On the Modified 𝑞-Bernoulli Numbers of Higher Order with Weight

Abstract

The purpose of this paper is to give some properties of the modified 𝑞-Bernoulli numbers and polynomials of higher order with weight. In particular, by using the bosonic 𝑝-adic 𝑞-integral on ℤ_𝑝, we derive new identities of 𝑞-Bernoulli numbers and polynomials with weight.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper ℤ_𝑝,ℚ_𝑝, and ℂ_𝑝 will, respectively, 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. Let ℕ be the set of natural numbers and ℤ₊=ℕ∪{0}. The 𝑝-adic norm of ℂ_𝑝 is defined by |𝑝|_𝑝=1/𝑝. When one talks of a 𝑞-extension, 𝑞 can be considered as an indeterminate, a complex number 𝑞∈ℂ, or a 𝑝-adic number 𝑞∈ℂ_𝑝. Throughout this paper we assume that 𝛼∈ℚ and 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<𝑝^{−1/(𝑝−1)} so that 𝑞^𝑥=exp(𝑥log𝑞).

Let 𝑈𝐷(ℤ_𝑝) be the space of uniformly differentiable functions on ℤ_𝑝. For 𝑓∈𝑈𝐷(ℤ_𝑝), the 𝑝-adic 𝑞-integral on ℤ_𝑝 is defined by Kim (see [1–3]) as follows:𝐼_𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁_𝑞^𝑝𝑁−1_𝑥=0𝑓(𝑥)𝑞^𝑥,(1.1) where [𝑥]_𝑞 is the 𝑞-number of 𝑥 which is defined by [𝑥]_𝑞=(1−𝑞^𝑥)/(1−𝑞).

From (1.1), we have𝑞^𝑛𝐼_𝑞𝑓_𝑛−𝐼_𝑞(𝑓)=(𝑞−1)^𝑛−1_𝑙=0𝑞^𝑙𝑓(𝑙)+𝑞−1log𝑞^𝑛−1_𝑙=0𝑞^𝑙𝑓^(𝑙),(1.2) where 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛) (see [2–4]).

As is well known, Bernoulli numbers are inductively defined by 𝐵₀=1,(𝐵+1)^𝑛−𝐵_𝑛=1if𝑛=1,0if𝑛>1,(1.3) with the usual convention about replacing 𝐵^𝑛 by 𝐵_𝑛 (see [3, 5]).

In [2, 5, 6], the 𝑞-Bernoulli numbers are defined by𝐵_0,𝑞=𝑞−1log𝑞,𝑞𝐵_𝑞+1^𝑛−𝐵_𝑛,𝑞=1if𝑛=1,0if𝑛>1,(1.4) with the usual convention about replacing 𝐵^𝑛_𝑞 by 𝐵_𝑛,𝑞. Note that lim_𝑞→1𝐵_𝑛,𝑞=𝐵_𝑛. In the viewpoint of (1.4), we consider the modified 𝑞-Bernoulli numbers with weight.

In this paper we study families of the modified 𝑞-Bernoulli numbers and polynomials of higher order with weight. In particular, by using the multivariate 𝑝-adic 𝑞-integral on ℤ_𝑝, we give new identities of the higher-order 𝑞-Bernoulli numbers and polynomials with weight.

2. Modified 𝑞-Bernoulli Numbers with Weight of Higher Order

For 𝑛∈ℤ₊, let us consider the following modified 𝑞-Bernoulli numbers with weight 𝛼 (see [1, 3]):𝐵^(𝛼)_𝑛,𝑞=_ℤ𝑝[𝑥]^𝑛_𝑞𝛼𝑞^−𝑥𝑑𝜇_𝑞(𝑥)=1(1−𝑞)^𝑛[𝛼]^𝑛_𝑞^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)^𝑙𝛼𝑙[𝛼𝑙]_𝑞,𝐵^(𝛼)_𝑛,𝑞(𝑥)=_ℤ𝑝[𝑥+𝑦]^𝑛_𝑞𝛼𝑞^−𝑦𝑑𝜇_𝑞(𝑦)=1(1−𝑞)^𝑛[𝛼]^𝑛_𝑞^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)^𝑙𝑞^𝛼𝑙𝑥𝛼𝑙[𝛼𝑙]_𝑞.(2.1) From (2.1), we note that𝐵^(𝛼)_𝑛,𝑞(𝑥)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠[𝑥]^𝑛−𝑙_𝑞𝛼𝑞^𝛼𝑙𝑥𝐵^(𝛼)_𝑙,𝑞(2.2) (see [1, 3]).

