Some Identities on the 𝑞-Bernoulli Numbers and Polynomials with Weight 0

Abstract

Recently, Kim (2011) has introduced the 𝑞-Bernoulli numbers with weight 𝛼. In this paper, we consider the 𝑞-Bernoulli numbers and polynomials with weight 𝛼=0 and give 𝑝-adic 𝑞-integral representation of Bernstein polynomials associated with 𝑞-Bernoulli numbers and polynomials with weight 0. From these integral representation on ℤ_𝑝, we derive some interesting identities on the 𝑞-Bernoulli numbers and polynomials with weight 0.

1. Introduction

Let 𝑝 be a fixed prime number. Throughout this paper, ℤ_𝑝,ℚ_𝑝, and ℂ_𝑝 will denote the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, and the completion of the algebraic closure of ℚ_𝑝, respectively. Let ℕ be the set of natural numbers and ℤ₊=ℕ∪{0}.

Let |⋅|_𝑝 be a 𝑝-adic norm with |𝑥|_𝑝=𝑝^−𝑟, where 𝑥=𝑝^𝑟𝑠/𝑡 and (𝑝,𝑠)=(𝑝,𝑡)=(𝑠,𝑡)=1, 𝑟∈ℚ. In this paper, we assume that 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<𝑝^{−1/(𝑝−1)} so that 𝑞^𝑥=exp(𝑥log𝑞), and [𝑥]_𝑞=(1−𝑞^𝑥)/(1−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 by Kim as follows: 𝐼_𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)=lim^𝑝_𝑁→∞^𝑁−1_𝑥=0𝑓(𝑥)𝜇_𝑞𝑥+𝑝^𝑁ℤ_𝑝=lim_𝑁→∞1𝑝^𝑁_𝑞^𝑝𝑁−1_𝑥=0𝑓(𝑥)𝑞^𝑥,(1.1) (see [1–5]). For 𝑛∈ℕ, let 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛). From (1.1), we note that 𝑞^𝑛𝐼_𝑞𝑓_𝑛−𝐼_𝑞𝑓=𝑞−1^𝑛−1_𝑙=0𝑞^𝑙𝑓(𝑙)+𝑞−1log𝑞^𝑛−1_𝑙=0𝑞^𝑙𝑓^′(𝑙),(1.2) where 𝑓^′(𝑙)=𝑑𝑓(𝑥)/𝑑𝑥|_𝑥=𝑙, (see [3, 6, 7]). In the special case, 𝑛=1, we get 𝑞_ℤ𝑝𝑓(𝑥+1)𝑑𝜇_𝑞(𝑥)−_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)=𝑞−1𝑓(0)+𝑞−1𝑓log𝑞^′(0).(1.3)

Throughout this paper, we assume that 𝛼∈ℚ.

The 𝑞-Bernoulli numbers with weight 𝛼 are defined by Kim [8] as follows: 𝛽^(𝛼)_0,𝑞=1,𝑞𝑞^𝛼𝛽_𝑞^(𝛼)+1^𝑛−𝛽^(𝛼)_𝑛,𝑞=𝛼[𝛼]_𝑞if0𝑛=1,if𝑛>1,(1.4) with the usual convention about replacing (𝛽_𝑞^(𝛼))^𝑛 with 𝛽^(𝛼)_𝑛,𝑞. From (1.4), we can derive the following equation: 𝛽^(𝛼)_𝑛,𝑞=11−𝑞^𝑛[𝛼]^𝑛_𝑞^𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝛼𝑙+1[𝛼𝑙+1]_𝑞=−𝑛𝛼[𝛼]_𝑞^∞_𝑚=0𝑞^{𝑚𝛼+𝑚}[𝑚]_𝑞^𝑛−1_𝛼+1−𝑞^∞_𝑚=0𝑞^𝑚[𝑚]^𝑛_𝑞𝛼.(1.5) By (1.1), (1.4), and (1.5), we get𝛽^(𝛼)_𝑛,𝑞=_ℤ𝑝[𝑥]^𝑛_𝑞𝛼𝑑𝜇_𝑞(𝑥)=11−𝑞^𝑛[𝛼]^𝑛_𝑞^𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝛼𝑙+1[𝛼𝑙+1]_𝑞.(1.6) The 𝑞-Bernoulli polynomials with weight 𝛼 are defined by 𝛽^(𝛼)_𝑛,𝑞(𝑥)=_ℤ𝑝𝑥+𝑦^𝑛_𝑞𝛼𝑑𝜇_𝑞𝑦=^𝑛_𝑙=0𝑛𝑙𝑞^𝛼𝑙𝑥[𝑥]_𝑞^𝑛−𝑙_𝛼𝛽^(𝛼)_𝑙,𝑞.(1.7) By (1.6) and (1.7), we easily see that 𝛽^(𝛼)_𝑛,𝑞(𝑥)=11−𝑞^𝑛[𝛼]^𝑛_𝑞^𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥𝛼𝑙+1[𝛼𝑙+1]_𝑞.(1.8)

