On the 𝑞-Bernoulli Numbers and Polynomials with Weight 𝜶

Abstract

We present a systemic study of some families of higher-order 𝑞-Bernoulli numbers and polynomials with weight 𝛼. From these studies, we derive some interesting identities on the 𝑞-Bernoulli numbers and polynomials with weight 𝛼.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper, ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 will denote the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of ℚ_𝑝, respectively. The 𝑝-adic norm of ℂ_𝑝 is defined as |𝑥|_𝑝=𝑝^−𝑟, where 𝑥=𝑝^𝑟𝑚/𝑛 with (𝑝,𝑚)=(𝑝,𝑛)=1, 𝑟∈ℚ and 𝑚,𝑛∈ℤ. Let ℕ and ℤ be the set of natural numbers and integers, respectively, ℤ₊=ℕ∪{0}. Let 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<𝑝^{−1/(𝑝−1)}. The notation of 𝑞-number is defined by [𝑥]_𝑤=(1−𝑤^𝑥)/(1−𝑤) and [𝑥]_𝑞=(1−𝑞^𝑥)/(1−𝑞), (see [1–13]).

As the well known definition, the Bernoulli polynomials are defined by 𝑡𝑒^𝑡−1𝑒^𝑥=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛𝑛!.(1.1) In the special case, 𝑥=0, 𝐵_𝑛(0)=𝐵_𝑛 are called the 𝑛th Bernoulli numbers. That is, the recurrence formula for the Bernoulli numbers is given by 𝐵₀=1,(𝐵+1)^𝑛−𝐵_𝑛=10if𝑛=1,if𝑛>1,(1.2) with the usual convention about replacing 𝐵^𝑖 with 𝐵_𝑖.

In [1, 2], 𝑞-extension of Bernoulli numbers are defined by Carlitz as follows:𝛽_0,𝑞=1,𝑞(𝑞𝛽+1)^𝑛−𝛽_𝑛,𝑞=10if𝑛=1,if𝑛>1,(1.3) with the usual convention about replacing 𝛽^𝑖 with 𝛽_𝑖,𝑞.

By (1.2) and (1.3), we get lim_𝑞→1𝛽_𝑖,𝑞=𝐵_𝑖. In this paper, we assume that 𝛼∈ℕ.

In [7], the 𝑞-Bernoulli numbers with weight 𝛼 are defined by Kim as follows:𝛽̃^(𝛼)0,𝑞=1,𝑞𝑞^𝛼𝛽̃^𝛼+1^𝑛−𝛽̃^{(𝛼)𝑛,𝑞}=[𝛼𝛼]_𝑞0if𝑛=1,if𝑛>1,(1.4) with the usual convention about replacing (𝛽̃^(𝛼))^𝑖 with 𝛽̃^{(𝛼)𝑖,𝑞}.

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 as 𝐼_𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁_𝑞𝑝^𝑁−1_𝑥=0𝑓(𝑥)𝑞^𝑥,(1.5) (see[4, 5]). From (1.5), we note that𝑞^𝑛𝐼_𝑞𝑓_𝑛=𝐼_𝑞(𝑓)+(𝑞−1)^𝑛−1_𝑙=0𝑞^𝑙𝑓(𝑙)+𝑞−1log𝑞^𝑛−1_𝑙=0𝑞^𝑙𝑓^(𝑙),(1.6) where 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛) and 𝑓^(𝑙)=(𝑑𝑓(𝑥)/𝑑𝑥)|_𝑥=𝑙.

By (1.4), (1.5), and (1.6), we set 𝛽̃^{(𝛼)𝑛,𝑞}=_ℤ𝑝[𝑥]^𝑛𝑞_𝛼𝑑𝜇_𝑞(𝑥),where𝑛∈ℤ₊,(1.7) (see[7]). The 𝑞-Bernoulli polynomials are also given by 𝛽̃^{(𝛼)𝑛,𝑞}(𝑥)=_ℤ𝑝[𝑥+𝑦]^𝑛𝑞_𝛼𝑑𝜇_𝑞(𝑥)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠[𝑥]_{𝑞𝑛−𝑙𝛼}𝑞^𝛼𝑙𝑥𝛽̃^{(𝛼)𝑙,𝑞}.(1.8)

The purpose of this paper is to derive a new concept of higher-order 𝑞-Bernoulli numbers and polynomials with weight 𝛼 from the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝. Finally, we present a systemic study of some families of higher-order 𝑞-Bernoulli numbers and polynomials with weight 𝛼.

