𝑞-Bernstein Polynomials Associated with 𝑞-Stirling Numbers and Carlitz's 𝑞-Bernoulli Numbers

Abstract

Recently, Kim (2011) introduced 𝑞-Bernstein polynomials which are different 𝑞-Bernstein polynomials of Phillips (1997). In this paper, we give a 𝑝-adic 𝑞-integral representation for 𝑞-Bernstein type polynomials and investigate some interesting identities of 𝑞-Bernstein type polynomials associated with 𝑞-extensions of the binomial distribution, 𝑞-Stirling numbers, and Carlitz's 𝑞-Bernoulli numbers.

1. Introduction

Let 𝑝 be a fixed prime number. Throughout this paper, ℤ_𝑝, ℚ_𝑝, ℂ, and ℂ_𝑝 denote the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, the complex number field, and the completion of the algebraic closure of ℚ_𝑝, respectively. Let ℕ be the set of natural numbers and ℤ₊=ℕ∪{0}. Let 𝜈_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{−𝜈𝑝(𝑝)}=1/𝑝.

When one talks of 𝑞-extensions, 𝑞 is variously considered as an indeterminate, a complex number 𝑞∈ℂ, or a 𝑝-adic number 𝑞∈ℂ_𝑝. If 𝑞∈ℂ then one normally assumes |𝑞|<1, and if 𝑞∈ℂ_𝑝 then one normally assumes |1−𝑞|_𝑝<1.

The 𝑞-bosonic natural numbers are defined by [𝑛]_𝑞=(1−𝑞^𝑛)/(1−𝑞)=1+𝑞+𝑞²+⋯+𝑞^𝑛−1 for 𝑛∈ℕ, and the 𝑞-factorial is defined by [𝑛]_𝑞!=[𝑛]_𝑞[𝑛−1]_𝑞⋯[2]_𝑞[1]_𝑞 (see [1–3]). For the 𝑞-extension of binomial coefficients, we use the following notation in the form of 𝑛𝑘_𝑞=[𝑛]_𝑞![𝑛−𝑘]_𝑞![𝑘]_𝑞!=[𝑛]_𝑞[𝑛−1]_𝑞⋯[𝑛−𝑘+1]_𝑞[𝑘]_𝑞!. (1.1)

Let 𝐶[0,1] denote the set of continuous functions on the real interval [0,1]. The Bernstein operator for 𝑓∈𝐶[0,1] is defined by 𝔹_𝑛(𝑓∣𝑥)=^𝑛_𝑘=0𝑘𝑓(𝑛)𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘=^𝑛_𝑘=0𝑘𝑓(𝑛)𝐵_𝑘,𝑛(𝑥), (1.2) where 𝑛,𝑘∈ℤ₊. The polynomials 𝐵_𝑘,𝑛(𝑥)=(^𝑛_𝑘)𝑥^𝑘(1−𝑥)^𝑛−𝑘 are called Bernstein polynomials of degree 𝑛 (see [4–8]). For 𝑓∈𝐶[0,1], 𝑞-Bernstein type operator of order 𝑛 for 𝑓 is defined by𝔹_𝑛,𝑞(𝑓∣𝑥)=^𝑛_𝑘=0𝑘𝑓(𝑛)𝑛𝑘[𝑥]^𝑘_𝑞[1−𝑥]^𝑛−𝑘_1/𝑞=^𝑛_𝑘=0𝑘𝑓(𝑛)𝐵_𝑘,𝑛(𝑥,𝑞), (1.3) where 𝑛,𝑘∈ℤ₊. Here 𝐵_𝑘,𝑛(𝑥,𝑞)=(^𝑛_𝑘)[𝑥]^𝑘_𝑞[1−𝑥]^𝑛−𝑘_1/𝑞 are called 𝑞-Bernstein type polynomials of degree 𝑛 (see [9]).