For 𝑘∈ℕ and 𝑛∈ℤ₊, by making use of the multivariate 𝑝-adic 𝑞-integral on ℤ_𝑝, we consider the following modified 𝑞-Bernoulli numbers with weight 𝛼 of order 𝑘, 𝐵^(𝑘,𝛼)_𝑛,𝑞:𝐵^(𝑘,𝛼)_𝑛,𝑞=_ℤ𝑝⋯_ℤ𝑝𝑥₁+⋯+𝑥_𝑘^𝑛_𝑞𝛼𝑞^{−𝑥1−⋯−𝑥𝑘}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘.(2.3) Note that 𝐵^(1,𝛼)_𝑛,𝑞=𝐵^(𝛼)_𝑛,𝑞 and lim_𝑞→1𝐵^(𝑘,𝛼)_𝑛,𝑞=𝐵^(𝑘)_𝑛, where 𝐵^(𝑘)_𝑛 are the 𝑛th ordinary Bernoulli numbers of order 𝑘.

For 𝑘,𝑁∈ℕ, we have1−𝑞1−𝑞^𝑝𝑁^{𝑘𝑝𝑁−1}_𝑖1=0⋯^𝑝𝑁−1_𝑖𝑘=0𝑖₁+⋯+𝑖_𝑘^𝑛_𝑞𝛼=1−𝑞1−𝑞^𝑝𝑁^𝑘11−𝑞^𝛼^{𝑛𝑝𝑁−1}_{𝑖1,…,𝑖𝑘=0}^𝑛_𝑗=0⎛⎜⎜⎝𝑛𝑗⎞⎟⎟⎠(−1)^𝑗𝑞^{𝛼(𝑖1+⋯+𝑖𝑘)𝑗}=1(1−𝑞)^𝑛[𝛼]^𝑛_𝑞^𝑛_𝑗=0⎛⎜⎜⎝𝑛𝑗⎞⎟⎟⎠(−1)^𝑗(1−𝑞)^𝑘1−𝑞^𝑝𝑁^𝑘⎛⎜⎝1−𝑞^{𝛼𝑝𝑁𝑗}1−𝑞^𝛼𝑗⋯1−𝑞^{𝛼𝑝𝑁𝑗}1−𝑞^𝛼𝑗⎞⎟_{𝑘−times}⎠.(2.4) By (1.1), (2.3), and (2.4), we get𝐵^(𝑘,𝛼)_𝑛,𝑞=1(1−𝑞)^𝑛[𝛼]^𝑛_𝑞^𝑛_𝑗=0⎛⎜⎜⎝𝑛𝑗⎞⎟⎟⎠(−1)^𝑗(𝛼𝑗)^𝑘[𝛼𝑗]^𝑘_𝑞.(2.5) Therefore, by (2.5), we obtain the following theorem.

Theorem 2.1. For 𝑛≥0, one has 𝐵^(𝑘,𝛼)_𝑛,𝑞=1(1−𝑞)^𝑛[𝛼]^𝑛_𝑞^𝑛_𝑗=0⎛⎜⎜⎝𝑛𝑗⎞⎟⎟⎠(−1)^𝑗(𝛼𝑗)^𝑘[𝛼𝑗]^𝑘_𝑞.(2.6)

Let us consider the modified 𝑞-Bernoulli and polynomials with weight 𝛼 of order 𝑘 as follows:𝐵^(𝑘,𝛼)_𝑛,𝑞(𝑥)=_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑘^𝑛_𝑞𝛼𝑞^{−𝑥1−⋯−𝑥𝑘}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘.(2.7) By the same method of (2.5), we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℤ₊, one has 𝐵^(𝑘,𝛼)_𝑛,𝑞(𝑥)=1(1−𝑞)^𝑛[𝛼]^𝑛_𝑞^𝑛_𝑗=0⎛⎜⎜⎝𝑛𝑗⎞⎟⎟⎠(−1)^𝑗𝑞^𝛼𝑥𝑗(𝛼𝑗)^𝑘[𝛼𝑗]^𝑘_𝑞.(2.8)