2. Higher Order 𝑞-Bernoulli Numbers with Weight 𝛼

Let 𝛽∈ℤ and 𝛼∈ℕ in this paper. For 𝑘∈ℕ and 𝑛∈ℤ₊, we consider the expansion of higher-order 𝑞-Bernoulli polynomials with weight 𝛼 as follows: 𝛽̃^{(𝛽,𝑘∣𝛼)𝑛,𝑞}(𝑥)=_ℤ𝑝⋯_ℤ𝑝𝑥₁+⋯+𝑥_𝑘+𝑥^𝑛𝑞_𝛼𝑞^{𝑥1(𝛽−1)+⋯+𝑥𝑘(𝛽−𝑘)}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘.(2.1)

From (2.1), we note that 𝛽̃^{(𝛽,𝑘∣𝛼)𝑛,𝑞}(𝑥)=(1−𝑞)^𝑘−𝑛[𝛼]^𝑛𝑞𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)_𝑙𝑞^𝛼𝑙𝑥∏^{𝑘−1𝑖=0}(𝛼𝑙+𝛽−𝑖)=1∏^{𝑘−1𝑖=0}1−𝑞^{𝛼𝑙+𝛽−𝑖}(1−𝑞)^𝑛[𝛼]^𝑛𝑞𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)_𝑙𝑞^𝛼𝑙𝑥^{𝑘𝛼𝑙+𝛽}𝑘!^{𝑘𝛼𝑙+𝛽}_𝑞[𝑘]_𝑞!,(2.2) where (^𝛼𝑙)_𝑞=((1−𝑞^𝛼)(1−𝑞^𝛼−1)⋯(1−𝑞^{𝛼−𝑙+1}))/((1−𝑞)(1−𝑞²)⋯(1−𝑞^𝑙)) and [𝑘]_𝑞!=[𝑘]_𝑞⋯[2]_𝑞[1]_𝑞.

Therefore, we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊ and 𝑘∈ℕ, we have 𝛽̃^{(𝛽,𝑘∣𝛼)𝑛,𝑞}(𝑥)=1(1−𝑞)^𝑛[𝛼]^𝑛𝑞𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)_𝑙𝑞^𝛼𝑙𝑥^{𝑘𝛼𝑙+𝛽}𝑘!^{𝛼𝑙+𝛽}_k_𝑞[𝑘]_𝑞!.(2.3)

In the special case, 𝑥=0, 𝛽̃^{(𝛽,𝑘∣𝛼)𝑛,𝑞}(0)=𝛽̃^{(𝛽,𝑘∣𝛼)𝑛,𝑞} are called the 𝑛th higher order 𝑞-Bernoulli numbers with weight 𝛼.

From (2.1) and (2.2), we can derive 𝛽̃^{(𝛽,𝑘∣𝛼)𝑛,𝑞}=(𝑞^𝛼−1)𝛽̃^{(𝛽−𝛼,𝑘∣𝛼)𝑛+1,𝑞}+𝛽̃^{(𝛽−𝛼,𝑘∣𝛼)𝑛,𝑞}.(2.4) By Theorem 2.1 and (2.4), we get 𝛽̃^{(𝑚𝛼,𝑘∣𝛼)0,𝑞}==_ℤ𝑝𝑚⋯_ℤ𝑝𝑞^{∑𝑘𝑗=1(𝛼𝑚−𝑗)𝑥𝑗}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘_𝑙=0⎛⎜⎜⎝𝑚𝑙⎞⎟⎟⎠(𝑞_𝛼−1)^𝑙_ℤ𝑝=_𝑚⋯_ℤ𝑝𝑥₁+⋯+𝑥_𝑘^𝑙𝑞_𝛼𝑞^{−∑𝑘𝑗=1𝑗𝑥𝑗}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘_𝑙=0⎛⎜⎜⎝𝑚𝑙⎞⎟⎟⎠(𝑞_𝛼−1)^𝑙𝛽̃^{(0,𝑘∣𝛼)𝑙,𝑞}=(1−𝑞)^𝑘∏^{𝑘−1𝑖=0}1−𝑞^{𝛼𝑚−𝑘+1+𝑖}=(1−𝑞)^𝑘𝑘_𝑙=0⎛⎜⎜⎝𝑘+𝑙𝑙−1⎞⎟⎟⎠_𝑞𝑞^{(𝛼𝑚−𝑘+1)𝑙}.(2.5) From (2.1), we have ^𝑖_𝑗=0⎛⎜⎜⎝𝑗𝑖⎞⎟⎟⎠(𝑞_𝛼−1)^𝑗_ℤ𝑝=_ℤ𝑝⋯_ℤ𝑝𝑥₁+⋯+𝑥_𝑘^{𝑞𝑛−𝑖+𝑗}_𝛼𝑞^{(𝛽−𝛼−1)𝑥1+⋯+(𝛽−𝛼−𝑘)𝑥𝑘}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘=_𝑖−1⋯_ℤ𝑝𝑥₁+⋯+𝑥_𝑘^{𝑞𝑛−𝑖}_𝛼𝑞^{(𝛽−1)𝑥1+⋯+(𝛽−𝑘)𝑥𝑘}𝑞^{𝛼(𝑥1+⋯+𝑥𝑘)(𝑖−1)}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘_𝑗=0⎛⎜⎜⎝𝑖−𝑗1⎞⎟⎟⎠(𝑞_𝛼−1)^𝑗𝛽̃^{(𝛽,𝑘∣𝛼)𝑛−𝑖+𝑗,𝑞}.(2.6)

Thus, we obtain the following theorem.

