On 𝑝-Adic Analogue of 𝑞-Bernstein Polynomials and Related Integrals

Abstract

Recently, Kim's work (in press) introduced 𝑞-Bernstein polynomials which are different Phillips' 𝑞-Bernstein polynomials introduced in the work by (Phillips, 1996; 1997). The purpose of this paper is to study some properties of several type Kim's 𝑞-Bernstein polynomials to express the 𝑝-adic 𝑞-integral of these polynomials on ℤ_𝑝 associated with Carlitz's 𝑞-Bernoulli numbers and polynomials. Finally, we also derive some relations on the 𝑝-adic 𝑞-integral of the products of several type Kim's 𝑞-Bernstein polynomials and the powers of them on ℤ_𝑝.

1. Introduction

Let 𝐶[0,1] denote the set of continuous functions on [0,1]. For 0<𝑞<1 and 𝑓∈𝐶[0,1], Kim introduced the 𝑞-extension of Bernstein linear operator of order 𝑛 for 𝑓 as follows: 𝔹_𝑛,𝑞𝑓∣𝑥=^𝑛_𝑘=0𝑓𝑘𝑛𝑛𝑘[𝑥]^𝑘_𝑞[1−𝑥]^𝑛−𝑘_1/𝑞=^𝑛_𝑘=0𝑓𝑘𝑛𝐵_𝑘,𝑛𝑥,𝑞, (1.1) where [𝑥]_𝑞=(1−𝑞^𝑥)/(1−𝑞) (see [1]). Here 𝔹_𝑛,𝑞(𝑓∣𝑥) is called Kim's 𝑞-Bernstein operator of order 𝑛 for 𝑓. For 𝑘,𝑛∈ℤ₊(=ℕ∪{0}), 𝐵_𝑘,𝑛(𝑥,𝑞)=(^𝑛_𝑘)[𝑥]^𝑘_𝑞[1−𝑥]^𝑛−𝑘_1/𝑞 are called the Kim's 𝑞-Bernstein polynomials of degree 𝑛 (see [2–6]).

In [7], Carlitz defined a set of numbers 𝜉_𝑘=𝜉_𝑘(𝑞) inductively by 𝜉₀=1,𝑞𝜉+1^𝑘−𝜉_𝑘=1if0𝑘=1,if𝑘>1, (1.2) with the usual convention of replacing 𝜉^𝑘 by 𝜉_𝑘. These numbers are 𝑞-analogues of ordinary Bernoulli numbers 𝐵_𝑘, but they do not remain finite for 𝑞=1. So he modified the definition as follows: 𝛽_0,𝑞=1,𝑞𝑞𝛽+1^𝑘−𝛽_𝑘,𝑞=1if0𝑘=1,if𝑘>1, (1.3) with the usual convention of replacing 𝛽^𝑘 by 𝛽_𝑘,𝑞 (see [7]). These numbers 𝛽_𝑛,𝑞 are called the 𝑛th Carlitz 𝑞-Bernoulli numbers. And Carlitz's 𝑞-Bernoulli polynomials are defined by 𝛽_𝑘,𝑞(𝑥)=𝑞^𝑥𝛽+[𝑥]_𝑞^𝑘=^𝑘_𝑖=0𝑘𝑖𝛽_𝑖,𝑞𝑞^𝑖𝑥[𝑥]_𝑞^𝑘−𝑖. (1.4) As 𝑞→1, we have 𝛽_𝑘,𝑞→𝐵_𝑘 and 𝛽_𝑘,𝑞(𝑥)→𝐵_𝑘(𝑥), where 𝐵_𝑘 and 𝐵_𝑘(𝑥) are the ordinary Bernoulli numbers and polynomials, respectively.

Let 𝑝 be a fixed prime number. Throughout this paper, ℤ, ℚ, ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 will denote the ring of rational integers, the field of rational numbers, the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers and the completion of algebraic closure of ℚ_𝑝, respectively. Let 𝜈_𝑝 be the normalized exponential valuation of ℂ_𝑝 such that |𝑝|_𝑝=𝑝^{−𝜈𝑝(𝑝)}=1/𝑝.