By Theorem 2.2, we get𝐵^(𝑘,𝛼)_{𝑛,𝑞−1}(𝑘−𝑥)=1(1−𝑞^−𝛼)^𝑛𝑛_𝑗=0⎛⎜⎜⎝𝑛𝑗⎞⎟⎟⎠(−1)^𝑗(𝛼𝑗)^𝑘[𝛼𝑗]^𝑘_𝑞−1𝑞^{−𝛼𝑗(𝑘−𝑥)}=(−1)^𝑛𝑞^𝛼𝑛(1−𝑞^𝛼)^𝑛𝑛_𝑗=0⎛⎜⎜⎝𝑛𝑗⎞⎟⎟⎠(−1)^𝑗𝑞⁻¹(𝑞−1)𝛼𝑗(𝑞^𝛼𝑗−1)𝑞^−𝛼𝑗^𝑘𝑞^{−𝛼𝑗(𝑘−𝑥)}=(−1)^𝑛𝑞^𝛼𝑛(1−𝑞^𝛼)^𝑛𝑛_𝑗=0⎛⎜⎜⎝𝑛𝑗⎞⎟⎟⎠(−1)^𝑗𝑞^𝛼𝑗𝑥𝑞^−𝑘(𝛼𝑗)^𝑘[𝛼𝑗]^𝑘_𝑞=(−1)^𝑛𝑞^{𝛼𝑛−𝑘}𝐵^(𝑘,𝛼)_𝑛,𝑞(𝑥).(2.9) Therefore, by (2.9), we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℤ₊, one has 𝐵^(𝑘,𝛼)_{𝑛,𝑞−1}(𝑘−𝑥)=(−1)^𝑛𝑞^{𝛼𝑛−𝑘}𝐵^(𝑘,𝛼)_𝑛,𝑞(𝑥),𝐵^(𝑘,𝛼)_{𝑛,𝑞−1}(𝑘)=(−1)^𝑛𝑞^{𝛼𝑛−𝑘}𝐵^(𝑘,𝛼)_𝑛,𝑞.(2.10)

From Theorem 2.3, we note thatlim_𝑞→1𝐵^(𝑘,𝛼)_{𝑛,𝑞−1}(𝑘−𝑥)=𝐵^(𝑘)_𝑛(𝑘−𝑥),lim_𝑞→1𝐵^(𝑘,𝛼)_{𝑛,𝑞−1}(𝑘)=(−1)^𝑛𝐵^(𝑘)_𝑛.(2.11) Thus, we have 𝐵^(𝑘)_𝑛(𝑘)=(−1)^𝑛𝐵^(𝑘)_𝑛, where 𝐵^(𝑘)_𝑛 are the 𝑛th Bernoulli numbers of order 𝑘.

From (2.3) and (2.7), we can derive the following equations:𝐵^(𝑙,𝛼)_𝑘,𝑞(𝑥)=lim_𝑁→∞1[𝑚]^𝑙_𝑞𝑝^𝑁^𝑙_𝑞𝑚^𝑚−1_{𝑖1,…,𝑖𝑙=0}^𝑝𝑁−1_{𝑛1,…,𝑛𝑙=0}𝑥+𝑖₁+⋯+𝑖_𝑙+𝑚𝑛₁+⋯+𝑛_𝑙^𝑘_𝑞𝛼=[𝑚]^𝑘_𝑞𝛼[𝑚]^𝑙_𝑞^𝑚−1_{𝑖1,…,𝑖𝑙=0}_ℤ𝑝⋯_ℤ𝑝𝑥+𝑖₁+⋯+𝑖_𝑙𝑚+𝑥₁+⋯+𝑥_𝑙^𝑘_𝑞𝛼𝑚×𝑞^{−𝑚𝑥1−⋯−𝑚𝑥𝑙}𝑑𝜇_𝑞𝑚𝑥₁⋯𝑑𝜇_𝑞𝑚𝑥_𝑘=[𝑚]^𝑘_𝑞𝛼[𝑚]^𝑙_𝑞^𝑚−1_{𝑖1,…,𝑖𝑙=0}𝐵^(𝑙,𝛼)_{𝑘,𝑞𝑚}𝑥+𝑖₁+⋯+𝑖_𝑙𝑚.(2.12) Therefore, by (2.12), we obtain the following theorem.