Theorem 2.2. For 𝑖∈ℕ, we have ^𝑖_𝑗=0⎛⎜⎜⎝𝑗𝑖⎞⎟⎟⎠(𝑞_𝛼−1)^𝑗𝛽̃^{(𝛽−𝛼,𝑘∣𝛼)𝑛−𝑖+𝑗,𝑞}=^𝑖−1_𝑗=0⎛⎜⎜⎝𝑖−𝑗1⎞⎟⎟⎠(𝑞_𝛼−1)^𝑗𝛽̃^{(𝛽,𝑘∣𝛼)𝑛−𝑖+𝑗,𝑞}.(2.7)

It is easy to show that

^𝑚∑_𝑗=0^𝑚𝑗(𝑞^𝛼−1)^𝑗𝛽̃^{(0,𝑘∣𝛼)𝑗,𝑞}=(1−𝑞)∑^𝑘𝑘_𝑙=0^{𝑙𝑘+𝑙−1}_𝑞𝑞^{(𝛼𝑚−𝑘+1)𝑙}=(1−𝑞)^𝑘∏^{𝑘−1𝑖=0}(1−𝑞^{𝛼𝑚−𝑘+1+𝑖}).

3. Polynomials 𝛽̃^{(0,𝑘∣𝛼)𝑛,𝑞}(𝑥)

In this section, we consider the polynomials 𝛽̃^{(0,𝑘|𝛼)𝑛,𝑞}(𝑥) as follows: 𝛽̃^{(0,𝑘∣𝛼)𝑛,𝑞}(𝑥)=_ℤ𝑝⋯_ℤ𝑝𝑥₁+⋯+𝑥_𝑘+𝑥^𝑛𝑞_𝛼𝑞^{−∑𝑘𝑗=1𝑗𝑥𝑗}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘.(3.1)

From (3.1), we can easily derive the following equation: 𝛽̃^{(0,𝑘|𝛼)𝑛,𝑞}(𝑥)=(1−𝑞)^𝑘(1−𝑞^𝛼)^𝑛𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)_𝑙𝑞^𝛼𝑙𝑥∏^𝑘𝑖=0(𝛼𝑙−𝑖)=1∏^{𝑘−1𝑖=0}1−𝑞^{𝛼𝑙−𝑖}(1−𝑞)^𝑛[𝛼]^𝑛𝑞𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)_𝑙𝑞^𝛼𝑙𝑥^𝑘𝛼𝑙𝑘!^𝑘𝛼𝑙_𝑞[𝑘]_𝑞!.(3.2) By (3.1) and (3.2), we get _ℤ𝑝⋯_ℤ𝑝𝑞^{∑𝑘𝑗=1(𝛼𝑛−𝑗)𝑥𝑗+𝛼𝑛𝑥}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠[𝛼]_𝑙𝑞(𝑞−1)^𝑙𝛽̃^{(0,𝑘∣𝛼)𝑙,𝑞}(𝑥),(3.3)_ℤ𝑝⋯=𝑞_{ℤ𝑝𝛼𝑛𝑥}𝑞^{∑𝑘𝑙=1(𝛼𝑛−𝑙)𝑥𝑙+𝛼𝑛𝑥}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘(1−𝑞)^𝑘∏^{𝑘−1𝑗=0}(𝛼𝑛−𝑗)∏^{𝑘−1𝑗=0}(1−𝑞^{𝛼𝑛−𝑗})=𝑞^𝛼𝑛𝑥(^𝑘𝛼𝑛)𝑘!(^𝑘𝛼𝑛)_𝑞[𝑘]_𝑞!.(3.4) Therefore, by (3.3) and (3.4), we obtain the following theorem.

Theorem 3.1. For 𝑛∈ℤ₊, we have (1−𝑞)^𝑛𝛽̃^{(0,𝑘∣𝛼)𝑛,𝑞}(𝑥)=1[𝛼]^𝑛𝑞𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)_𝑙𝑞^𝛼𝑙𝑥^𝑘𝛼𝑙𝑘!^𝑘𝛼𝑙_𝑞[𝑘]_𝑞!.(3.5) Moreover, ^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠[𝛼]_𝑙𝑞(𝑞−1)^𝑙𝛽̃^{(0,𝑘∣𝛼)𝑙,𝑞}(𝑥)=𝑞^𝛼𝑛𝑥(^𝑘𝛼𝑛)𝑘!(^𝑘𝛼𝑛)_𝑞[𝑘]_𝑞!.(3.6)