Let 𝑞 be regarded as either a complex number 𝑞∈ℂ or a 𝑝-adic number 𝑞∈ℂ_𝑝. If 𝑞∈ℂ, we assume |𝑞|<1, and if 𝑞∈ℂ_𝑝, we normally assume |1−𝑞|_𝑝<1.

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 quotient 𝐹_𝑓(𝑥,𝑦)=(𝑓(𝑥)−𝑓(𝑦))/(𝑥−𝑦) has a limit 𝑓′(𝑎) as (𝑥,𝑦)→(𝑎,𝑎) (see [1, 3, 8–13]).

For 𝑓∈UD(ℤ_𝑝), let us begin with the expression1𝑝^𝑁_{𝑞0≤𝑥<𝑝𝑁}𝑞^𝑥𝑓(𝑥)=_{0≤𝑥<𝑝𝑁}𝑓(𝑥)𝜇_𝑞𝑥+𝑝^𝑁ℤ_𝑝, (1.5) representing a 𝑞-analogue of the Riemann sums for 𝑓 (see [11]). The integral of 𝑓 on ℤ_𝑝 is defined as the limit as 𝑛→∞ of the sums (if exists). The 𝑝-adic 𝑞-integral on a function 𝑓∈UD(ℤ_𝑝) is defined by 𝐼_𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁_𝑞^𝑝𝑁−1_𝑥=0𝑓(𝑥)𝑞^𝑥, (1.6) (see [11]).

As was shown in [3], Carlitz's 𝑞-Bernoulli numbers can be represented by 𝑝-adic 𝑞-integral on ℤ_𝑝 as follows:_ℤ𝑝[𝑥]^𝑚_𝑞𝑑𝜇_𝑞(𝑥)=𝛽_𝑚,𝑞,for𝑚∈ℤ₊. (1.7) Also, Carlitz's 𝑞-Bernoulli polynomials 𝛽_𝑘,𝑞(𝑥) can be represented𝛽_𝑚,𝑞(𝑥)=_ℤ𝑝𝑥+𝑦^𝑚_𝑞𝑑𝜇_𝑞𝑦,for𝑚∈ℤ₊, (1.8) (see [3]).

In this paper, we consider the 𝑝-adic analogue of Kim's 𝑞-Bernstein polynomials on ℤ_𝑝 and give some properties of the several type Kim's 𝑞-Bernstein polynomials to represent the 𝑝-adic 𝑞-integral on ℤ_𝑝 of these polynomials. Finally, we derive some relations on the 𝑝-adic 𝑞-integral of the products of several type Kim's 𝑞-Bernstein polynomials and the powers of them on ℤ_𝑝.

2. 𝑞-Bernstein Polynomials Associated with 𝑝-Adic 𝑞-Integral on ℤ_𝑝

In this section, we assume that 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1.

From (1.5), (1.7) and (1.8), we note that_ℤ𝑝1−𝑥+𝑥₁^𝑛_1/𝑞𝑑𝜇_1/𝑞𝑥₁=𝑞^𝑛𝑞−1^𝑛𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥𝑙+1𝑞^𝑙+1,−1_ℤ𝑝𝑥+𝑥₁^𝑛_𝑞𝑑𝜇_𝑞𝑥₁=11−𝑞^𝑛𝑛−1_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥𝑙+11−𝑞^𝑙+1. (2.1) By (2.1), we get (−1)^𝑛𝑞^𝑛_ℤ𝑝𝑥+𝑥₁^𝑛_𝑞𝑑𝜇_𝑞𝑥₁=_ℤ𝑝1−𝑥+𝑥₁^𝑛_1/𝑞𝑑𝜇_1/𝑞𝑥₁. (2.2) Therefore, we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊, one has _ℤ𝑝1−𝑥+𝑥₁^𝑛_1/𝑞𝑑𝜇_1/𝑞𝑥₁=(−1)^𝑛𝑞^𝑛_ℤ𝑝𝑥+𝑥₁^𝑛_𝑞𝑑𝜇_𝑞𝑥₁. (2.3)

By the definition of Carlitz's 𝑞-Bernoulli numbers and polynomials, we get 𝑞²𝛽_𝑛,𝑞(2)−(𝑛+1)𝑞²+𝑞=𝑞𝑞𝛽+1^𝑛=𝛽_𝑛,𝑞if𝑛>1. (2.4) Thus, we have the following proposition.