Let 𝐶(ℤ_𝑝) be the set of continuous functions on ℤ_𝑝. For 𝑓∈𝐶(ℤ_𝑝), the 𝑝-adic analogue of Bernstein operator of order 𝑛 for 𝑓 is given by 𝔹_𝑛,𝑞𝑓∣𝑥=^𝑛_𝑘=0𝑓𝑘𝑛𝐵_𝑘,𝑛(𝑥)=^𝑛_𝑘=0𝑓𝑘𝑛𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘,(1.9) where 𝑛,𝑘∈ℤ₊ (see [1, 9, 10]). For 𝑛,𝑘∈ℤ₊, the 𝑝-adic Bernstein polynomials of degree 𝑛 are defined by 𝐵_𝑘,𝑛(𝑥)=(^𝑛_𝑘)𝑥^𝑘(1−𝑥)^𝑛−𝑘 for 𝑥∈ℤ_𝑝, (see [1, 10, 11]).

In this paper, we consider Bernstein polynomials to express the 𝑝-adic 𝑞-integral on ℤ_𝑝 and investigate some interesting identities of Bernstein polynomials associated with the 𝑞-Bernoulli numbers and polynomials with weight 0 by using the expression of 𝑝-adic 𝑞-integral on ℤ_𝑝 of these polynomials.

2. 𝑞-Bernoulli Numbers with Weight 0 and Bernstein Polynomials

In the special case, 𝛼=0, the 𝑞-Bernoulli numbers with weight 0 will be denoted by 𝛽⁽⁰⁾_𝑛,𝑞=𝛽_𝑛,𝑞. From (1.4), (1.5), and (1.6), we note that ^∞_𝑛=0𝛽_𝑛,𝑞𝑡^𝑛=𝑛!^∞_{𝑛=0ℤ𝑝}𝑥^𝑛𝑑𝜇_𝑞(𝑥)𝑡^𝑛=𝑛!_ℤ𝑝𝑒^𝑥𝑡𝑑𝜇_𝑞(𝑥)=𝑞−1log𝑞𝑡+log𝑞𝑞𝑒^𝑡.−1(2.1)

It is easy to show that 𝑡+log𝑞𝑞𝑒^𝑡=𝑡−1𝑞−11−𝑞⁻¹𝑒^𝑡−𝑞⁻¹+log𝑞𝑞−11−𝑞⁻¹𝑒^𝑡−𝑞⁻¹=𝑡𝑞−1^∞_𝑛=0𝐻_𝑛𝑞⁻¹𝑡^𝑛+𝑛!log𝑞𝑞−1^∞_𝑛=0𝐻_𝑛𝑞⁻¹𝑡^𝑛=1𝑛!𝑞−1^∞_𝑛=1𝑛𝐻_𝑛−1𝑞⁻¹𝑡^𝑛+𝑛!log𝑞𝑞−1^∞_𝑛=0𝐻_𝑛𝑞⁻¹𝑡^𝑛,𝑛!(2.2) where 𝐻_𝑛(𝑞⁻¹) are the 𝑛th Frobenius-Euler numbers.