Let 𝑑∈ℕ. Then, we have _ℤ𝑝⋯=[𝑑]_ℤ𝑝_𝑛𝑞𝛼𝑥+𝑥₁+⋯+𝑥_𝑘^𝑛𝑞_𝛼𝑞^{−∑𝑘𝑗=1𝑗𝑥𝑗}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘[𝑑]^{𝑘𝑞𝑑−1}_{𝑎1,⋯,𝑎𝑘=0}𝑞^{−∑𝑘𝑗=2(𝑗−1)𝑎𝑗}×_ℤ𝑝⋯_ℤ𝑝𝑥+∑^𝑘𝑗=1𝑎_𝑗𝑑+^𝑘_𝑖=1𝑥_𝑖^𝑛𝑞_𝛼𝑑𝑞^{∑−𝑑𝑘𝑗=1𝑗𝑥𝑗}𝑑𝜇_𝑞𝑑𝑥₁⋯𝑑𝜇_𝑞^𝑑𝑥_𝑘.(3.7) Thus, by (3.1) and (3.7), we obtain the following theorem.

Theorem 3.2. For 𝑑,𝑘∈ℕ, and 𝑛∈ℤ₊, we have 𝛽̃^{(0,𝑘∣𝛼)𝑛,𝑞}(𝑥)=[𝑑]^𝑛𝑞𝛼[𝑑]^{𝑘𝑞𝑑−1}_{𝑎1,…,𝑎𝑘=0}𝑞^{−∑𝑘𝑗=2(𝑗−1)𝑎𝑗}𝛽̃^{(0,𝑘∣𝛼)𝑛,𝑞}_𝛼𝑥+𝑎¹+⋯+𝑎_𝑘𝑑.(3.8)

From (3.1), we note that 𝛽̃^{(0,𝑘|𝛼)𝑛,𝑞}(𝑥)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠[𝑥]_{𝑞𝑛−𝑙𝛼}𝛽̃_{(0,𝑘∣𝛼)𝑛,𝑞}𝑞^𝛼𝑙𝑥𝛽̃^{(0,𝑘∣𝛼)𝑙,𝑞},(𝑥+𝑦)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠[𝑦]_{𝑞𝑛−𝑙𝛼}𝑞^𝛼𝑙𝑥𝛽̃^{(0,𝑘∣𝛼)𝑙,𝑞}(𝑥).(3.9)

4. Polynomials 𝛽̃^{(ℎ,1|𝛼)𝑛,𝑞}(𝑥)

For ℎ∈ℤ, let us define weighted (ℎ,𝑞)-Bernoulli polynomials 𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}(𝑥) as follows: 𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}(𝑥)=_ℤ𝑝𝑥+𝑥₁^𝑛𝑞_𝛼𝑞^{𝑥1(ℎ−1)}𝑑𝜇_𝑞𝑥₁.(4.1) By (4.1), we easily see that 𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}(𝑥)=1[𝛼]^𝑛𝑞(1−𝑞)^𝑛𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)_𝑙𝑞^𝛼𝑙𝑥𝛼𝑙+ℎ[𝛼𝑙+ℎ]_𝑞.(4.2) Therefore, by (4.2), we obtain the following theorem.

Theorem 4.1. For ℎ∈ℤ and 𝑛∈ℤ₊, we have 𝛽̃^{(ℎ,1|𝛼)𝑛,𝑞}(𝑥)=1[𝛼]^𝑛𝑞(1−𝑞)^𝑛𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)_𝑙𝑞^𝛼𝑙𝑥𝛼𝑙+ℎ[𝛼𝑙+ℎ]_𝑞.(4.3)

From (4.1), we can derive the following equation: 𝑞^𝛼𝑥_ℤ𝑝𝑥+𝑥₁^𝑛𝑞_𝛼=(𝑞_𝛼𝑞^{𝑥1(ℎ−1)}𝑑𝜇_𝑞𝑥₁−1)_ℤ𝑝𝑥+𝑥₁^𝑞𝑛+1_𝛼𝑞^{𝑥1(ℎ−𝛼−1)}𝑑𝜇_𝑞𝑥₁+_ℤ𝑝𝑥+𝑥₁^𝑛𝑞_𝛼𝑞^{𝑥1(ℎ−𝛼−1)}𝑑𝜇_𝑞𝑥₁.(4.4) By (4.4), we easily get 𝑞^𝛼𝑥𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}(𝑥)=(𝑞^𝛼−1)𝛽̃^{(ℎ−𝛼−1,1∣𝛼)𝑛+1,𝑞}(𝑥)+𝛽̃^{(ℎ−𝛼−1,1∣𝛼)𝑛,𝑞}(𝑥).(4.5) From (4.1), we have 𝛽̃^{(ℎ,1|𝛼)𝑛,𝑞}(𝑥)=_ℤ𝑝𝑥+𝑥₁^𝑛𝑞_𝛼𝑞^{𝑥1(ℎ−1)}𝑑𝜇_𝑞𝑥₁=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠[𝑥]_{𝑞𝑛−𝑙𝛼}𝑞^𝛼𝑙𝑥𝛽̃^{(ℎ,1∣𝛼)𝑙,𝑞},(4.6) where 𝛽̃^{(ℎ,1∣𝛼)𝑙,𝑞}(0)=𝛽̃^{(ℎ,1∣𝛼)𝑙,𝑞}.

