Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

Abstract

By the properties of 𝑝-adic invariant integral on ℤ_𝑝, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties of 𝑝-adic invariant integral on ℤ_𝑝, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

1. Introduction

Let 𝑝 be a fixed prime number. Throughout this paper, the symbols ℤ, ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 will denote the ring of rational integers, the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of ℚ_𝑝, respectively. 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)}=1/𝑝. Let UD(ℤ_𝑝) be the space of uniformly differentiable function on ℤ_𝑝. 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),(1.1) (see [1]). From the definition (1.1), we have 𝐼(𝑓₁)=𝐼(𝑓)+𝑓^′(0),where𝑓^′(0)=𝑑𝑓(𝑥)|𝑑𝑥_𝑥=0,𝑓₁(𝑥)=𝑓(𝑥+1).(1.2) Let 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛), (𝑛∈ℕ). Then we can derive the following equation from (1.2): 𝐼(𝑓_𝑛)=𝐼(𝑓)+^𝑛−1_𝑖=0𝑓^′(𝑖),(1.3) (see [1]). It is well known that the ordinary Bernoulli polynomials 𝐵_𝑛(𝑥) are defined as 𝑡𝑒^𝑡𝑒−1^𝑥𝑡=^∞_𝑛=0𝐵_𝑛𝑡(𝑥)^𝑛𝑛!,(1.4) (see [1–25]), and the Bernoulli number 𝐵_𝑛 are defined as 𝐵_𝑛=𝐵_𝑛(0).

Let 𝑑 be a fixed positive integer. For 𝑛∈ℕ, we set 𝑋=𝑋_𝑑=lim_←𝑁(ℤ/𝑑𝑝^𝑁ℤ),𝑋₁=ℤ_𝑝;𝑋^∗=_{0<𝑎<𝑑𝑝,(𝑎,𝑝)=1}(𝑎+𝑑𝑝ℤ_𝑝);𝑎+𝑑𝑝^𝑁ℤ_𝑝={𝑥∈𝑋∣𝑥≡𝑎(mod𝑑𝑝^𝑁)},(1.5) where 𝑎∈ℤ lies in 0≤𝑎<𝑑𝑝^𝑁. It is easy to see that _𝑋𝑓(𝑥)𝑑𝑥=_ℤ𝑝𝑓(𝑥)𝑑𝑥,for𝑓∈UD(ℤ_𝑝).(1.6)

In [14], the Witt's formula for the Bernoulli numbers are given by _ℤ𝑝𝑥^𝑛𝑑𝑥=𝐵_𝑛,𝑛∈ℤ₊.(1.7) Let 𝜒 be the Dirichlet's character with conductor 𝑑∈ℕ. Then the generalized Bernoulli polynomials attached to 𝜒 are defined as ^𝑑_𝑎=1𝜒(𝑎)𝑡𝑒^𝑎𝑡𝑒^𝑑𝑡𝑒−1^𝑥𝑡=^∞_𝑛=0𝐵_𝑛,𝜒𝑡(𝑥)^𝑛𝑛!,(1.8) (see [22]), and the generalized Bernoulli numbers attached to 𝜒, 𝐵_𝑛,𝜒 are defined as 𝐵_𝑛,𝜒=𝐵_𝑛,𝜒(0).

In this paper, we investigate the interesting identities of symmetry for the generalized Bernoulli numbers and polynomials attached to 𝜒 by using the properties of 𝑝-adic invariant integral on ℤ_𝑝. Finally, we will give relationship between the power sum polynomials and the generalized Bernoulli numbers attached to 𝜒.