By (2.1) and (2.2), we get 𝛽_𝑛,𝑞=1if𝑛𝑛=0,𝐻log𝑞^𝑛−1𝑞⁻¹+𝐻_𝑛𝑞⁻¹if𝑛>0.(2.3) Therefore, we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊, we have 𝛽_𝑛,𝑞=1if𝑛𝑛=0,𝐻log𝑞^𝑛−1𝑞⁻¹+𝐻_𝑛𝑞⁻¹if𝑛>0,(2.4) where 𝐻_𝑛(𝑞⁻¹) are the 𝑛th Frobenius-Euler numbers.

From (1.5), (1.6), and (1.7), we have 𝛽_0,𝑞=1,𝑞𝛽_𝑞+1^𝑛−𝛽_𝑛,𝑞=𝑞−1log𝑞if0𝑛=1,if𝑛>1,(2.5) with the usual convention about replacing (𝛽_𝑞)^𝑛 with 𝛽_𝑛,𝑞. By (1.7), the 𝑛th 𝑞-Bernoulli polynomials with weight 0 are given by 𝛽_𝑛,𝑞(𝑥)=_ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇_𝑞𝑦=^𝑛_𝑙=0𝑛𝑙𝑥^𝑛−𝑙𝛽_𝑙,𝑞.(2.6) From (2.6), we can derive the following function equation: 𝑞−1log𝑞𝑡+log𝑞𝑞𝑒^𝑡𝑒−1^𝑥𝑡=^∞_𝑛=0𝛽_𝑛,𝑞(𝑥)𝑡^𝑛.𝑛!(2.7) Thus, by (2.7), we get that 𝛽_{𝑛,𝑞−1}(1−𝑥)=(−1)^𝑛𝛽_𝑛,𝑞(𝑥),for𝑛∈ℤ₊.(2.8) By the definition of 𝑝-adic 𝑞-integral on ℤ_𝑝, we see that _ℤ𝑝(1−𝑥)^𝑛𝑑𝜇_𝑞(𝑥)=(−1)^𝑛_ℤ𝑝(𝑥−1)^𝑛𝑑𝜇_𝑞(𝑥)=(−1)^𝑛𝛽_𝑛,𝑞(−1).(2.9) Therefore, by (2.8) and (2.9), we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℤ₊, we have (−1)^𝑛𝛽_𝑛,𝑞(𝑥)=𝛽_{𝑛,𝑞−1}(1−𝑥).(2.10) In particular, 𝑥=−1, we get _ℤ𝑝1−𝑦^𝑛𝑑𝜇_𝑞𝑦=(−1)^𝑛𝛽_𝑛,𝑞(−1)=𝛽_{𝑛,𝑞−1}(2).(2.11)

From (2.5), we can derive the following equation: 𝑞²𝛽_𝑛,𝑞(2)=𝑞²+𝑛𝑞𝑞−1𝛽log𝑞−𝑞+_𝑛,𝑞,if𝑛>1.(2.12) Therefore, by (2.12), we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℕ with 𝑛>1, we have 𝛽_𝑛,𝑞(2)𝑛=1+𝑞𝑞−1−1log𝑞𝑞+1𝑞²𝛽_𝑛,𝑞.(2.13)

Taking the 𝑝-adic 𝑞-integral on ℤ_𝑝 for one Bernstein polynomials in (1.9), we get _ℤ𝑝𝐵_𝑘,𝑛(𝑥)𝑑𝜇_𝑞(𝑥)=𝑛𝑘^ℤ𝑝𝑥^𝑘(1−𝑥)^𝑛−𝑘𝑑𝜇_𝑞(𝑥)=𝑛𝑘^{𝑛−𝑘𝑙=0}𝑙()𝑛−𝑘−1^𝑙_ℤ𝑝𝑥^𝑘+𝑙𝑑𝜇_𝑞(𝑥)=𝑛𝑘^{𝑛−𝑘𝑙=0}𝑙()𝑛−𝑘−1^𝑙𝛽_{𝑘+𝑙,𝑞}.(2.14) From the symmetry of Bernstein polynomials, we note that _ℤ𝑝𝐵_𝑘,𝑛(𝑥)𝑑𝜇_𝑞(𝑥)=_ℤ𝑝𝐵_{𝑛−𝑘,𝑛}(1−𝑥)𝑑𝜇_𝑞(𝑥)=𝑛𝑘^𝑘𝑙=0𝑘𝑙(−1)^𝑘+𝑙_ℤ𝑝(1−𝑥)^𝑛−𝑙𝑑𝜇_𝑞(𝑥).(2.15)