By (4.6), we get the following recurrence formula: 𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}(𝑥)=𝑞^𝛼𝑥𝛽̃^{𝑞(ℎ,1∣𝛼)}+[𝑥]_𝑞𝛼^𝑛,for𝑛≥1,(4.7) with the usual convention about replacing (𝛽̃^{𝑞(ℎ,1∣𝛼)})^𝑛 with 𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}.

From (1.6), we note that 𝑞𝐼_𝑞𝑓₁=𝐼_𝑞(𝑓)+(𝑞−1)𝑓(0)+𝑞−1log𝑞𝑓^(0).(4.8) For ℎ∈ℤ₊, by (4.8), we have 𝑞^ℎ_ℤ𝑝𝑓(𝑥+1)𝑞^{(ℎ−1)𝑥}𝑑𝜇_𝑞(𝑥)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)+(𝑞−1)ℎ𝑓(0)+𝑞−1log𝑞𝑓^(0).(4.9) If ℎ∈{−1,−2,−3,…}, then we get 𝑞^ℎ_ℤ𝑝𝑓(𝑥+1)𝑞^{(ℎ−1)𝑥}𝑑𝜇_𝑞(𝑥)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)+(1−𝑞)ℎ𝑓(0)+𝑞−1log𝑞𝑓^(0).(4.10) Let ℎ∈ℤ₊. By (4.9), we get 𝑞^ℎ_ℤ𝑝𝑥+𝑥₁=(𝑞−1)ℎ+[𝑥1]^𝑛𝑞_{𝛼𝑛𝑞𝛼}𝑞^{(ℎ−1)𝑥1}𝑑𝜇_𝑞𝑥₁−_ℤ𝑝𝑥+𝑥₁^𝑛𝑞_𝛼𝑞^{(ℎ−1)𝑥1}𝑑𝜇_𝑞𝑥₁+𝛼[𝛼]_𝑞[𝑥]^𝑞𝑛−1_𝛼𝑞^𝛼𝑥.(4.11) From (4.6) and (4.11), we note that 𝑞^ℎ𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}(𝑥+1)−𝛽̃^{(ℎ,1|𝛼)𝑛,𝑞}(𝑥)=(𝑞−1)ℎ[𝑥]^𝑛𝑞_𝛼+𝑛𝛼[𝛼]_𝑞[𝑥]^𝑞𝑛−1_𝛼𝑞^𝛼𝑥.(4.12) If we take 𝑥=0 in (4.12), then we have 𝛽̃^{(ℎ,1|𝛼)0,𝑞}=ℎ[ℎ]_𝑞,𝑞^ℎ𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}(1)−𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}=[𝛼𝛼]_𝑞0if𝑛=1,if𝑛>1.(4.13) Therefore, by (4.12) and (4.13), we obtain the following theorem.

Theorem 4.2. For ℎ∈ℤ₊, we have 𝛽̃^{(ℎ,1∣𝛼)0,𝑞}=ℎ[ℎ]_𝑞,𝑞^ℎ𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}(1)−𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}=[𝛼𝛼]_𝑞0if𝑛=1,if𝑛>1.(4.14)

By (4.7) and Theorem 4.2, we obtain the following corollary.

Corollary 4.3. For ℎ∈ℤ₊, we have 𝛽̃^{(ℎ,1∣𝛼)0,𝑞}=ℎ[ℎ]_𝑞,𝑞^ℎ𝑞^𝛼𝛽̃^{𝑞(ℎ,1∣𝛼)}+1^𝑛−𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}=[𝛼𝛼]_𝑞0if𝑛=1,if𝑛>1,(4.15) with the usual convention about replacing (𝛽̃^{𝑞(ℎ,1∣𝛼)})^𝑛 with 𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}.

From (4.1), we have 𝛽̃^{(ℎ,1∣𝛼)0,𝑞}=_ℤ𝑝𝑞^{𝑥1(ℎ−1)}𝑑𝜇_𝑞𝑥₁=ℎ[ℎ]_𝑞,ifℎ∈ℤ₊.(4.16) It is not difficult to show that 𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}₋₁(1−𝑥)==(−_ℤ𝑝1)1_𝑛−𝑥+𝑥₁^𝑛𝑞_−𝛼𝑞^{−𝑥1(ℎ−1)}𝑑𝜇_𝑞−1𝑥₁𝑞^{𝛼𝑛+ℎ−1}_ℤ𝑝𝑥+𝑥₁_𝑞𝛼=(−1)_𝑛𝑞^{𝑥1(ℎ−1)}𝑑𝜇_𝑞𝑥₁𝑞^{𝛼𝑛+ℎ−1}𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}(𝑥).(4.17) Therefore, by (4.17), we obtain the following theorem.