Proposition 2.2. For 𝑛∈ℕ with 𝑛>1, one has 𝛽_𝑛,𝑞(2)=1𝑞²𝛽_𝑛,𝑞1+𝑛+1−𝑞. (2.5)

It is easy to show that [1−𝑥]^𝑛_1/𝑞=1−[𝑥]_𝑞^𝑛=(−1)^𝑛𝑞^𝑛[𝑥−1]^𝑛_𝑞. (2.6) Hence, we have _ℤ𝑝[1−𝑥]^𝑛_1/𝑞𝑑𝜇_𝑞(𝑥)=(−1)^𝑛𝑞^𝑛_ℤ𝑝[𝑥−1]^𝑛_𝑞𝑑𝜇_𝑞(𝑥). (2.7) By (1.8), we get_ℤ𝑝[1−𝑥]^𝑛_1/𝑞𝑑𝜇_𝑞(𝑥)=(−1)^𝑛𝑞^𝑛𝛽_𝑛,𝑞(−1). (2.8) By Theorem 2.1 and (2.8), we see that_ℤ𝑝[1−𝑥]^𝑛_1/𝑞𝑑𝜇_𝑞(𝑥)=(−1)^𝑛𝑞^𝑛𝛽_𝑛,𝑞(−1)=𝛽_𝑛,1/𝑞(2). (2.9) From (2.9) and Proposition 2.2, we have_ℤ𝑝[1−𝑥]^𝑛_1/𝑞𝑑𝜇_𝑞(𝑥)=𝛽_𝑛,1/𝑞(2)=𝑞²𝛽_𝑛,1/𝑞+𝑛+1−𝑞. (2.10) By (1.7) and (2.10), we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℕ with 𝑛>1, one has _ℤ𝑝[1−𝑥]^𝑛_1/𝑞𝑑𝜇_𝑞(𝑥)=𝑞²_ℤ𝑝[𝑥]^𝑛_1/𝑞𝑑𝜇_1/𝑞(𝑥)+𝑛+1−𝑞. (2.11)

Taking the 𝑝-adic 𝑞-integral on ℤ_𝑝 for one Kim's 𝑞-Bernstein polynomials, we get_ℤ𝑝𝐵_𝑘,𝑛𝑥,𝑞𝑑𝜇_𝑞(𝑥)=𝑛𝑘^ℤ𝑝[𝑥]^𝑘_𝑞[1−𝑥]^𝑛−𝑘_1/𝑞𝑑𝜇_𝑞(𝑥)=𝑛𝑘^{𝑛−𝑘𝑙=0}𝑙()𝑛−𝑘−1^𝑙_ℤ𝑝[𝑥]_𝑞^𝑘+𝑙𝑑𝜇_𝑞(𝑥)=𝑛𝑘^{𝑛−𝑘𝑙=0}𝑙()𝑛−𝑘−1^𝑙𝛽_{𝑘+𝑙,𝑞}, (2.12) and, by the 𝑞-symmetric property of 𝐵_𝑘,𝑛(𝑥,𝑞), we see that_ℤ𝑝𝐵_𝑘,𝑛𝑥,𝑞𝑑𝜇_𝑞(𝑥)=_ℤ𝑝𝐵_{𝑛−𝑘,𝑛}11−𝑥,𝑞𝑑𝜇_𝑞(𝑥)=𝑛𝑘^𝑘𝑙=0𝑘𝑙(−1)^𝑘+𝑙_ℤ𝑝[1−𝑥]^𝑛−𝑙_1/𝑞𝑑𝜇_𝑞(𝑥). (2.13)

For 𝑛>𝑘+1, by Theorem 2.3 and (2.13), one has_ℤ𝑝𝐵_𝑘,𝑛𝑥,𝑞𝑑𝜇_𝑞(𝑥)=𝑛𝑘^𝑘𝑙=0(−1)^𝑘+𝑙𝑘𝑙𝑛−𝑙+1−𝑞+𝑞²_ℤ𝑝[𝑥]^𝑛−𝑙_1/𝑞𝑑𝜇_1/𝑞(𝑥=𝑛𝑘_𝑘𝑙=0)(−1)^𝑘+𝑙𝑘𝑙𝑛−𝑙+1−𝑞+𝑞²𝛽_{𝑛−𝑙,1/𝑞}. (2.14)