We say that 𝑓 is uniformly differentiable function at a point 𝑎∈ℤ_𝑝 and write 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). For 𝑓∈UD(ℤ_𝑝), the 𝑝-adic 𝑞-integral on ℤ_𝑝 is defined by 𝐼_𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)=lim_𝑁→∞1[𝑝^𝑁]_𝑞^𝑝𝑁−1_𝑥=0𝑓(𝑥)𝑞^𝑥 (1.4) (see [10]). Carlitz's 𝑞-Bernoulli numbers can be represented by a 𝑝-adic 𝑞-integral on ℤ_𝑝 as follows:_ℤ𝑝[𝑥]^𝑛_𝑞𝑑𝜇_𝑞(𝑥)=lim_𝑁→∞1[𝑝^𝑁]_𝑞^𝑝𝑁−1_𝑥=0[𝑥]^𝑛_𝑞𝑞^𝑥=𝛽_𝑛,𝑞 (1.5) (see [10, 11]). The 𝑘th order factorial of [𝑥]_𝑞 is defined by [𝑥]_𝑘,𝑞=[𝑥]_𝑞[𝑥−1]_𝑞⋯[𝑥−𝑘+1]_𝑞=(1−𝑞^𝑥)(1−𝑞^𝑥−1)⋯(1−𝑞^{𝑥−𝑘+1})(1−𝑞)^𝑘 (1.6) and is called the 𝑞-factorial of 𝑥 of order 𝑘 (see [10]).

In this paper, we give a 𝑝-adic 𝑞-integral representation for 𝑞-Bernstein type polynomials and derive some interesting identities for the 𝑞-Bernstein type polynomials associated with the 𝑞-extension of binomial distributions, 𝑞-Stirling numbers, and Carlitz's 𝑞-Bernoulli numbers.

2. 𝑞-Bernstein Polynomials

In this section, we assume that 0<𝑞<1. Let ℙ_𝑛,𝑞={∑_𝑖𝑎_𝑖[𝑥]^𝑖_𝑞∣𝑎_𝑖∈ℝ} be the space of 𝑞-polynomials of degree less than or equal to 𝑛.

We claim that the 𝑞-Bernstein type polynomials of degree 𝑛 defined by (1.3) are a basis for ℙ_𝑛,𝑞.

First, we see that the 𝑞-Bernstein type polynomials of degree 𝑛 span the space of 𝑞-polynomials. That is, any 𝑞-polynomials of degree less than or equal to 𝑛 can be written as a linear combination of the 𝑞-Bernstein type polynomials of degree 𝑛.

For 𝑛,𝑘∈ℤ₊ and 𝑥∈[0,1], we have𝐵_𝑘,𝑛(𝑥,𝑞)=^𝑛_𝑙=𝑘𝑛𝑙𝑙𝑘(−1)^𝑙−𝑘[𝑥]^𝑙_𝑞 (2.1) (see [9]). If there exist constants 𝐶₀,𝐶₁,…,𝐶_𝑛 such that 𝐶₀𝐵_0,𝑛(𝑥,𝑞)+𝐶₁𝐵_1,𝑛(𝑥,𝑞)+⋯+𝐶_𝑛𝐵_𝑛,𝑛(𝑥,𝑞)=0 holds for all 𝑥, then we can derive the following equation from (2.1): 0=𝐶₀𝐵_0,𝑛(𝑥,𝑞)+𝐶₁𝐵_1,𝑛(𝑥,𝑞)+⋯+𝐶_𝑛𝐵_𝑛,𝑛(𝑥,𝑞)=𝐶₀^𝑛_𝑖=0(−1)^𝑖𝑛𝑖𝑖0[𝑥]^𝑖_𝑞+𝐶₁^𝑛_𝑖=1(−1)^𝑖−1𝑛𝑖𝑖1[𝑥]^𝑖_𝑞+⋯+𝐶_𝑛^𝑛_𝑖=𝑛(−1)^𝑖−𝑛𝑛𝑖𝑖𝑛[𝑥]^𝑖_𝑞=𝐶₀+{¹_𝑖=0𝐶_𝑖(−1)^𝑖−1𝑛11𝑖}[𝑥]_𝑞+⋯+{^𝑛_𝑖=0𝐶_𝑖(−1)^𝑖−𝑛𝑛𝑛𝑛𝑖}[𝑥]^𝑛_𝑞. (2.2) Since the power basis is a linearly independent set, it follows that 𝐶₀=0,_1𝑖=0𝐶_𝑖(−1)^𝑖−1𝑛11𝑖=0,⋮⋮_𝑛𝑖=0𝐶_𝑖(−1)^𝑖−𝑛𝑛𝑛𝑛𝑖=0, (2.3) which implies that 𝐶₀=𝐶₁=⋯=𝐶_𝑛=0 (𝐶₀ is clearly zero, substituting this in the second equation gives 𝐶₁=0, substituting these two into the third equation gives 𝐶₂=0, and so on). Hence, we have the following theorem.