Theorem 4.4. For ℎ,𝑛∈ℤ₊, we have 𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}₋₁(1−𝑥)=(−1)^𝑛𝑞^{𝛼𝑛+ℎ−1}𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}(𝑥).(4.18)

For 𝑥=1 in Theorem 4.4, we get 𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}₋₁==((−−11))^𝑛_𝑛𝑞^{𝛼𝑛+ℎ−1}𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}(1)𝑞^𝛼𝑛−1𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}if𝑛>1.(4.19) Therefore, by (4.19), we obtain the following corollary.

Corollary 4.5. For ℎ∈ℤ₊ and 𝑛∈ℕ with 𝑛>1, we have 𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}₋₁=(−1)^𝑛𝑞^𝛼𝑛−1𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}.(4.20)

Let 𝑑∈ℕ. By (4.1), we see that _ℤ𝑝𝑞^{(ℎ−1)𝑥1}𝑥+𝑥₁^𝑛𝑞_𝛼𝑑𝜇_𝑞𝑥₁=[𝑑]^𝑛𝑞𝛼[𝑑]_{𝑞𝑑−1𝑎=0}𝑞^ℎ𝑎_ℤ𝑝𝑥+𝑑𝑎+𝑥¹^𝑛𝑞_𝛼𝑑𝑞^{𝑥1(ℎ−1)𝑑}𝑑𝜇_𝑞𝛼𝑥₁.(4.21) By (4.1) and (4.21), we obtain the following equation: 𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}(𝑥)=[𝑑]^𝑛𝑞𝛼[𝑑]_{𝑞𝑑−1𝑎=0}𝑞^ℎ𝑎𝛽̃^{(ℎ,1∣𝛼)𝑛,𝑞}𝑥+𝑎𝑑,(4.22) where 𝑑∈ℕ and ℎ∈ℤ₊.

5. Polynomials 𝛽̃^{(ℎ,𝑘|𝛼)𝑛,𝑞}(𝑥) and ℎ=𝑘

From (2.1), we note that 𝛽̃^{(ℎ,𝑘|𝛼)𝑛,𝑞}(𝑥)==_ℤ𝑝⋯1_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑘^𝑛𝑞_𝛼𝑞^{(ℎ−1)𝑥1+⋯+(ℎ−𝑘)𝑥𝑘}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘(1−𝑞)^𝑛[𝛼]^𝑛𝑞𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)_𝑙𝑞^𝛼𝑙𝑥(𝛼𝑙+ℎ)⋯(𝛼𝑙+ℎ−𝑘+1)[𝛼𝑙+ℎ]_𝑞⋯[𝛼𝑙+ℎ−𝑘+1]_𝑞,(5.1)𝑞^ℎ_ℤ𝑝=⋯_ℤ𝑝_ℤ𝑝𝑥+1+𝑥₁+⋯+𝑥_𝑘^𝑛𝑞_𝛼𝑞^{(ℎ−1)𝑥1+⋯+(ℎ−𝑘)𝑥𝑘}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘+(𝑞⋯−1)ℎ_ℤ𝑝𝑥_ℤ𝑝+𝑥₁+⋯+𝑥_𝑘^𝑛𝑞_𝛼𝑞^{(ℎ−1)𝑥1+⋯+(ℎ−𝑘)𝑥𝑘}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘+𝑛𝛼⋯_ℤ𝑝𝑥+𝑥₂+⋯+𝑥_𝑘^𝑛𝑞_𝛼𝑞^{(ℎ−1)𝑥2+⋯+(ℎ−𝑘)𝑥𝑘}𝑑𝜇_𝑞𝑥₂⋯𝑑𝜇_𝑞𝑥_𝑘[𝛼]_𝑞𝑞^𝛼𝑥_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₂+⋯+𝑥_𝑘^𝑞𝑛−1_𝛼×𝑞^{𝛼(ℎ−1)𝑥2+⋯+(ℎ−𝑘+1)𝑥𝑘}𝑑𝜇_𝑞𝑥₂⋯𝑑𝜇_𝑞𝑥_𝑘.(5.2) From (5.1), we have 𝑞^ℎ𝛽̃^{(ℎ,𝑘∣𝛼)𝑛,𝑞}(𝑥+1)=𝛽̃^{(ℎ,𝑘∣𝛼)𝑛,𝑞}(𝑥)+(𝑞−1)ℎ𝛽̃^{(ℎ−1,𝑘−1∣𝛼)𝑛,𝑞}(𝑥)+𝑞^𝛼𝑥𝑛𝛼[𝛼]_𝑞𝛽̃^{(ℎ,𝑘−1∣𝛼)𝑛,𝑞}(𝑥).(5.3) Therefore, by (5.3), we obtain the following theorem.