2. Symmetry of Power Sum and the Generalized Bernoulli Polynomials

Let 𝜒 be the Dirichlet character with conductor 𝑑∈ℕ. From (1.3), we note that _𝑋𝜒(𝑥)𝑒^𝑥𝑡𝑡∑𝑑𝑥=^{𝑑−1𝑖=0}𝜒(𝑖)𝑒^𝑖𝑡𝑒^𝑑𝑡=−1^∞_𝑛=0𝐵_𝑛,𝜒𝑡^𝑛𝑛!,(2.1) where 𝐵_𝑛,𝜒(𝑥) are the 𝑛th generalized Bernoulli numbers attached to 𝜒. Now, we also see that the generalized Bernoulli polynomials attached to 𝜒 are given by _𝑋𝜒(𝑦)𝑒^{(𝑥+𝑦)𝑡}𝑡∑𝑑𝑦=^{𝑑−1𝑖=0}𝜒(𝑖)𝑒^𝑖𝑡𝑒^𝑑𝑡𝑒−1^𝑥𝑡=^∞_𝑛=0𝐵_𝑛,𝜒𝑡(𝑥)^𝑛𝑛!.(2.2) By (2.1) and (2.2), we easily see that _𝑋𝜒(𝑥)𝑥^𝑛𝑑𝑥=𝐵_𝑛,𝜒,_𝑋𝜒(𝑦)(𝑥+𝑦)^𝑛𝑑𝑦=𝐵_𝑛,𝜒(𝑥).(2.3) From (2.2), we have 𝐵_𝑛,𝜒(𝑥)=^𝑛_ℓ=0𝑛ℓ𝐵_ℓ,𝜒𝑥^𝑛−ℓ.(2.4) From (2.2), we can also derive _𝑋𝜒(𝑥)𝑒^𝑥𝑡𝑑𝑥=^𝑑−1_𝑖=0𝑡𝜒(𝑖)𝑒^𝑑𝑡𝑒−1^{(𝑖/𝑑)𝑑𝑡}=^∞_𝑛=0(𝑑^{𝑛−1𝑑−1}_𝑖=0𝜒(𝑖)𝐵_𝑛(𝑖𝑑𝑡))^𝑛𝑛!.(2.5) Therefore, we obtain the following lemma.

Lemma 2.1. For 𝑛∈ℤ₊, one has _𝑋𝜒(𝑥)𝑥^𝑛𝑑𝑥=𝐵_𝑛,𝜒=𝑑^{𝑛−1𝑑−1}_𝑖=0𝜒(𝑖)𝐵_𝑖(𝑖𝑑).(2.6)

We observe that 1𝑡(_𝑋𝜒(𝑥)𝑒^{(𝑛𝑑+𝑥)𝑡}𝑑𝑥−_𝑋𝑒^𝑥𝑡∫𝜒(𝑥)𝑑𝑥)=𝑛𝑑_𝑋𝜒(𝑥)𝑒^𝑥𝑡𝑑𝑥∫_𝑋𝑒^{𝑛𝑑𝑥𝑡}=𝑒𝑑𝑥^𝑛𝑑𝑡−1𝑒^𝑑𝑡(−1^𝑑−1_𝑖=0𝜒(𝑖)𝑒^𝑖𝑡).(2.7) Thus, we have 1𝑡(_𝑋𝜒(𝑥)𝑒^{(𝑛𝑑+𝑥)𝑡}𝑑𝑥−_𝑋𝜒(𝑥)𝑒^𝑥𝑡𝑑𝑥)=^∞_𝑘=0(^𝑛𝑑−1_ℓ=0𝜒(ℓ)ℓ^𝑘)𝑡^𝑘𝑘!.(2.8) Let us define the 𝑝-adic functional 𝑇_𝑘(𝜒,𝑛) as follows: 𝑇_𝑘(𝜒,𝑛)=^𝑛_ℓ=0𝜒(ℓ)ℓ^𝑘,for𝑘∈ℤ₊.(2.9) By (2.8) and (2.9), we see that 1𝑡(_𝑋𝜒(𝑥)𝑒^{(𝑛𝑑+𝑥)𝑡}𝑑𝑥−_𝑋𝜒(𝑥)𝑒^𝑥𝑡𝑑𝑥)=^∞_𝑛=0(𝑇_𝑘𝑡(𝜒,𝑛𝑑−1))^𝑘𝑘!.(2.10) By using Taylor expansion in (2.10), we have _𝑋𝜒(𝑥)(𝑑𝑛+𝑥)^𝑘𝑑𝑥−_𝑋𝜒(𝑥)𝑥^𝑘𝑑𝑥=𝑘𝑇_𝑘−1(𝜒,𝑛𝑑−1),for𝑘,𝑛,𝑑∈ℕ.(2.11) That is, 𝐵_𝑘,𝜒(𝑛𝑑)−𝐵_𝑘,𝜒=𝑘𝑇_𝑘−1(𝜒,𝑛𝑑−1).(2.12) Let 𝑤₁,𝑤₂,𝑑∈ℕ. Then we consider the following integral equation: 𝑑∬_𝑋𝜒(𝑥₁)𝜒(𝑥₂)𝑒^{(𝑤1𝑥1+𝑤2𝑥2)𝑡}𝑑𝑥₁𝑑𝑥₂∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑥𝑡}=𝑑𝑥𝑡(𝑒^{𝑑𝑤1𝑤2𝑡}−1)(𝑒^{𝑤1𝑑𝑡}−1)(𝑒^{𝑤2𝑑𝑡}(−1)^𝑑−1_𝑎=0𝜒(𝑎)𝑒^{𝑤1𝑎𝑡})(^𝑑−1_𝑏=0𝜒(𝑏)𝑒^{𝑤2𝑏𝑡}).(2.13) From (2.7) and (2.10), we note that 𝑑𝑤₁∫_𝑋𝜒(𝑥)𝑒^𝑥𝑡𝑑𝑥∫_𝑋𝑒^{𝑑𝑤1𝑥𝑡}=𝑑𝑥^∞_𝑘=0(𝑇_𝑘(𝜒,𝑑𝑤₁𝑡−1))^𝑘𝑘!.(2.14) Let us consider the 𝑝-adic functional 𝑇_𝜒(𝑤₁,𝑤₂) as follows: 𝑇_𝜒(𝑤₁,𝑤₂)=𝑑∬_𝑋𝜒(𝑥₁)𝜒(𝑥₂)𝑒^{(𝑤1𝑥1+𝑤2𝑥2+𝑤1𝑤2𝑥)𝑡}𝑑𝑥₁𝑑𝑥₂∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑥3𝑡}𝑑𝑥₃.(2.15) Then we see that 𝑇_𝜒(𝑤₁,𝑤₂) is symmetric in 𝑤₁ and 𝑤₂, and 𝑇_𝜒(𝑤₁,𝑤₂)=𝑡(𝑒^{𝑑𝑤1𝑤2𝑡}−1)𝑒^{𝑤1𝑤2𝑥𝑡}(𝑒^{𝑤1𝑑𝑡}−1)(𝑒^{𝑤2𝑑𝑡}(−1)^𝑑−1_𝑎=0𝜒(𝑎)𝑒^{𝑤1𝑎𝑡})(^𝑑−1_𝑏=0𝜒(𝑏)𝑒^{𝑤2𝑏𝑡}).(2.16) By (2.15) and (2.16), we have