Theorem 2.1. The 𝑞-Bernstein type polynomials of degree 𝑛 are a basis for ℙ_𝑛,𝑞.

Let us consider a 𝑞-polynomial 𝑃_𝑞(𝑥)∈ℙ_𝑛,𝑞 as a linear combination of 𝑞-Bernstein type basis functions as follows:𝑃_𝑞(𝑥)=𝐶₀𝐵_0,𝑛(𝑥,𝑞)+𝐶₁𝐵_1,𝑛(𝑥,𝑞)+⋯+𝐶_𝑛𝐵_𝑛,𝑛(𝑥,𝑞). (2.4) We can write (2.4) as a dot product of two values:𝑃_𝑞(𝑥)=(𝐵_0,𝑛(𝑥,𝑞),𝐵_1,𝑛(𝑥,𝑞),…,𝐵_𝑛,𝑛𝐶(𝑥,𝑞))⁰𝐶₁⋮𝐶_𝑛. (2.5) From (2.5), we can derive the following equation: 𝑃_𝑞(𝑥)=(1,[𝑥]_𝑞,…,[𝑥]^𝑛_𝑞)𝑏₀₀𝑏00⋯0₁₀𝑏₁₁𝑏0⋯0⋮⋮⋮⋱⋮_𝑛0𝑏_𝑛1𝑏_𝑛2⋯𝑏_𝑛𝑛𝐶₀𝐶₁⋮𝐶_𝑛, (2.6) where the 𝑏_𝑖𝑗 are the coefficients of the power basis that are used to determine the respective 𝑞-Bernstein type polynomials.

From (1.3) and (2.1), we note that 𝐵_0,2(𝑥,𝑞)=[1−𝑥]²_1/𝑞=²_𝑙=02𝑙(−1)^𝑙[𝑥]^𝑙_𝑞=1−2[𝑥]_𝑞+[𝑥]²_𝑞,𝐵_1,221(𝑥,𝑞)=[𝑥]_𝑞[1−𝑥]_1/𝑞=2[𝑥]_𝑞−2[𝑥]²_𝑞,𝐵_2,222(𝑥,𝑞)=[𝑥]²_𝑞=[𝑥]²_𝑞. (2.7)

In the quadratic case (𝑛=2), the matrix representation is 𝑃_𝑞(𝑥)=(1,[𝑥]_𝑞,[𝑥]²_𝑞)100𝐶−2201−21⁰𝐶₁𝐶₂. (2.8) In the cubic case (𝑛=3), the matrix representation is 𝑃_𝑞(𝑥)=(1,[𝑥]_𝑞,[𝑥]²_𝑞,[𝑥]³_𝑞)1000𝐶−33003−630−13−31⁰𝐶₁𝐶₂𝐶₃. (2.9) In many applications of 𝑞-Bernstein polynomials, a matrix formulation for the 𝑞-Bernstein type polynomials seems to be useful.

Remark 2.2 (see [12]). All results of this section for 𝑞=1 are well known in classical case (see Bernstein Polynomials by Joy).