Theorem 5.1. For ℎ,𝑛∈ℤ₊ and 𝑘∈ℕ, we have 𝑞^ℎ𝛽̃^{(ℎ,𝑘∣𝛼)𝑛,𝑞}(𝑥+1)−𝛽̃^{(ℎ,𝑘∣𝛼)𝑛,𝑞}(𝑥)=(𝑞−1)ℎ𝛽̃^{(ℎ−1,𝑘−1∣𝛼)𝑛,𝑞}(𝑥)+𝑛𝑞^𝛼𝑥𝛼[𝛼]_𝑞𝛽̃^{(ℎ,𝑘−1∣𝛼)𝑛,𝑞}(𝑥).(5.4)

It is easy to show that 𝑞^𝛼𝑥_ℤ𝑝=(⋯𝑞_𝛼_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑘^𝑛𝑞_𝛼𝑞^{ℎ𝑥1+(ℎ−1)𝑥2+⋯+(ℎ+1−𝑘)𝑥𝑘}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘+−_ℤ𝑝1)_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑘^𝑞𝑛+1_𝛼𝑞^{(ℎ−𝛼)𝑥1+⋯+(ℎ−𝛼+1−𝑘)𝑥𝑘}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘=(𝑞_𝛼⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑘^𝑛𝑞_𝛼𝑞^{(ℎ−𝛼)𝑥1+⋯+(ℎ−𝛼+1−𝑘)𝑥𝑘}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘−1)𝛽̃^{(ℎ+1−𝛼,𝑘∣𝛼)𝑛+1,𝑞}(𝑥)+𝛽̃^{(ℎ+1−𝛼,𝑘∣𝛼)𝑛,𝑞}(𝑥).(5.5) Thus, by (5.5), we obtain the following proposition.

Proposition 5.2. For ℎ,𝑛∈ℤ₊, we have 𝑞^𝛼𝑥𝛽̃^{(ℎ+1,𝑘∣𝛼)𝑛,𝑞}(𝑥)=(𝑞^𝛼−1)𝛽̃^{(ℎ+1−𝛼,𝑘∣𝛼)𝑛+1,𝑞}(𝑥)+𝛽̃^{(ℎ+1−𝛼,𝑘∣𝛼)𝑛,𝑞}(𝑥).(5.6)

For 𝑑∈ℕ, we get _ℤ𝑝⋯_ℤ𝑝𝑥+^𝑘_𝑗=1𝑥_𝑗^𝑛𝑞_𝛼=[𝑑]_𝑛𝑞𝛼𝑞^{∑𝑘𝑗=1(ℎ−𝑗)𝑥𝑗}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘[𝑑]^{𝑘𝑞𝑑−1}_{𝑎1,…,𝑎𝑘=0}𝑞^{ℎ∑𝑘𝑗=1𝑎𝑗−∑𝑘𝑗=2(𝑗−1)𝑎𝑗}×_ℤ𝑝⋯_ℤ𝑝𝑥+∑^𝑘𝑗=1𝑎_𝑗𝑑+^𝑘_𝑗=1𝑥_𝑗^𝑛𝑞_𝛼𝑑𝑞^{𝑑∑𝑘𝑗=1(ℎ−𝑗)𝑥𝑗}𝑑𝜇_𝑞𝑑𝑥₁⋯𝑑𝜇_𝑞^𝑑𝑥_𝑘.(5.7) Thus, we obtain 𝛽̃^{(ℎ,𝑘∣𝛼)𝑛,𝑞}(𝑥)=[𝑑]^𝑛𝑞𝛼[𝑑]^{𝑘𝑞𝑑−1}_{𝑎1,…,𝑎𝑘=0}𝑞^{ℎ∑𝑘𝑗=1𝑎𝑗−∑𝑘𝑗=2(𝑗−1)𝑎𝑗}𝛽̃^{(ℎ,𝑘∣𝛼)𝑛,𝑞}_𝑑𝑥+𝑎¹+⋯+𝑎_𝑘𝑑.(5.8) Equation (5.8) is multiplication formula for the 𝑞-Bernoulli polynomials of order (ℎ,𝑘) with weight 𝛼.

