A Note on the Modified 𝑞-Bernoulli Numbers and Polynomials with Weight 𝛼

Abstract

A systemic study of some families of the modified 𝑞-Bernoulli numbers and polynomials with weight 𝛼 is presented by using the 𝑝-adic 𝑞-integration ℤ_𝑝. The study of these numbers and polynomials yields an interesting 𝑞-analogue related to Bernoulli numbers and polynomials.

1. Introduction

Let 𝑝 be a fixed prime number. Throughout this paper ℤ_𝑝,ℚ_𝑝,ℂ, and ℂ_𝑝 will, respectively, denote the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, the complex number field, and the completion of the algebraic closure of ℚ_𝑝. Let 𝑣_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{−𝑣𝑝(𝑝)}=1/𝑝. When one talks of 𝑞-extension, 𝑞 is variously considered as an indeterminate, a complex 𝑞∈ℂ, or a 𝑝-adic number 𝑞∈ℂ_𝑝. If 𝑞∈ℂ, one normally assumes |𝑞|<1. If 𝑞∈ℂ_𝑝, then we assume |𝑞−1|_𝑝<𝑝^{−1/(𝑝−1)} so that 𝑞^𝑥=exp(𝑥log𝑞) for |𝑥|_𝑝≤1.

The 𝑞-number [𝑥]_𝑞 is defined by [𝑥]_𝑞=1−𝑞^𝑥,1−𝑞(1.1) see [1–10].

We say that 𝑓 is a uniformly differentiable function at a point 𝑎∈ℤ_𝑝 and denote this property by 𝑓∈𝑈𝐷(ℤ_𝑝), if the difference quotients 𝐹_𝑓=𝑓(𝑥)𝑦𝑥,𝑦−𝑓𝑥−𝑦(1.2) have a limit 𝑙=𝑓^′(𝑎) as (𝑥,𝑦)→(𝑎,𝑎). c.f. [11].

For 𝑓∈𝑈𝐷(ℤ_𝑝,ℂ_𝑝)={𝑓∣𝑓∶ℤ_𝑝→ℂ_𝑝isuniformlydifferentiablefunctions}, the 𝑝-adic 𝑞-integral on ℤ_𝑝 is defined by 𝐼_𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁_𝑞^𝑝𝑁−1_𝑥=0𝑓(𝑥)𝑞^𝑥,(1.3) (see [3]).

From (1.3), we can easily derive the following: 𝑞^𝑛𝐼_𝑞𝑓_𝑛=𝐼_𝑞𝑓+𝑞−1^𝑛−1_𝑙=0𝑓(𝑙)𝑞^𝑙+𝑞−1log𝑞^𝑛−1_𝑙=0𝑞^𝑙𝑓^′(𝑙),(1.4) where 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛), (see [5, 12]).

In [1, 2], Carlitz defined a set of numbers 𝐵_𝑘,𝑞 inductively by 𝐵_0,𝑞=1,𝑞𝐵_𝑞+1^𝑘−𝐵_𝑘,𝑞=1,if𝑘=1,0,if𝑘>1,(1.5) with the usual convention about replacing 𝐵^𝑘_𝑞 by 𝐵_𝑘,𝑞.

These numbers are the 𝑞-extension of ordinary Bernoulli numbers. But they do not remain finite when 𝑞=1. So, Carlitz modified (1.5) as follows: 𝛽_0,𝑞=1,𝑞𝑞𝛽+1^𝑘−𝛽_𝑘,𝑞=1,if𝑘=1,0,if𝑘>1,(1.6) with the usual convention of replacing 𝛽^𝑘 by 𝛽_𝑘,𝑞.

In [1], Carlitz also considered the extension of Carlitz’s 𝑞-Bernoulli numbers as follows: 𝛽^ℎ_0,𝑞=ℎ[ℎ]_𝑞,𝑞^ℎ𝑞𝛽^ℎ+1^𝑘−𝛽^ℎ_𝑘,𝑞=1,if𝑘=1,0,if𝑘>1,(1.7) with the usual convention of replacing (𝛽^ℎ)^𝑘 by 𝛽^ℎ_𝑘,𝑞.

In this paper, we construct the modified 𝑞-Bernoulli numbers with weight 𝛼, which are different Carlitz’s 𝑞-Bernoulli numbers, by using 𝑝-adic 𝑞-integral equation. From (1.4), we derive some interesting identities and relations on the modified 𝑞-Bernoulli numbers and polynomials.