Let 𝑛>𝑘+1. Then, by Theorem 2.3 and (2.15), we get _ℤ𝑝𝐵_𝑘,𝑛(𝑥)𝑑𝜇_𝑞(𝑥)=𝑛𝑘^𝑘𝑙=0𝑘𝑙(−1)^𝑘+𝑙1−𝑛−𝑙𝑞⁻¹𝑞⁻¹−1log𝑞−𝑞+𝑞²𝛽_{𝑛−𝑙,𝑞−1}=1+𝑛𝑞−1log𝑞−𝑞+𝑞²𝛽_{𝑛,𝑞−1}if𝑛𝑘=0,1−𝑞log𝑞+𝑛𝑞²𝛽_{𝑛,𝑞−1}+𝑛𝑞²𝛽_{𝑛−1,𝑞−1}if𝑛𝑘𝑞𝑘=1,_2𝑘𝑙=0𝑘𝑙(−1)^𝑘+𝑙𝛽_{𝑛−𝑙,𝑞−1}if𝑘>1.(2.16)

By comparing the coefficients on the both sides of (2.14) and (2.16), we obtain the following theorem.

Theorem 2.4. For 𝑛,𝑘∈ℤ₊ with 𝑛>𝑘+1, we have ^{𝑛−1𝑙=0}𝑙()𝑛−1−1^𝑙𝛽_1+𝑙,𝑞=1−𝑞log𝑞+𝑞²𝛽_{𝑛,𝑞−1}+𝑞²𝛽_{𝑛−1,𝑞−1},_{𝑛−𝑘𝑙=0}𝑙()𝑛−𝑘−1^𝑙𝛽_{𝑘+𝑙,𝑞}=𝑞^2𝑘_𝑙=0𝑘𝑙(−1)^𝑘+𝑙𝛽_{𝑛−𝑙,𝑞−1},if𝑘>1.(2.17) In particular, when 𝑘=0, we have ^𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝛽_𝑙,𝑞=1+𝑛𝑞−1log𝑞−𝑞+𝑞²𝛽_{𝑛,𝑞−1}.(2.18)

Let 𝑚,𝑛,𝑘∈ℤ₊ with 𝑚+𝑛>2𝑘+1. Then we see that _ℤ𝑝𝐵_𝑘,𝑛(𝑥)𝐵_𝑘,𝑚(𝑥)𝑑𝜇_𝑞(𝑥)=𝑛𝑘𝑚𝑘^ℤ𝑝𝑥^2𝑘(1−𝑥)^{𝑛+𝑚−2𝑘}𝑑𝜇_𝑞(𝑥)=𝑛𝑘𝑚𝑘^2𝑘𝑙=0𝑙()2𝑘−1^𝑙+2𝑘_ℤ𝑝(1−𝑥)^{𝑛+𝑚−l}𝑑𝜇_𝑞(𝑥)=𝑛𝑘𝑚𝑘^2𝑘𝑙=0𝑙()2𝑘−1^𝑙+2𝑘1−(𝑛+𝑚−𝑙)1−𝑞log𝑞−𝑞+𝑞²𝛽_{𝑛+𝑚−𝑙,𝑞−1}=1+(𝑛+𝑚)𝑞−1log𝑞−𝑞+𝑞²𝛽_{𝑛+𝑚,𝑞−1}if𝑛𝑘𝑚𝑘𝑞𝑘=0,_22𝑘𝑙=0𝑙()2𝑘−1^l+2𝑘𝛽_{𝑛+𝑚−𝑙,𝑞−1}if𝑘>0.(2.19) For 𝑚,𝑛,𝑘∈ℤ₊, we have _ℤ𝑝𝐵_𝑘,𝑛(𝑥)𝐵_𝑘,𝑚(𝑥)𝑑𝜇_𝑞(𝑥)=𝑛𝑘𝑚𝑘^ℤ𝑝𝑥^2𝑘(1−𝑥)^{𝑛+𝑚−2𝑘}𝑑𝜇_𝑞(𝑥)=𝑛𝑘𝑚𝑘^{𝑛+𝑚−2𝑘𝑙=0}𝑙()𝑛+𝑚−2𝑘−1^𝑙_ℤ𝑝𝑥^2𝑘+𝑙𝑑𝜇_𝑞(𝑥)=𝑛𝑘𝑚𝑘^{𝑛+𝑚−2𝑘𝑙=0}𝑙()𝑛+𝑚−2𝑘−1^𝑙𝛽_{𝑙+2𝑘,𝑞}.(2.20)

By comparing the coefficients on the both sides of (2.19) and (2.20), we obtain the following theorem.

