_𝑘=0(^𝑛−1_𝑖=0𝑖^𝑘𝑞^𝑖)𝑡^𝑘𝑘!.(2.8)

In (2.2), it is not difficult to show that 1log𝑞+𝑡(_ℤ𝑝𝑞^𝑛+𝑥𝑒^{(𝑛+𝑥)𝑡}𝑑𝑥−_ℤ𝑝𝑞^𝑥𝑒^𝑥𝑡𝑑𝑥)=𝑛∫^ℤ𝑝𝑒^𝑥𝑡𝑞^𝑥𝑑𝑥∫_ℤ𝑝𝑒^𝑛𝑥𝑡𝑞^𝑛𝑥𝑑𝑥.(2.9) For each integer 𝑘≥0, let 𝑆_𝑘,𝑞(𝑛)=0^𝑘+1^𝑘𝑞+2^𝑘𝑞²+⋯+𝑞^𝑛𝑛^𝑘.(2.10)

From (2.8) and (2.9), we derive 1log𝑞+𝑡(∫^ℤ𝑝𝑞^𝑛+𝑥𝑒^{(𝑛+𝑥)𝑡}𝑑𝑥−∫_ℤ𝑝𝑞^𝑥𝑒^𝑥𝑡𝑑𝑥)=𝑛∫^ℤ𝑝𝑒^𝑥𝑡𝑞^𝑥𝑑𝑥∫_ℤ𝑝𝑒^𝑛𝑥𝑡𝑞^𝑛𝑥𝑑𝑥=^∞_𝑘∑₌₀𝑆_𝑘,𝑞(𝑛−1)𝑡^𝑘𝑘!.(2.11)

From (2.11), we note that 𝐵_𝑘,𝑞(𝑛)−𝐵_𝑘,𝑞=𝑘𝑆_{𝑘−1,𝑞}(𝑛−1)+log𝑞𝑆_𝑘,𝑞(𝑛−1),where𝑘,𝑛∈ℕ.(2.12) Let 𝑤₁,𝑤₂∈ℕ, then we have ∬_ℤ𝑝𝑒^{(𝑤1𝑥1+𝑤2𝑥2)𝑡}𝑞^{𝑤1𝑥1+𝑤2𝑥2}𝑑𝑥₁𝑑𝑥₂∫_ℤ𝑝𝑒^{𝑤1𝑤2𝑥𝑡}𝑞^{𝑤1𝑤2𝑥}𝑑𝑥=(𝑡+log𝑞)𝑞^𝑤1𝑤2𝑒^{𝑤1𝑤2𝑡}−1(𝑞^𝑤1𝑒^𝑤1𝑡−1)(𝑞^𝑤2𝑒^𝑤2𝑡−1).(2.13) By (2.11), we see that 𝑤₁∫_ℤ𝑝𝑒^𝑥𝑡𝑞^𝑥𝑑𝑥∫_ℤ𝑝𝑞^𝑤1𝑥𝑒^{𝑤1𝑥𝑡}𝑑𝑥=^∞_𝑙=0(^𝑤1−1_𝑘=0𝑘^𝑙𝑞^𝑘)𝑡^𝑙𝑙!=^∞_𝑙=0𝑆_𝑙,𝑞(𝑤₁−1)𝑡^𝑙𝑙!.(2.14)

Let 𝑇(𝑤₁,𝑤₂;𝑥,𝑡)=∬^ℤ𝑝𝑞^{𝑤1𝑥1+𝑤2𝑥2}𝑒^{(𝑤1𝑥1+𝑤2𝑥2+𝑤1𝑤2𝑥)𝑡}𝑑𝑥₁𝑑𝑥₂∫_ℤ𝑝𝑒^{𝑤1𝑤2𝑥3𝑡}𝑞^{𝑤1𝑤2𝑥3}𝑑𝑥₃,(2.15) then we have 𝑇(𝑤₁,𝑤₂;𝑥,𝑡)=(𝑡+log𝑞)𝑒^{𝑤1𝑤2𝑥𝑡}(𝑞^𝑤1𝑤2𝑒^{𝑤1𝑤2𝑡}−1)(𝑞^𝑤1𝑒^𝑤1𝑡−1)(𝑞^𝑤2𝑒^𝑤2𝑡−1).(2.16)