2. The Modified 𝑞-Bernoulli Numbers and Polynomials with Weight 𝛼

In this section, we assume 𝛼∈ℚ. Now, we define the modified 𝑞-Bernoulli numbers with weight 𝛼 (=𝐵^(𝛼)_𝑛,𝑞 ) as follows: 𝐵^(𝛼)_𝑛,𝑞=_ℤ𝑝𝑞^−𝑥[𝑥]^𝑛_𝑞𝛼𝑑𝜇_𝑞(𝑥)=11−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝛼𝑙[𝛼𝑙]_𝑞.(2.1) Thus, by (2.1), we have 𝐵^(𝛼)_𝑛,𝑞=11−𝑞^𝛼𝑛𝑞−1𝛼log𝑞−𝑛[𝛼]_𝑞^∞_𝑚=0𝑞^𝛼𝑚[𝑚]_𝑞^𝑛−1_𝛼.(2.2) Therefore, by (2.1) and (2.2), we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊=ℕ∪{0}, one has 𝐵^(𝛼)_𝑛,𝑞=𝛼1−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑙[𝛼𝑙]_𝑞=11−𝑞^𝛼𝑛𝑞−1𝛼log𝑞−𝑛[𝛼]_𝑞^∞_𝑚=0𝑞^𝛼𝑚[𝑚]_𝑞^𝑛−1_𝛼.(2.3)

Let us define the generating function of the modified 𝑞-Bernoulli numbers with weight 𝛼 as follows: 𝐹_𝑞^(𝛼)(𝑡)=^∞_𝑛=0𝐵^(𝛼)_𝑛,𝑞𝑡^𝑛.𝑛!(2.4) Then, by (2.3) and (2.4), we get 𝐹_𝑞^(𝛼)(𝑡)=𝑞−1𝑒log𝑞^{(1/(1−𝑞𝛼))𝑡}𝛼−𝑡[𝛼]_𝑞^∞_𝑚=0𝑞^𝛼𝑚𝑒^{[𝑚]𝑞𝛼𝑡}.(2.5) In the viewpoint of (2.1), we define the modified 𝑞-Bernoulli numbers with weight 𝛼 as follows: 𝐵^(𝛼)_𝑛,𝑞(𝑥)=_ℤ𝑝𝑞^−𝑦𝑥+𝑦^𝑛_𝑞𝛼𝑑𝜇_𝑞𝑦=^𝑛_𝑙=0𝑛𝑙[𝑥]_𝑞^𝑛−𝑙_𝛼𝑞^𝛼𝑙𝑥𝐵^(𝛼)_𝑙,𝑞=[𝑥]_𝑞𝛼+𝑞^𝛼𝑥𝐵_𝑞^(𝛼)𝑛,for𝑛∈ℤ₊,(2.6) with the usual convention of replacing (𝐵_𝑞^(𝛼))^𝑛 by 𝐵^(𝛼)_𝑛,𝑞.

From (2.6), we note that 𝐵^(𝛼)_𝑛,𝑞(𝑥)=𝛼1−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥𝑙[𝛼𝑙]_𝑞=11−𝑞^𝛼𝑛𝑞−1𝛼log𝑞−𝑛[𝛼]_𝑞𝑞^∞𝛼𝑥_𝑚=0𝑞^𝛼𝑚[𝑚+𝑥]_𝑞^𝑛−1_𝛼.(2.7) Therefore, by (2.7), we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℤ₊, one has 𝐵^(𝛼)_𝑛,𝑞(𝑥)=𝛼1−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥𝑙[𝛼𝑙]_𝑞=11−𝑞^𝛼𝑛𝑞−1𝛼log𝑞−𝑛[𝛼]_𝑞𝑞^∞𝛼𝑥_𝑚=0𝑞^𝛼𝑚[𝑚+𝑥]_𝑞^𝑛−1_𝛼.(2.8)

Let 𝐹_𝑞^(𝛼)(𝑡,𝑥)=∑^∞_𝑛=0𝐵^(𝛼)_𝑛,𝑞(𝑥)(𝑡^𝑛/𝑛!) be the generating function of the modified 𝑞-Bernoulli polynomials with weight 𝛼.

Then, by (2.7), we get 𝐹_𝑞^(𝛼)(𝑡,𝑥)=𝑞−1𝑒log𝑞^{(1/(1−𝑞𝛼))𝑡}𝛼−𝑡[𝛼]_𝑞^∞_𝑚=0𝑞^{𝛼(𝑚+𝑥)}𝑒^{[𝑚+𝑥]𝑞𝛼𝑡}.(2.9) Therefore, by (2.9), we obtain the following corollary

Corollary 2.3. Let 𝐹_𝑞^(𝛼)(𝑡,𝑥)=∑^∞_𝑛=0𝐵^(𝛼)_𝑛,𝑞(𝑥)(𝑡^𝑛/𝑛!). Then one has 𝐹_𝑞^(𝛼)(𝑡,𝑥)=𝑞−1𝑒log𝑞^{(1/(1−𝑞𝛼))𝑡}𝛼−𝑡[𝛼]_𝑞^∞_𝑚=0𝑞^{𝛼(𝑚+𝑥)}𝑒^{[𝑚+𝑥]𝑞𝛼𝑡}.(2.10)

In particular, 𝐹_𝑞^(𝛼)(𝑡,0)=𝐹_𝑞^(𝛼)(𝑡).