𝑇_𝜒(𝑤₁,𝑤₂1)=(𝑤_1𝑋𝜒(𝑥₁)𝑒^{𝑤1(𝑥1+𝑤2𝑥)𝑡}𝑑𝑥₁)(𝑑𝑤₁∫_𝑋𝜒(𝑥₂)𝑒^{𝑤2𝑥2𝑡}𝑑𝑥₂∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑥𝑡})1𝑑𝑥=(𝑤₁^∞_𝑖=0𝐵_𝑖,𝜒(𝑤₂𝑤𝑥)^𝑖1𝑡^𝑖𝑖!)(^∞_𝑘=0𝑇_𝑘(𝜒,𝑑𝑤₁𝑤−1)^𝑘2𝑡^𝑘)=1𝑘!𝑤₁(^∞_ℓ=0(^ℓ_𝑖=0𝐵_𝑖,𝜒(𝑤₂𝑥)𝑇_ℓ−𝑖(𝜒,𝑑𝑤₁−1)𝑤^𝑖₁𝑤₂^ℓ−𝑖ℓ!)𝑡𝑖!(ℓ−𝑖)!^ℓ)=ℓ!_∞ℓ=0(^ℓ_𝑖=0ℓ𝑖𝐵_𝑖,𝜒(𝑤₂𝑥)𝑇_ℓ−𝑖(𝜒,𝑑𝑤₁−1)𝑤₁^𝑖−1𝑤₂^ℓ−𝑖)𝑡^ℓ.ℓ!(2.17) From the symmetric property of 𝑇_𝜒(𝑤₁,𝑤₂) in 𝑤₁ and 𝑤₂, we note that 𝑇_𝜒(𝑤₁,𝑤₂1)=(𝑤_2𝑋𝜒(𝑥₂)𝑒^{𝑤2(𝑥2+𝑤1𝑥)𝑡}𝑑𝑥₂)(𝑑𝑤₂∫_𝑋𝜒(𝑥₁)𝑒^{𝑤1𝑥1𝑡}𝑑𝑥₁∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑥𝑡})1𝑑𝑥=(𝑤₂^∞_𝑖=0𝐵_𝑖,𝜒(𝑤₁𝑤𝑥)^𝑖2𝑡^𝑖𝑖!)(^∞_𝑘=0𝑇_𝑘(𝜒,𝑑𝑤₂𝑤−1)^𝑘1𝑡^𝑘)=1𝑘!𝑤₂(^∞_ℓ=0(^ℓ_𝑖=0𝐵_𝑖,𝜒(𝑤₁𝑥)𝑤^𝑖₂𝑇_ℓ−𝑖(𝜒,𝑑𝑤₂−1)𝑤₁^ℓ−𝑖ℓ!)𝑡𝑖!(ℓ−𝑖)!^ℓ)=ℓ!_∞ℓ=0(^ℓ_𝑖=0ℓ𝑖𝑤₂^𝑖−1𝑤₁^ℓ−𝑖𝐵_𝑖,𝜒(𝑤₁𝑥)𝑇_ℓ−𝑖(𝜒,𝑑𝑤₂𝑡−1))^ℓ.ℓ!(2.18) By comparing the coefficients on the both sides of (2.17) and (2.18), we obtain the following theorem.