From (2.15) we derive 𝑇(𝑤₁,𝑤₂;𝑥,𝑡)=(1𝑤₁_ℤ𝑝𝑒^{𝑤1(𝑥1+𝑤2𝑥)𝑡}𝑞^𝑤1𝑥1𝑑𝑥₁)(𝑤¹∫_ℤ𝑝𝑒^{𝑤2𝑥2𝑡}𝑞^𝑤2𝑥2𝑑𝑥₂∫_ℤ𝑝𝑒^{𝑤1𝑤2𝑥𝑡}𝑞^{𝑤1𝑤2𝑥}𝑑𝑥).(2.17)

By (2.5), (2.14), and (2.17), we see that 𝑇(𝑤₁,𝑤₂;𝑥,𝑡)=1𝑤₁(^∞_𝑖=0𝐵_{𝑖,𝑞𝑤1}(𝑤₂𝑥)𝑤^𝑖1𝑡^𝑖𝑖!)(^∞_𝑙=0𝑆_{𝑙,𝑞𝑤2}(𝑤₁−1)𝑤^𝑙2𝑡^𝑙=𝑙!_∞)_𝑛=0(^𝑛_𝑖=0𝑛𝑖𝐵^{𝑖,𝑞𝑤1}(𝑤₂𝑥)𝑆_{𝑛−𝑖,𝑞𝑤2}(𝑤₁−1)𝑤^1𝑖−1𝑤^2𝑛−𝑖)𝑡^𝑛𝑛!.(2.18) By the symmetry of 𝑝-adic invariant integral on ℤ_𝑝, we also see that 𝑇(𝑤₁,𝑤₂;𝑥,𝑡)=(1𝑤₂_ℤ𝑝𝑒^{𝑤2(𝑥2+𝑤1𝑥)𝑡}𝑞^𝑤2𝑥2𝑑𝑥₂)(𝑤²∫_ℤ𝑝𝑒^{𝑤1𝑥1𝑡}𝑞^𝑤1𝑥1𝑑𝑥₁=∫_ℤ𝑝𝑒^{𝑤1𝑤2𝑥𝑡}𝑞^{𝑤1𝑤2𝑥}𝑑𝑥_∞)_𝑛=0(^𝑛_𝑖=0𝑛𝑖𝐵^{𝑖,𝑞𝑤2}(𝑤₁𝑥)𝑆_{𝑛−𝑖,𝑞𝑤1}(𝑤₂−1)𝑤^2𝑖−1𝑤^1𝑛−𝑖)𝑡^𝑛𝑛!.(2.19) By comparing the coefficients 𝑡^𝑛/𝑛! on the both sides of (2.18) and (2.19), we obtain the following theorem.

Theorem 2.1. For all 𝑤₁,𝑤₂(∈ℕ), we have ^𝑛∑_𝑖=0𝑛𝑖𝐵^{𝑖,𝑞𝑤1}(𝑤₂𝑥)𝑆_{𝑛−𝑖,𝑞𝑤2}(𝑤₁−1)𝑤^1𝑖−1𝑤^2𝑛−𝑖=∑^𝑛_𝑖=0𝑛𝑖𝐵^{𝑖,𝑞𝑤2}(𝑤₁𝑥)𝑆_{𝑛−𝑖,𝑞𝑤1}(𝑤₂−1)𝑤^2𝑖−1𝑤^1𝑛−𝑖,(2.20) where (^𝑛𝑖) is the binomial coefficient.

If we take 𝑤₂=1 in Theorem 2.1, then we have 𝐵_𝑛,𝑞(𝑤₁𝑥)=^𝑛_𝑖=0𝑛𝑖𝐵^{𝑖,𝑞𝑤1}(𝑥)𝑆_{𝑛−𝑖,𝑞}(𝑤₁−1)𝑤^1𝑖−1.(2.21)

Therefore, we obtain the following corollary.