Let 𝑚,𝑛,𝑘∈ℤ₊ with 𝑚+𝑛>2𝑘+1. Then the 𝑝-adic 𝑞-integral for the multiplication of two Kim's 𝑞-Bernstein polynomials on ℤ_𝑝 can be given by the following relation:_ℤ𝑝𝐵_𝑘,𝑛𝑥,𝑞𝐵_𝑘,𝑚𝑥,𝑞𝑑𝜇_𝑞(𝑥)=𝑛𝑘𝑚𝑘^ℤ𝑝[𝑥]_𝑞^2𝑘[1−𝑥]^{𝑛+𝑚−2𝑘}_1/𝑞𝑑𝜇_𝑞(𝑥)=𝑛𝑘𝑚𝑘^2𝑘𝑙=0𝑙()2𝑘−1^𝑙+2𝑘𝑞_ℤ𝑝[1−𝑥]^{𝑛+𝑚−𝑙}_1/𝑞𝑑𝜇_𝑞(𝑥). (2.15)

By Theorem 2.3 and (2.15), we get_ℤ𝑝𝐵_𝑘,𝑛𝑥,𝑞𝐵_𝑘,𝑚𝑥,𝑞𝑑𝜇_𝑞(𝑥)=𝑛𝑘𝑚𝑘^2𝑘𝑙=0𝑙()2𝑘−1^𝑙+2𝑘𝑛+𝑚−𝑙+1−𝑞+𝑞²_ℤ𝑝[𝑥]^{𝑛+𝑚−𝑙}_1/𝑞𝑑𝜇_1/𝑞(𝑥=𝑛𝑘𝑚𝑘_2𝑘𝑙=0)𝑙()2𝑘−1^𝑙+2𝑘𝑛+𝑚−𝑙+1−𝑞+𝑞²𝛽_{𝑛+𝑚−𝑙,1/𝑞}. (2.16)

By the simple calculation, we easily get_ℤ𝑝𝐵_𝑘,𝑛𝑥,𝑞𝐵_𝑘,𝑚𝑥,𝑞𝑑𝜇_𝑞(𝑥)=𝑛𝑘𝑚𝑘^ℤ𝑝[𝑥]_𝑞^2𝑘[1−𝑥]^{𝑛+𝑚−2𝑘}_1/𝑞𝑑𝜇_𝑞(𝑥)=𝑛𝑘𝑚𝑘^{𝑛+𝑚−2𝑘𝑙=0}𝑙()𝑛+𝑚−2𝑘−1^𝑙_ℤ𝑝[𝑥]_𝑞^𝑙+2𝑘𝑑𝜇_𝑞(𝑥)=𝑛𝑘𝑚𝑘^{𝑛+𝑚−2𝑘𝑙=0}𝑙()𝑛+𝑚−2𝑘−1^𝑙𝛽_{𝑙+2𝑘,𝑞}. (2.17) Continuing this process, we obtain_ℤ𝑝^𝑠_𝑖=1𝐵_{𝑘,𝑛𝑖}𝑥,𝑞𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^ℤ𝑝[𝑥]_𝑞^𝑠𝑘[1−𝑥]^{𝑛1+⋯+𝑛𝑠−𝑠𝑘}_1/𝑞𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑠𝑘𝑙=0}𝑙()𝑠𝑘−1^{𝑠𝑘+𝑙}_ℤ𝑝[1−𝑥]^{𝑛1+⋯+𝑛𝑠−𝑙}_1/𝑞𝑑𝜇_𝑞(𝑥). (2.18)