Theorem 2.2. For 𝑤₁,𝑤₂,𝑑∈ℕ, one has ^ℓ_𝑖=0ℓ𝑖𝐵_𝑖,𝜒(𝑤₂𝑥)𝑇_ℓ−𝑖(𝜒,𝑑𝑤₁−1)𝑤₁^𝑖−1𝑤₂^ℓ−𝑖=^ℓ_𝑖=0ℓ𝑖𝐵_𝑖,𝜒(𝑤₁𝑥)𝑇_ℓ−𝑖(𝜒,𝑑𝑤₂−1)𝑤₂^𝑖−1𝑤₁^ℓ−𝑖.(2.19)

Let 𝑥=0 in Theorem 2.2. Then we have ^ℓ_𝑖=0ℓ𝑖𝐵_𝑖,𝜒𝑇_ℓ−𝑖(𝜒,𝑑𝑤₁−1)𝑤₁^𝑖−1𝑤₂^ℓ−𝑖=^ℓ_𝑖=0ℓ𝑖𝐵_𝑖,𝜒𝑇_ℓ−𝑖(𝜒,𝑑𝑤₂−1)𝑤₂^𝑖−1𝑤₁^ℓ−𝑖.(2.20) By (2.14) and (2.16), we also see that 𝑇_𝜒(𝑤₁,𝑤₂𝑒)=(^{𝑤1𝑤2𝑥𝑡}𝑤_1𝑋𝜒(𝑥₁)𝑒^{𝑤1𝑥1𝑡}𝑑𝑥₁)(𝑑𝑤₁∫_𝑋𝜒(𝑥₂)𝑒^{𝑤2𝑥2𝑡}𝑑𝑥₂∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑥𝑡})𝑒𝑑𝑥=(^{𝑤1𝑤2𝑥𝑡}𝑤_1𝑋𝜒(𝑥₁)𝑒^{𝑤1𝑥1𝑡}𝑑𝑥₁𝑒)(^{𝑑𝑤1𝑤2𝑡}−1𝑒^{𝑤2𝑑𝑡}−1)(^𝑑−1_𝑖=0𝜒(𝑖)𝑒^{𝑤2𝑖𝑡})𝑒=(^{𝑤1𝑤2𝑥𝑡}𝑤_1𝑋𝜒(𝑥₁)𝑒^{𝑤1𝑥1𝑡}𝑑𝑥₁)(^𝑤1−1_ℓ=0^𝑑−1_𝑖=0𝑒^{𝑤2(𝑖+ℓ𝑑)𝑡}𝑒𝜒(𝑖+ℓ𝑑))=(^{𝑤1𝑤2𝑥𝑡}𝑤_1𝑋𝜒(𝑥₁)𝑒^{𝑤1𝑥1𝑡}𝑑𝑥₁)(^{𝑑𝑤1−1}_𝑖=0𝑒^{𝑤2𝑖𝑡}=1𝜒(𝑖))𝑤₁^{𝑑𝑤1−1}_𝑖=0𝜒(𝑖)_𝑋𝜒(𝑥₁)𝑒^{𝑤1(𝑥1+𝑤2𝑥+(𝑤2/𝑤1)𝑖)𝑡}𝑑𝑥₁=1𝑤₁^{𝑑𝑤1−1}_𝑖=0𝜒(𝑖)^∞_𝑘=0𝐵_𝑘,𝜒(𝑤₂𝑤𝑥+²𝑤₁𝑤𝑖)^𝑘1𝑡^𝑘=𝑘!_∞𝑘=0(^{𝑑𝑤1−1}_𝑖=0𝜒(𝑖)𝐵_𝑘,𝜒(𝑤₂𝑤𝑥+²𝑤₁𝑖)𝑤₁^𝑘−1)𝑡^𝑘.𝑘!(2.21) From the symmetric property of 𝑇_𝜒(𝑤₁,𝑤₂) in 𝑤₁ and 𝑤₂, we can also derive the following equation: 𝑇_𝜒(𝑤₁,𝑤₂𝑒)=(^{𝑤1𝑤2𝑥𝑡}𝑤_2𝑋𝜒(𝑥₂)𝑒^{𝑤2𝑥2𝑡}𝑑𝑥₂)(𝑑𝑤₂∫_𝑋𝜒(𝑥₁)𝑒^{𝑤1𝑥1𝑡}𝑑𝑥₁∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑥𝑡})𝑒𝑑𝑥=(^{𝑤1𝑤2𝑥𝑡}𝑤_2𝑋𝜒(𝑥₂)𝑒^{𝑤2𝑥2𝑡}𝑑𝑥₂𝑒)(^{𝑑𝑤1𝑤2𝑡}−1𝑒^{𝑤1𝑑𝑡}−1)(^𝑑−1_𝑖=0𝜒(𝑖)𝑒^{𝑤1𝑖𝑡})𝑒=(^{𝑤1𝑤2𝑥𝑡}𝑤_2𝑋𝜒(𝑥₂)𝑒^{𝑤2𝑥2𝑡}𝑑𝑥₂)(^𝑤2−1_ℓ=0𝑒^{𝑤1𝑑ℓ𝑡})(^𝑑−1_𝑖=0𝜒(𝑖)𝑒^{𝑤1𝑖𝑡})=1𝑤₂^{𝑑𝑤2−1}_𝑖=0𝜒(𝑖)_𝑋𝜒(𝑥₂)𝑒^{𝑤2(𝑥2+𝑤1𝑥+(𝑤1/𝑤2)𝑖)𝑡}𝑑𝑥₂=1𝑤₂^{𝑑𝑤2−1}_𝑖=0𝜒(𝑖)^∞_𝑘=0𝐵_𝑘,𝜒(𝑤₁𝑤𝑥+¹𝑤₂𝑤𝑖)^𝑘2𝑡^𝑘=𝑘!_∞𝑘=0{^{𝑑𝑤2−1}_𝑖=0𝜒(𝑖)𝐵_𝑘,𝜒(𝑤₁𝑤𝑥+¹𝑤₂𝑖)𝑤₂^𝑘−1}𝑡^𝑘.𝑘!(2.22) By comparing the coefficients on the both sides of (2.21) and (2.22), we obtain the following theorem.