Corollary 2.2. For 𝑛≥0, we have 𝐵_𝑛,𝑞(𝑤₁𝑥)=^𝑛_𝑖=0𝑛𝑖𝐵^{𝑖,𝑞𝑤1}(𝑥)𝑆_{𝑛−𝑖,𝑞}(𝑤₁−1)𝑤^1𝑖−1.(2.22)

By (2.17), (2.18), and (2.19), we also see that 𝑇(𝑤₁,𝑤₂;𝑥,𝑡)=(𝑒^{𝑤1𝑤2𝑥𝑡}𝑤₁_ℤ𝑝𝑒^{𝑤1𝑥1𝑡}𝑞^𝑤1𝑥1𝑑𝑥₁)(𝑤¹∫_ℤ𝑝𝑒^{𝑤2𝑥2𝑡}𝑞^𝑤2𝑥2𝑑𝑥₂=(𝑒_{𝑤1𝑤2𝑥𝑡}∫_ℤ𝑝𝑒^{𝑤1𝑤2𝑥𝑡}𝑞^{𝑤1𝑤2𝑥}𝑑𝑥)𝑤₁_ℤ𝑝𝑒^{𝑤1𝑥1𝑡}𝑞^𝑤1𝑥1𝑑𝑥₁)(^𝑤1−1_𝑖=0=1𝑞^𝑤2𝑖𝑒^{𝑤2𝑖𝑡})𝑤_1𝑤¹⁻¹_𝑖=0𝑞^𝑤2𝑖_ℤ𝑝𝑒^{(𝑥1+𝑤2𝑥+(𝑤2/𝑤1)𝑖)𝑡𝑤1}𝑞^𝑥𝑤1𝑑𝑥₁=^∞_𝑛=0(^𝑤1−1_𝑖=0𝐵_{𝑛,𝑞𝑤1}(𝑤₂𝑥+𝑤²𝑤₁𝑖)𝑤^1𝑛−1𝑞^𝑤2𝑖)𝑡^𝑛𝑛!.(2.23)

From the symmetry of 𝑇(𝑤₁,𝑤₂;𝑥,𝑡), we can also derive 𝑇(𝑤₁,𝑤₂;𝑥,𝑡)=^∞_𝑛=0(^𝑤2−1_𝑖=0𝐵_{𝑛,𝑞𝑤2}(𝑤₁𝑥+𝑤¹𝑤₂𝑖)𝑤^2𝑛−1𝑞^𝑤1𝑖)𝑡^𝑛𝑛!.(2.24)

By comparing the coefficients 𝑡^𝑛/𝑛! on the both sides of (2.23) and (2.24), we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℤ₊, 𝑤₁,𝑤₂∈ℕ, we have ^𝑤¹⁻¹_𝑖=0𝐵_{𝑛,𝑞𝑤1}(𝑤₂𝑥+𝑤²𝑤₁𝑖)𝑤^1𝑛−1𝑞^𝑤2𝑖=^𝑤2−1_𝑖=0𝐵_{𝑛,𝑞𝑤2}(𝑤₁𝑥+𝑤¹𝑤₂𝑖)𝑤^2𝑛−1𝑞^𝑤1𝑖.(2.25)

Remark 2.4. Setting 𝑤₂=1 in Theorem 2.3, we get the multiplication theorem for the 𝑞-Bernoulli polynomials as follows: 𝐵_𝑛,𝑞(𝑤₁𝑥)=𝑤^{1𝑤𝑛−11−1}_𝑖=0𝐵_{𝑛,𝑞𝑤1}(𝑥+𝑖𝑤₁)𝑞^𝑖.(2.26)
I cannot obtain the extended formulae of Theorems 2.1 and 2.3 related to the Carlitz's 𝑞-Bernoulli numbers and polynomials. So, we suggest the following two questions.

Question. Find the extended formulae of Theorems 2.1 and 2.3, which are related to the Carlitz's 𝑞-Bernoulli numbers and polynomials.

Question. Find the twisted formulae of Theorems 2.1 and 2.3, which are related to the twisted Carlitz's 𝑞-Bernoulli polynomials.

Remark 2.5. In [12], 𝑞-Volkenborn integral is defined by 𝐼_𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)=lim_𝑁→∞1[𝑝^𝑁]_𝑞𝑝^𝑁−1_𝑥=0𝑓(𝑥)𝑞^𝑥.(2.27)
Thus, we note that Carlitz's 𝑞-Bernoulli numbers can be written by 𝛽_𝑛,𝑞=_ℤ𝑝[𝑥]^𝑛𝑞𝑑𝜇_𝑞(𝑥),Witt'stypeformula.(2.28)

Acknowledgments

The author wishes to express his sincere gratitude to the referees for their valuable suggestions and comments. The present Research has been conducted by the research Grant of Kwangwoon University in 2008.