From Corollary 2.3, we can derive the following equation:𝐹^𝛼_𝑞(𝑡,1)−𝐹^𝛼_𝑞(𝑡)𝛼=𝑡[𝛼]_𝑞.(2.11)

By (2.5) and (2.11), we get𝐵^(𝛼)_0,𝑞=𝑞−1,𝐵log𝑞^{(𝛼)𝑛,𝑞}(1)−𝐵^(𝛼)_𝑛,𝑞=𝛼[𝛼]_𝑞,if𝑛=1,0,if𝑛>1.(2.12)

Therefore, by (2.12), we obtain the following theorem.

Theorem 2.4. For 𝑛∈ℤ₊, one has 𝐵^(𝛼)_0,𝑞=𝑞−1,𝐵log𝑞^{(𝛼)𝑛,𝑞}(1)−𝐵^(𝛼)_𝑛,𝑞=𝛼[𝛼]_𝑞,if𝑛=1,0,if𝑛>1.(2.13)

By using (2.6), we obtain the following corollary.

Corollary 2.5. For 𝑛∈ℤ₊, one has 𝐵^(𝛼)_0,𝑞=𝑞−1,𝑞log𝑞^𝛼𝐵_𝑞^(𝛼)+1^𝑛−𝐵^(𝛼)_𝑛,𝑞=𝛼[𝛼]_𝑞,if𝑛=1,0,if𝑛>1.(2.14) with the usual convention of replacing (𝐵_𝑞^(𝛼))^𝑛 by 𝐵^(𝛼)_𝑛,𝑞.

From (1.4), we can derive the following equation:_ℤ𝑝𝑓(𝑥+𝑛)𝑞^−𝑥𝑑𝜇_𝑞(𝑥)=_ℤ𝑝𝑓(𝑥)𝑞^−𝑥𝑑𝜇_𝑞(𝑥)+𝑞−1log𝑞^𝑛−1_𝑙=0𝑓^′(𝑙).(2.15)

Thus, by (1.6), (2.6), and (2.15), we get𝐵^(𝛼)_𝑚,𝑞(𝑛)−𝐵^(𝛼)_𝑚,𝑞=𝛼[𝛼]_𝑞𝑚^𝑛−1_𝑙=0[𝑙]_𝑞^𝑚−1_𝛼𝑞^𝛼𝑙,𝑛∈ℕ,𝑚∈ℤ₊.(2.16)

Therefore, by (2.16), we obtain the following theorem.

Theorem 2.6. For 𝑛∈ℕ,𝑚∈ℤ₊, one has 𝐵^(𝛼)_𝑚,𝑞(𝑛)−𝐵^(𝛼)_𝑚,𝑞=𝛼[𝛼]_𝑞𝑚^𝑛−1_𝑙=0[𝑙]_𝑞^𝑚−1_𝛼𝑞^𝛼𝑙.(2.17)

From (2.6), we note that𝐵^(𝛼)_𝑛,𝑞(𝑥)=_ℤ𝑝𝑥+𝑦^𝑛_𝑞𝛼𝑞^−𝑦𝑑𝜇_𝑞𝑦=lim_𝑁→∞1𝑝^𝑁_𝑞^𝑝𝑁−1_𝑦=0𝑥+𝑦^𝑛_𝑞𝛼=1−𝑞1−𝑞^𝑑𝑑−1_𝑎=0lim_𝑁→∞1𝑝^𝑁_𝑞𝑑^𝑝𝑁−1_𝑦=0𝑎+𝑥+𝑑𝑦^𝑛_𝑞𝛼=[𝑑]^𝑛_𝑞𝛼[𝑑]_𝑞^𝑑−1_{𝑎=0ℤ𝑝}𝑎+𝑥𝑑+𝑦^𝑛_𝑞𝛼𝑑𝑞^−𝑑𝑦𝑑𝜇_𝑞𝑑𝑦=[𝑑]^𝑛_𝑞𝛼[𝑑]_𝑞^𝑑−1_𝑎=0𝐵^(𝛼)_{𝑛,𝑞𝑑}𝑥+𝑎𝑑.(2.18)

Therefore, by (2.18), we obtain the following distribution relation for the modified 𝑞-Bernoulli polynomials with weight 𝛼.