Let 𝑠∈ℕ and 𝑛₁,…,𝑛_𝑠, 𝑘∈ℤ₊ with 𝑛₁+𝑛₂+⋯+𝑛_𝑠>𝑠𝑘+1. By Theorem 2.3 and (2.18), we get_ℤ𝑝^𝑠_𝑖=1𝐵_{𝑘,𝑛𝑖}𝑥,𝑞𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑠𝑘𝑙=0}𝑙()𝑠𝑘−1^{𝑠𝑘+𝑙𝑠}_𝑖=1𝑛_𝑖−𝑙+1−𝑞+𝑞²_ℤ𝑝[𝑥]^{𝑛1+⋯+𝑛𝑠−𝑙}_1/𝑞𝑑𝜇_1/𝑞(𝑥=_𝑠𝑖=1)𝑛_𝑖𝑘^{𝑠𝑘𝑙=0}𝑙()𝑠𝑘−1^{𝑠𝑘+𝑙𝑠}_𝑖=1𝑛_𝑖−𝑙+1−𝑞+𝑞²𝛽_{𝑛1+⋯+𝑛𝑠−𝑙,1/𝑞}. (2.19) From the definition of binomial coefficient, we note that_ℤ𝑝^𝑠_𝑖=1𝐵_{𝑘,𝑛𝑖}𝑥,𝑞𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^ℤ𝑝[𝑥]_𝑞^𝑠𝑘[1−𝑥]^{𝑛1+⋯+𝑛𝑠−𝑠𝑘}_1/𝑞𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑛1+⋯+𝑛𝑠−𝑠𝑘𝑙=0}𝑛₁+⋯+𝑛_𝑠𝑙()−𝑠𝑘−1^𝑙_ℤ𝑝[𝑥]_𝑞^{𝑠𝑘+𝑙}𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑛1+⋯+𝑛𝑠−𝑠𝑘𝑙=0}𝑛₁+⋯+𝑛_𝑠𝑙)−𝑠𝑘(−1^𝑙𝛽_{𝑠𝑘+𝑙,𝑞}, (2.20) where 𝑠∈ℕ and 𝑛₁,…,𝑛_𝑠, 𝑘∈ℤ₊.

By (2.19) and (2.20), we obtain the following theorem.

Theorem 2.4. (I) For 𝑠∈ℕ and 𝑛₁,…,𝑛_𝑠, 𝑘∈ℕ with 𝑛₁+𝑛₂+⋯+𝑛_𝑠>𝑠𝑘+1, one has _ℤ𝑝^𝑠_𝑖=1𝐵_{𝑘,𝑛𝑖}𝑥,𝑞𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑠𝑘𝑙=0}𝑙()𝑠𝑘−1^{𝑠𝑘+𝑙𝑠}_𝑖=1𝑛_𝑖−𝑙+1−𝑞+𝑞²𝛽_{𝑛1+⋯+𝑛𝑠−𝑙,1/𝑞}. (2.21)
 (II) For 𝑠∈ℕ and 𝑛₁,…,𝑛_𝑠, 𝑘∈ℤ₊, one has_ℤ𝑝^𝑠_𝑖=1𝐵_{𝑘,𝑛𝑖}𝑥,𝑞𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑛1+⋯+𝑛𝑠−𝑠𝑘𝑙=0}𝑛₁+⋯+𝑛_𝑠𝑙()−𝑠𝑘−1^𝑙𝛽_{𝑠𝑘+𝑙,𝑞}. (2.22)

By Theorem 2.4, we obtain the following corollary.

Corollary 2.5. For 𝑠∈ℕ and 𝑛₁,…,𝑛_𝑠, 𝑘∈ℕ with 𝑛₁+𝑛₂+⋯+𝑛_𝑠>𝑠𝑘+1, one has ^{𝑠𝑘𝑙=0}𝑙()𝑠𝑘−1^{𝑠𝑘+𝑙𝑠}_𝑖=1𝑛_𝑖−𝑙+1−𝑞+𝑞²𝛽_{𝑛1+⋯+𝑛𝑠−𝑙,1/𝑞}=^{𝑛1+⋯+𝑛𝑠−𝑠𝑘}_𝑙=0𝑛₁+⋯+𝑛_𝑠𝑙()−𝑠𝑘−1^𝑙𝛽_{𝑠𝑘+𝑙,𝑞}. (2.23)