Let us define 𝛽̃^{(𝑘,𝑘∣𝛼)𝑛,𝑞}(𝑥)=𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}(𝑥). Then we see that 𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}(𝑥)=1(1−𝑞)^𝑛[𝛼]^𝑛𝑞𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)_𝑙𝑞^𝛼𝑙𝑥(𝛼𝑙+𝑘)⋯(𝛼𝑙+1)[𝛼𝑙+𝑘]_𝑞⋯[𝛼𝑙+1]_𝑞,(5.9)_ℤ𝑝⋯=(−_ℤ𝑝1)𝑘_𝑛−𝑥+𝑥₁+⋯𝑥_𝑘^𝑛𝑞_−𝛼𝑞^{−(𝑘−1)𝑥1−⋯−(𝑘−𝑘)𝑥𝑘}𝑑𝜇_𝑞−1𝑥₁⋯𝑑𝜇_𝑞⁻¹𝑥_𝑘𝑞^{2𝛼𝑛−𝑘+𝑘+1}(1−𝑞)^𝑛[𝛼]^𝑛𝑞𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)_𝑙𝑞^𝛼𝑙𝑥(𝛼𝑙+𝑘)⋯(𝛼𝑙+1)[𝛼𝑙+𝑘]_𝑞⋯[𝛼𝑙+1]_𝑞=(−1)^𝑛𝑞^{2𝛼𝑛−𝑘+𝑘+1}𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}(𝑥).(5.10) Therefore, by (5.9) and (5.10), we obtain the following theorem.

Theorem 5.3. For 𝑛∈ℤ₊ and 𝑘∈ℕ, we have 𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}₋₁(𝑘−𝑥)=(−1)^𝑛𝑞^{2𝛼𝑛−𝑘+𝑘+1}𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}(𝑥).(5.11)

Let 𝑥=𝑘 in Theorem 5.3. Then we see that 𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}₋₁=(−1)^𝑛𝑞^{2𝛼𝑛−𝑘+𝑘+1}𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}(𝑘).(5.12) From Theorem 5.1, we can derive the following equation: 𝑞^𝑘𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}(𝑥+1)−𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}(𝑥)=𝑘(𝑞−1)𝛽̃^{(𝑘−1∣𝛼)𝑛,𝑞}(𝑥)+𝑛𝑞^𝛼𝑥𝛼[𝛼]_𝑞𝛽̃^{(𝑘,𝑘−1∣𝛼)𝑛,𝑞}(𝑥).(5.13) By (5.9), we easily get 𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}=1(1−𝑞)^𝑛[𝛼]^𝑛𝑞𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)_𝑙(𝛼𝑙+𝑘)⋯(𝛼𝑙+1)[𝛼𝑙+𝑘]_𝑞⋯[𝛼𝑙+1]_𝑞.(5.14) From the definition of 𝑝-adic 𝑞-integral on ℤ_𝑝, we note that ^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(𝑞_𝛼−1)^𝑙_ℤ𝑝=(𝛼𝑛+𝑘)⋯(⋯𝛼𝑛+_ℤ𝑝1)𝑥₁+⋯+𝑥_𝑘^𝑙𝑞_𝛼𝑞^{∑𝑘𝑙=0(𝑘−𝑙)𝑥𝑙}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘[𝛼𝑛+𝑘]_𝑞⋯[𝛼𝑛+1]_𝑞=(^𝑘𝛼𝑛)𝑘!(^𝑘𝛼𝑛)_𝑞[𝑘]_𝑞!.(5.15) Thus, by (5.15), we get ^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(𝑞_𝛼−1)^𝑙𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}=(^𝑘𝛼𝑛)𝑘!(^𝑘𝛼𝑛)_𝑞[𝑘]_𝑞!.(5.16) By the definition of polynomial 𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}(𝑥), we see that 𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}(𝑥)==_ℤ𝑝𝑛⋯_ℤ𝑝𝑥+𝑥₁+⋯𝑥_𝑘^𝑛𝑞_𝛼𝑞^{(𝑘−1)𝑥1+⋯+(𝑘−𝑘)𝑥𝑘}𝑑𝜇_𝑞𝑥₁⋯𝑑𝜇_𝑞𝑥_𝑘_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝑞_𝛼𝑙𝑥𝛽̃^{(𝑘∣𝛼)𝑙,𝑞}[𝑥]^{𝑞𝑛−𝑙}=𝑞^𝛼𝑥𝛽̃^{𝑞(𝑘∣𝛼)}+[𝑥]_𝑞𝛼^𝑛,where𝑛∈ℤ₊,(5.17) with the usual convention about replacing (𝛽̃^{𝑞(𝑘∣𝛼)})^𝑛 with 𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}.

Let 𝑥=0 in (5.13). Then we have 𝑞^𝑘𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}(1)−𝛽̃^{(𝑘∣𝛼)𝑛,𝑞}=𝑘(𝑞−1)𝛽̃^{(𝑘−1∣𝛼)𝑛,𝑞}+𝑛𝛼[𝛼]_𝑞𝛽̃^{(𝑘,𝑘−1∣𝛼)𝑛,𝑞}.(5.18)

Acknowledgments

The present research has been conducted by the Research Grant of Kwangwoon University in 2011. The authors would like to express their gratitude to referees for their valuable suggestions and comments.