Theorem 2.4. For 𝑘∈ℤ₊ and 𝑙,𝑚∈ℕ, one has 𝐵^(𝑙,𝛼)_𝑘,𝑞(𝑥)=[𝑚]^𝑘_𝑞𝛼[𝑚]^𝑙_𝑞^𝑚−1_{𝑖1,…,𝑖𝑙=0}𝐵^(𝑙,𝛼)_{𝑘,𝑞𝑚}𝑥+𝑖₁+⋯+𝑖_𝑙𝑚.(2.13)

In particular,𝐵^(𝑙,𝛼)_𝑘,𝑞(𝑚𝑥)=[𝑚]^𝑘_𝑞𝛼[𝑚]^𝑙_𝑞^𝑚−1_{𝑖1,…,𝑖𝑙=0}𝐵^(𝑙,𝛼)_{𝑘,𝑞𝑚}𝑥+𝑖₁+⋯+𝑖_𝑙𝑚.(2.14)

From (1.2), we can derive the following integral:_ℤ𝑝𝑓(𝑥+1)𝑞^−𝑥𝑑𝜇_𝑞(𝑥)=_ℤ𝑝𝑓(𝑥)𝑞^−𝑥𝑑𝜇_𝑞(𝑥)+𝑞−1log𝑞𝑓^(0),_ℤ𝑝𝑓(𝑥+2)𝑞^−𝑥𝑑𝜇_𝑞(𝑥)=_ℤ𝑝𝑓₁(𝑥)𝑞^−𝑥𝑑𝜇_𝑞(𝑥)+𝑞−1log𝑞𝑓^(1)=_ℤ𝑝𝑓(𝑥)𝑞^−𝑥𝑑𝜇_𝑞(𝑥)+𝑞−1log𝑞𝑓^(0)+𝑓^(1).(2.15) Continuing this process, we obtain_ℤ𝑝𝑓(𝑥+𝑛)𝑞^−𝑥𝑑𝜇_𝑞(𝑥)=_ℤ𝑝𝑓(𝑥)𝑞^−𝑥𝑑𝜇_𝑞(𝑥)+𝑞−1log𝑞^𝑛−1_𝑙=0𝑓^(𝑙).(2.16) By (2.16), we get_ℤ𝑝[𝑥+𝑛]^𝑚_𝑞𝛼𝑞^−𝑥𝑑𝜇_𝑞(𝑥)=_ℤ𝑝[𝑥]^𝑚_𝑞𝛼𝑞^−𝑥𝑑𝜇_𝑞(𝑥)+𝑚𝛼[𝛼]_𝑞^𝑛−1_𝑙=0[𝑙]^𝑚−1_𝑞𝛼𝑞^𝛼𝑙.(2.17) Therefore, by (2.1) and (2.17), we obtain the following theorem.

Theorem 2.5. For 𝑛∈ℕ and 𝑚∈ℤ₊, one has 𝐵^(𝛼)_𝑚,𝑞(𝑛)−𝐵^(𝛼)_𝑚,𝑞=𝑚𝛼[𝛼]_𝑞^𝑛−1_𝑙=0[𝑙]^𝑚_𝑞𝛼𝑞^𝛼𝑙.(2.18)

In an analogues manner as the previous investigation [7–10], we can define a further generalization of modified 𝑞-Bernoulli numbers with weight. Let 𝜒 be the Dirichlet character with conductor 𝑑∈ℕ. Then the generalized 𝑞-Bernoulli numbers with weight attached to 𝜒 can be defined as follows:𝐵^(𝛼)_{𝑛,𝜒,𝑞}=_𝑋𝜒(𝑥)[𝑥]^𝑛_𝑞𝛼𝑞^−𝑥𝑑𝜇_𝑞(𝑥)=[𝑑]^𝑛_𝑞𝛼[𝑑]_𝑞^𝑑−1_𝑎=0𝜒(𝑎)𝐵^(𝛼)_{𝑛,𝑞𝑑}𝑎𝑑.(2.19) We expect to investigate these objects in future papers. This definition 𝐵^(𝛼)_𝑛,𝑞 was also given in a previous paper (see [9]).

Acknowledgments

The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2011.