Theorem 2.7. For 𝑑∈ℕ,𝑛∈ℤ₊, one has 𝐵^(𝛼)_𝑛,𝑞(𝑥)=[𝑑]^𝑛_𝑞𝛼[𝑑]_𝑞^𝑑−1_𝑎=0𝐵^(𝛼)_{𝑛,𝑞𝑑}𝑥+𝑎𝑑.(2.19)

To derive the relation of reflection symmetry of the modified 𝑞-Bernoulli polynomials with weight 𝛼, we evaluate the following 𝑝-adic 𝑞-integral on ℤ_𝑝:𝐵^(𝛼)_{𝑛,𝑞−1}(1−𝑥)=_ℤ𝑝1−𝑥+𝑥₁^𝑛_𝑞−𝛼𝑞^𝑥1𝑑𝜇_𝑞−1𝑥₁=11−𝑞^{−𝛼𝑛𝑛}_𝑙=0𝑛𝑙(−1)^𝑙𝑞^{𝛼𝑙𝑥−1}𝛼𝑙[𝛼𝑙]_𝑞=(−1)^𝑛𝑞𝑞^𝛼𝑛1−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥𝛼𝑙[𝛼𝑙]_𝑞=𝑞^𝛼𝑛−1(−1)^𝑛𝐵^(𝛼)_𝑛,𝑞(𝑥).(2.20)

Therefore, by (2.20), we obtain the following reflection symmetry relation of the modified 𝑞-Bernoulli polynomials with weight 𝛼.

Theorem 2.8. For 𝑛∈ℤ₊, one has 𝐵^(𝛼)_{𝑛,𝑞−1}(1−𝑥)=𝑞^𝛼𝑛−1(−1)^𝑛𝐵^(𝛼)_𝑛,𝑞(𝑥).(2.21)

From (1.3), we note that1𝑞_ℤ𝑝[1−𝑥]^𝑛_𝑞−𝛼𝑞^−𝑥𝑑𝜇_𝑞(𝑥)=(−1)^𝑛𝑞^𝛼𝑛−1_ℤ𝑝[𝑥−1]^𝑛_𝑞𝛼𝑞^−𝑥𝑑𝜇_𝑞(𝑥)=(−1)^𝑛𝑞^𝛼𝑛−1𝐵^(𝛼)_𝑛,𝑞()=𝐵−1_{(𝛼)𝑛,𝑞−1}(2),(2.22) and, by (2.6), we get𝐵^(𝛼)_𝑛,𝑞(2)=𝑞^2𝛼𝐵_𝑞^(𝛼)+[2]_𝑞𝛼^𝑛=𝑞^𝛼𝑞^𝛼𝐵_𝑞^(𝛼)+1+1^𝑛=^𝑛_𝑙=0𝑛𝑙𝑞^𝛼𝑙𝑞^𝛼𝐵_𝑞^(𝛼)+1^𝑙=𝐵^(𝛼)_0,𝑞+𝑛𝑞^𝛼𝑞^𝛼𝐵_𝑞^(𝛼)+1¹+^𝑛_𝑙=2𝑛𝑙𝑞^𝛼𝑙𝑞^𝛼𝐵_𝑞^(𝛼)+1^𝑙=𝑞−1log𝑞+𝑛𝑞^𝛼𝛼[𝛼]_𝑞+𝐵^(𝛼)_1,𝑞+^𝑛_𝑙=2𝑛𝑙𝑞^𝛼𝑙𝐵^(𝛼)_𝑙,𝑞=𝑛𝑞^𝛼𝛼[𝛼]_𝑞+^𝑛_𝑙=0𝑛𝑙𝑞^𝛼𝑙𝐵^(𝛼)_𝑙,𝑞.(2.23)

Let 𝑛∈ℕ with 𝑛≥2. Then, by (2.12) and (2.23), we obtain the following theorem.

Theorem 2.9. For 𝑛∈ℕ with 𝑛≥2, one has 𝐵^(𝛼)_𝑛,𝑞(2)−𝑛𝑞^𝛼𝛼[𝛼]_𝑞=𝑞^𝛼𝐵_𝑞^(𝛼)+1^𝑛=𝐵^(𝛼)_𝑛,𝑞.(2.24) In particular, 1𝑞_ℤ𝑝[1−𝑥]^𝑛_𝑞−𝛼𝑞^−𝑥𝑑𝜇_𝑞(𝑥)=𝐵^(𝛼)_{𝑛,𝑞−1}(2)=𝑛𝑞𝛼[𝛼]_𝑞+𝐵^(𝛼)_{𝑛,𝑞−1}.(2.25)

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.