3. 𝑞-Bernstein Polynomials, 𝑞-Stirling Numbers, and 𝑞-Bernoulli Numbers

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

For 𝑓∈UD(ℤ_𝑝), let us consider the 𝑝-adic analogue of 𝑞-Bernstein type operator of order 𝑛 on ℤ_𝑝 as follows: 𝔹_𝑛,𝑞(𝑓∣𝑥)=^𝑛_𝑘=0𝑘𝑓(𝑛)𝑛𝑘[𝑥]^𝑘_𝑞[1−𝑥]^𝑛−𝑘_1/𝑞=^𝑛_𝑘=0𝑘𝑓(𝑛)𝐵_𝑘,𝑛(𝑥,𝑞). (3.1) Here 𝐵_𝑘,𝑛(𝑥,𝑞) is the 𝑞-Bernstein type polynomials of degree 𝑛 on ℤ_𝑝 defined by 𝐵_𝑘,𝑛𝑛𝑘(𝑥,𝑞)=[𝑥]^𝑘_𝑞[1−𝑥]^𝑛−𝑘_1/𝑞, (3.2) for 𝑛,𝑘∈ℤ₊ and 𝑥∈ℤ_𝑝.

Let (𝐸ℎ)(𝑥)=ℎ(𝑥+1) be the shift operator. Then the 𝑞-difference operator is defined byΔ^𝑛_𝑞∶=(𝐸−𝐼)^𝑛_𝑞=^𝑛_𝑖=1(𝐸−𝑞^𝑖−1𝐼), (3.3) where (𝐼ℎ)(𝑥)=ℎ(𝑥). From (3.3), we derive the following equation:Δ^𝑛_𝑞𝑓(0)=^𝑛_𝑘=0𝑛𝑘_𝑞(−1)^𝑘𝑞^(𝑘2)𝑓(𝑛−𝑘). (3.4) By (3.4), we easily see that 𝑓(𝑥)=_𝑛≥0𝑥𝑛_𝑞Δ^𝑛_𝑞𝑓(0) (3.5) (see [10, 11]).

The 𝑞-Stirling number of the first kind is defined by^𝑛_𝑘=1(1+[𝑘]_𝑞𝑧)=^𝑛_𝑘=0𝑆_1,𝑞(𝑛,𝑘)𝑧^𝑘, (3.6) and the 𝑞-Stirling number of the second kind is also defined by^𝑛_𝑘=1(11+[𝑘]_𝑞𝑧)=^𝑛_𝑘=0𝑆_2,𝑞(𝑛,𝑘)𝑧^𝑘. (3.7) By (3.3), (3.4), (3.6), and (3.7), we get 𝑆_2,𝑞𝑞(𝑛,𝑘)=^𝑘2)−([𝑘]_𝑞!^𝑘_𝑗=0(−1)^𝑗𝑞^(𝑗2)𝑘𝑗_𝑞[𝑘−𝑗]^𝑛_𝑞=𝑞^𝑘2)−([𝑘]_𝑞!Δ^𝑘_𝑞0^𝑛, (3.8) for 𝑛,𝑘∈ℤ₊ (see [10, 13]).

From the definition of 𝑞-Bernstein type polynomials of degree 𝑛 on ℤ_𝑝, we easily see that_ℤ𝑝𝐵_𝑘,𝑛(𝑥,𝑞)𝑑𝜇_𝑞(𝑥)=^𝑛−𝑘_𝑙=0𝑙𝑛𝑘𝑛−𝑘(−1)^𝑙_ℤ𝑝[𝑥]_𝑞^𝑙+𝑘𝑑𝜇_𝑞(𝑥). (3.9) By (1.5) and (3.9), we obtain the following proposition.

Proposition 3.1. For 𝑛,𝑘∈ℤ₊, one has _ℤ𝑝𝐵_𝑘,𝑛(𝑥,𝑞)𝑑𝜇_𝑞(𝑥)=^𝑛−𝑘_𝑙=0𝑙𝑛𝑘𝑛−𝑘(−1)^𝑙𝛽_{𝑙+𝑘,𝑞}, (3.10) where 𝛽_{𝑙+𝑘,𝑞} are the (𝑙+𝑘)th Carlitz's 𝑞-Bernoulli numbers.