Theorem 2.3. For 𝑤₁,𝑤₂,𝑑∈ℕ, one has ^{𝑑𝑤1−1}_𝑖=0𝜒(𝑖)𝐵_𝑘,𝜒(𝑤₂𝑤𝑥+²𝑤₁𝑖)𝑤₁^𝑘−1=^{𝑑𝑤2−1}_𝑖=0𝜒(𝑖)𝐵_𝑘,𝜒(𝑤₁𝑤𝑥+¹𝑤₂𝑖)𝑤₂^𝑘−1.(2.23)

Remark 2.4. Let 𝑥=0 in Theorem 2.3. Then we see that ^{𝑑𝑤1−1}_𝑖=0𝜒(𝑖)𝐵_𝑘,𝜒(𝑤₂𝑤₁𝑖)𝑤₁^𝑘−1=^{𝑑𝑤2−1}_𝑖=0𝜒(𝑖)𝐵_𝑘,𝜒(𝑤₁𝑤₂𝑖)𝑤₂^𝑘−1.(2.24)
If we take 𝑤₂=1, then we have ^{𝑑𝑤1−1}_𝑖=0𝜒(𝑖)𝐵_𝑘,𝜒(𝑖𝑤₁)𝑤₁^𝑘−1=^𝑑−1_𝑖=0𝜒(𝑖)𝐵_𝑘,𝜒(𝑤₁𝑖).(2.25)

Remark 2.5. Let 𝜒 be trivial character. Then we can easily derive the “multiplication theorem for Bernoulli polynomials’’ from Theorems 2.2 and 2.3 (see [14]).

Acknowledgment

The present research has been conducted by the research grant of the Kwangwoon University in 2009.