Theorem 2.5. For 𝑚,𝑛,𝑘∈ℤ₊ with 𝑚+𝑛>2𝑘+1, we have ^{𝑛+𝑚𝑙=0}𝑙()𝑛+𝑚−1^𝑙𝛽_𝑙,𝑞=1+(𝑛+𝑚)𝑞−1log𝑞−𝑞+𝑞²𝛽_{𝑛+𝑚,𝑞−1}.(2.21) In particular, when 𝑘≠0, we have ^{𝑛+𝑚−2𝑘𝑙=0}𝑙()𝑛+𝑚−2𝑘−1^𝑙𝛽_{𝑙+2𝑘,𝑞}=𝑞^22𝑘_𝑙=0𝑙()2𝑘−1^𝑙+2𝑘𝛽_{𝑛+𝑚−𝑙,𝑞−1}.(2.22)

For 𝑠∈ℕ, let 𝑘,𝑛₁,…,𝑛_𝑠∈ℤ₊ with 𝑛₁+𝑛₂+⋯+𝑛_𝑠>𝑠𝑘+1. By the same method above, we get _ℤ𝑝^𝑠_𝑖=1𝐵_{𝑘,𝑛𝑖}(𝑥)𝑑𝜇_𝑞(𝑥)=1+^𝑠_𝑖=1𝑛_𝑖𝑞−1log𝑞−𝑞+𝑞²𝛽_{𝑛1+𝑛2+⋯+𝑛𝑠,𝑞−1}if𝑘=0,_𝑠𝑖=1𝑛_𝑖𝑘𝑞^2𝑠𝑘_𝑙=0𝑙()𝑠𝑘−1^{𝑙+𝑠𝑘}𝛽_{𝑛1+𝑛2+⋯+𝑛𝑠−𝑙,𝑞−1}if𝑘>0.(2.23) From the binomial theorem, we note that _ℤ𝑝^𝑠_𝑖=1𝐵_{𝑘,𝑛𝑖}(𝑥)𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑛1+⋯+𝑛𝑠−𝑠𝑘𝑙=0}𝑛₁+⋯+𝑛_𝑠𝑙()−𝑠𝑘−1^𝑙𝛽_{𝑙+𝑠𝑘,𝑞}.(2.24)

By comparing the coefficients on the both sides of (2.23) and (2.24), we obtain the following theorem.

Theorem 2.6. For 𝑠∈ℕ, let 𝑘,𝑛₁,…,𝑛_𝑠∈ℤ₊ with 𝑛₁+𝑛₂+⋯+𝑛_𝑠>𝑠𝑘+1. Then, we have ^{𝑛1+⋯+𝑛𝑠}_𝑙=0𝑛₁+⋯+𝑛_𝑠𝑙(−1)^𝑙𝛽_𝑙,𝑞=1+^𝑠_𝑖=1𝑛_𝑖𝑞−1log𝑞−𝑞+𝑞²𝛽_{𝑛1+⋯+𝑛𝑠,𝑞−1}.(2.25) In particular, when 𝑘≠0, we have ^{𝑛1+⋯+𝑛𝑠−𝑠𝑘}_𝑙=0𝑛₁+⋯+𝑛_𝑠𝑙()−𝑠𝑘−1^𝑙𝛽_{𝑙+𝑠𝑘,𝑞}=𝑞^2𝑠𝑘_𝑙=0𝑙()𝑠𝑘−1^{𝑙+𝑠𝑘}𝛽_{𝑛1+⋯+𝑛𝑠−𝑙,𝑞−1}.(2.26)

Acknowledgment

The present research has been conducted by the Research Grant of Kwangwoon University in 2011.