From the definition of 𝑞-Bernstein polynomial, we note that^𝑛_𝑘=𝑖(^𝑘_𝑖)(^𝑛_𝑖)𝐵_𝑘,𝑛(𝑥,𝑞)=^𝑖_𝑘=0𝑞^(𝑘2)𝑥𝑘_𝑞[𝑘]_𝑞!𝑆_2,𝑞(𝑘,𝑖−𝑘), (3.11) where 𝑖∈ℕ. From the definition of 𝑞-binomial coefficient, we have𝑘𝑛+1_𝑞=𝑛𝑘−1_𝑞+𝑞^𝑘𝑛𝑘_𝑞=𝑞^𝑛−𝑘𝑛𝑘−1_𝑞+𝑛𝑘_𝑞. (3.12) By (3.12), we see that_ℤ𝑝𝑥𝑛_𝑞𝑑𝜇_𝑞(𝑥)=(−1)^𝑛[𝑛+1]_𝑞𝑞^{2)(𝑛+1)−(𝑛+1} (3.13) (see [10, 11]). From (1.5), (3.11), and (3.13), we obtain the following theorem.

Theorem 3.2. For 𝑛,𝑘∈ℤ₊ and 𝑖∈ℕ, one has ^𝑛_𝑘=𝑖^𝑛−𝑘_𝑙=0(^𝑘_𝑖)(^𝑛_𝑖)𝑙𝑛𝑘𝑛−𝑘(−1)^𝑙𝛽_{𝑙+𝑘,𝑞}=^𝑖_𝑘=0𝑞^(𝑘2)[𝑘]_𝑞!𝑆_2,𝑞(𝑘,𝑖−𝑘)(−1)^𝑘[𝑘+1]_𝑞𝑞^{2)(𝑘+1)−(𝑘+1}. (3.14)

It is easy to see that, for 𝑖∈ℕ,^𝑛_𝑘=𝑖(^𝑘_𝑖)(^𝑛_𝑖)𝐵_𝑘,𝑛(𝑥,𝑞)=[𝑥]^𝑖_𝑞. (3.15) By (3.11) and (3.15), we easily get [𝑥]^𝑖_𝑞=^𝑖_𝑘=0𝑞^(𝑘2)𝑥𝑘_𝑞[𝑘]_𝑞!𝑆_2,𝑞(𝑘,𝑖−𝑘) (3.16) (see [10]). Thus, we have_ℤ𝑝[𝑥]^𝑖_𝑞𝑑𝜇_𝑞(𝑥)=^𝑖_𝑘=0𝑞^(𝑘2)[𝑘]_𝑞!𝑆_2,𝑞(𝑘,𝑖−𝑘)_ℤ𝑝𝑥𝑘_𝑞𝑑𝜇_𝑞(𝑥)=𝑞^𝑖_𝑘=0[𝑘]_𝑞!𝑆_2,𝑞(𝑘,𝑖−𝑘)(−1)^𝑘[𝑘+1]_𝑞. (3.17) By (1.5) and (3.17), we obtain the following corollary.

Corollary 3.3. For 𝑛,𝑘∈ℤ₊ and 𝑖∈ℕ, one has 𝛽_𝑖,𝑞=𝑞^𝑖_𝑘=0[𝑘]_𝑞!𝑆_2,𝑞(𝑘,𝑖−𝑘)(−1)^𝑘[𝑘+1]_𝑞. (3.18)