Let 𝑠∈ℕ and 𝑚₁,…,𝑚_𝑠, 𝑛₁,…,𝑛_𝑠, 𝑘∈ℤ₊ with 𝑚₁𝑛₁+⋯+𝑚_𝑠𝑛_𝑠>(𝑚₁+⋯+𝑚_𝑠)𝑘+1. Then one has_ℤ𝑝^𝑠_𝑖=1𝐵^𝑚𝑖_{𝑘,𝑛𝑖}𝑥,𝑞𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑚𝑖𝑘∑𝑠𝑖=1𝑚𝑖𝑙=0}𝑘^𝑠_𝑖=1𝑚_𝑖𝑙(−1)^{𝑘∑𝑠𝑖=1𝑚𝑖−𝑙}×_ℤ𝑝[1−𝑥]^{∑𝑠𝑖=1𝑛𝑖𝑚𝑖}_𝑞^−𝑙𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑚𝑖𝑘∑𝑠𝑖=1𝑚𝑖𝑙=0}𝑘^𝑠_𝑖=1𝑚_𝑖𝑙(−1)^{𝑘∑𝑠𝑖=1𝑚𝑖−𝑙}×^𝑠_𝑖=1𝑚_𝑖𝑛_𝑖−𝑙+1−𝑞+𝑞²_ℤ𝑝[𝑥]^{∑𝑠𝑖=1𝑛𝑖𝑚𝑖−𝑙}_1/𝑞𝑑𝜇_1/𝑞(𝑥=_𝑠𝑖=1)𝑛_𝑖𝑘^{𝑚𝑖𝑘∑𝑠𝑖=1𝑚𝑖𝑙=0}𝑘^𝑠_𝑖=1𝑚_𝑖𝑙(−1)^{𝑘∑𝑠𝑖=1𝑚𝑖−𝑙}×^𝑠_𝑖=1𝑚_𝑖𝑛_𝑖−𝑙+1−𝑞+𝑞²𝛽_{𝑛1𝑚1+⋯+𝑛𝑠𝑚𝑠−𝑙,1/𝑞}. (2.24)

From the definition of binomial coefficient, one has_ℤ𝑝^𝑠_𝑖=1𝐵^𝑚𝑖_{𝑘,𝑛𝑖}𝑥,𝑞𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑚𝑖∑𝑠𝑖=1𝑛𝑖𝑚𝑖−𝑘∑𝑠𝑖=1𝑚𝑖𝑙=0𝑠𝑖=1}𝑛_𝑖𝑚_𝑖−𝑘^𝑠_𝑖=1𝑚_𝑖𝑙(−1)^𝑙×_ℤ𝑝[𝑥]^{(𝑚1+⋯+𝑚𝑠}_𝑞^)𝑘+𝑙𝑑𝜇_𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑚𝑖∑𝑠𝑖=1𝑛𝑖𝑚𝑖−𝑘∑𝑠𝑖=1𝑚𝑖𝑙=0𝑠𝑖=1}𝑛_𝑖𝑚_𝑖−𝑘^𝑠_𝑖=1𝑚_𝑖𝑙×(−1)^𝑙𝛽_{(𝑚1+⋯+𝑚𝑠)𝑘+𝑙,𝑞}. (2.25)

By (2.24) and (2.25), we obtain the following theorem.

Theorem 2.6. For 𝑠∈ℕ and 𝑚₁,…,𝑚_𝑠, 𝑛₁,…,𝑛_𝑠, 𝑘∈ℤ₊ with 𝑚₁𝑛₁+⋯+𝑚_𝑠𝑛_𝑠>(𝑚₁+⋯+𝑚_𝑠)𝑘+1, one has ^𝑘∑𝑠_𝑖=1^𝑚𝑖_𝑙=0𝑘^𝑠_𝑖=1𝑚_𝑖𝑙(−1)^{𝑘∑𝑠𝑖=1𝑚𝑖−𝑙𝑠}_𝑖=1𝑚_𝑖𝑛_𝑖−𝑙+1−𝑞+𝑞²𝛽_{𝑛1𝑚1+⋯+𝑛𝑠𝑚𝑠−𝑙,1/𝑞}=^{∑𝑠𝑖=1𝑛𝑖𝑚𝑖−𝑘∑𝑠𝑖=1𝑚𝑖}_𝑙=0^𝑠𝑖=1𝑛_𝑖𝑚_𝑖−𝑘^𝑠_𝑖=1𝑚_𝑖𝑙(−1)^𝑙𝛽_{(𝑚1+⋯+𝑚𝑠)𝑘+𝑙,𝑞}. (2.26)

Acknowledgment

This paper was supported by the research grant of Kwangwoon University in 2010.