It is known that𝑆_2,𝑞1(𝑛,𝑘)=(1−𝑞)^𝑘𝑘_𝑗=0(−1)^𝑘−𝑗𝑗𝑘+𝑛𝑘−𝑗𝑗+𝑛_𝑞 (3.19) (see [10]) and 𝑛𝑘_𝑞=^𝑛_𝑗=0𝑛𝑗(𝑞−1)^𝑗−𝑘𝑆_2,𝑞(𝑘,𝑗−𝑘). (3.20) By a simple calculation, we have that𝑞^𝑛𝑥=^𝑛_𝑘=0(𝑞−1)^𝑘𝑞^(𝑘2)𝑛𝑘_𝑞[𝑥]_𝑘,𝑞=^𝑛_𝑚=0{^𝑛_𝑘=𝑚(𝑞−1)^𝑘𝑛𝑘_𝑞𝑆_1,𝑞(𝑘,𝑚)}[𝑥]^𝑚_𝑞,𝑞^𝑛𝑥=^𝑛_𝑚=0𝑛𝑚(𝑞−1)^𝑚[𝑥]^𝑚_𝑞. (3.21) From (3.21), we note that 𝑛𝑚=^𝑛_𝑘=𝑚(𝑞−1)^−𝑚+𝑘𝑛𝑘_𝑞𝑆_1,𝑞(𝑘,𝑚) (3.22) (see [10]).

Thus, we obtain the following proposition.

Proposition 3.4. For 𝑛,𝑘∈ℤ₊, one has 𝐵_𝑘,𝑛𝑛𝑘(𝑥,𝑞)=[𝑥]^𝑘_𝑞[1−𝑥]^𝑛−𝑘_1/𝑞=^𝑛_𝑚=k(𝑞−1)^−𝑘+𝑚𝑛𝑚_𝑞𝑆_1,𝑞(𝑚,𝑘)[𝑥]^𝑘_𝑞[1−𝑥]^𝑛−𝑘_1/𝑞. (3.23)

From the definition of the 𝑞-Stirling numbers of the first kind, we get𝑞^(𝑛2)𝑥𝑛_𝑞[𝑛]_𝑞!=[𝑥]_𝑛,𝑞𝑞^(𝑛2)=^𝑛_𝑘=0𝑆_1,𝑞(𝑛,𝑘)[𝑥]^𝑘_𝑞. (3.24) By (3.11) and (3.24), we obtain the following theorem.

Theorem 3.5. For 𝑛,𝑘∈ℤ₊ and 𝑖∈ℕ, one has ^𝑛_𝑘=𝑖(^𝑘_𝑖)(^𝑛_𝑖)𝐵_𝑘,𝑛(𝑥,𝑞)=^𝑖𝑘_𝑘=0𝑙=0𝑆_1,𝑞(𝑘,𝑙)𝑆_2,𝑞(𝑘,𝑖−𝑘)[𝑥]^𝑙_𝑞. (3.25)

By (3.15) and Theorem 3.5, we obtain the following corollary.

Corollary 3.6. For 𝑖∈ℤ₊, one has 𝛽_𝑖,𝑞=^𝑖𝑘_𝑘=0𝑙=0𝑆_1,𝑞(𝑘,𝑙)𝑆_2,𝑞(𝑘,𝑖−𝑘)𝛽_𝑙.𝑞. (3.26)

The 𝑞-Bernoulli polynomials of order 𝑘∈ℤ₊ are defined by 𝛽^(𝑘)_𝑛,𝑞1(𝑥)=(1−𝑞)^𝑛𝑛_𝑖=0𝑛𝑖(−1)^𝑖𝑞^𝑖𝑥_ℤ𝑝⋯_ℤ𝑝𝑞^{∑𝑘𝑙=1(𝑘−𝑙+𝑖)𝑥𝑙}𝑑𝜇_𝑞(𝑥₁)⋯𝑑𝜇_𝑞(𝑥_𝑘). (3.27) Thus, we have 𝛽^(𝑘)_𝑛,𝑞1(𝑥)=(1−𝑞)^𝑛𝑛_𝑖=0(−1)^𝑖(^𝑛_𝑖)(𝑖+𝑘)⋯(𝑖+1)[𝑖+𝑘]_𝑞⋯[𝑖+1]_𝑞𝑞^𝑖𝑥 (3.28) (see [10]). The inverse 𝑞-Bernoulli polynomials of order 𝑘 are defined by𝛽^(−𝑘)_𝑛,𝑞1(𝑥)=(1−𝑞)^𝑛𝑛_𝑖=0(−1)^𝑖(^𝑛_𝑖)𝑞^𝑖𝑥∫_ℤ𝑝⋯∫_ℤ𝑝𝑞^{∑𝑘𝑙=1(𝑘−𝑙+𝑖)𝑥l}𝑑𝜇_𝑞(𝑥₁)⋯𝑑𝜇_𝑞(𝑥_𝑘). (3.29) In the special case 𝑥=0, 𝛽^(𝑘)_𝑛,𝑞(0)=𝛽^(𝑘)_𝑛,𝑞 are called the 𝑛th 𝑞-Bernoulli numbers of order 𝑘, and 𝛽^(−𝑘)_𝑛,𝑞(0)=𝛽^(−𝑘)_𝑛,𝑞 are also called the inverse 𝑞-Bernoulli numbers of order 𝑘 (see [10]).

From (3.29), we have𝛽^(−𝑛)_𝑘,𝑞=1(1−𝑞)^𝑘𝑘_𝑗=0(−1)^𝑗𝑘𝑗[𝑗+𝑛]_𝑞⋯[𝑗+1]_𝑞=1(𝑗+𝑛)⋯(𝑗+1)(1−𝑞)^𝑘𝑘_𝑗=0(−1)^𝑗(^𝑘+𝑛_𝑛−𝑗)(_𝑛^𝑘+𝑛)𝑛𝑗+𝑛_𝑞[𝑛]_𝑞!=𝑛![𝑛]_𝑞!(_𝑛^𝑘+𝑛{1)𝑛!(1−𝑞)^𝑘𝑘_𝑗=0(−1)^𝑗𝑛𝑘+𝑛𝑛−𝑗𝑗+𝑛_𝑞}. (3.30) By (3.19) and (3.30), we get𝑛![𝑛]_𝑞!𝑛𝛽𝑘+𝑛_{(−𝑛)𝑘,𝑞}=𝑆_2,𝑞(𝑛,𝑘). (3.31) Therefore, by (3.11) and (3.31), we obtain the following theorem.

Theorem 3.7. For 𝑖,𝑛,𝑘∈ℤ₊, one has ^𝑛_𝑘=𝑖(^𝑘_𝑖)(^𝑛_𝑖)𝐵_𝑘,𝑛(𝑥,𝑞)=^𝑖_𝑘=0𝑞^(𝑘2)𝑖𝑘𝑥𝑘𝑘!_𝑞𝛽^(−𝑘)_{𝑖−𝑘,𝑞}. (3.32)

It is easy to show that 𝑞^(𝑛2)𝑥𝑛_𝑞=1[𝑛]_𝑞!^𝑛−1_𝑘=0([𝑥]_𝑞−[𝑘]_𝑞1)=[𝑛]_𝑞!^𝑛_𝑘=0(−1)^𝑘[𝑥]_𝑞^𝑛−𝑘𝑆_1,𝑞(𝑛−1,𝑘). (3.33) Thus, we have that ^𝑛_𝑘=𝑖(^𝑘_𝑖)(^𝑛_𝑖)𝐵_𝑘,𝑛(𝑥,𝑞)=^𝑖𝑘_𝑘=0𝑗=0(−1)^𝑗[𝑥]_𝑞^𝑘−𝑗𝑆_1,𝑞(𝑘−1,𝑗)𝑘![𝑘]_𝑞!𝑖𝑘𝛽^(−𝑘)_{𝑖−𝑘,𝑞}, (3.34) where 𝑛,𝑘,𝑖∈ℤ₊.

Acknowledgments

This paper was supported by the research grant of Kwangwoon University in 2010, and the authors would like to thank the referees for their careful reading and valuable comments.