A Note on the Modified 𝑞-Bernstein Polynomials

Abstract

We propose the modified 𝑞-Bernstein polynomials of degree 𝑛 which are different 𝑞-Bernstein polynomials of Phillips (1997). From these modified 𝑞-Bernstein polynomials of degree 𝑛, we derive some recurrence formulae for the modified 𝑞-Bernstein polynomials.

1. Introduction

Let 𝐶[0,1] denote the set of continuous function on [0,1]. For 𝑓∈𝐶[0,1], Bernstein introduced the following well-known linear positive operators in [1]: 𝐵_𝑛(𝑓∶𝑥)∶=^𝑛_𝑘=0𝑘𝑓(𝑛)𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘=^𝑛_𝑘=0𝑘𝑓(𝑛)𝐵_𝑘,𝑛(𝑥), (1.1) where (^𝑛_𝑘)=𝑛(𝑛−1)⋯(𝑛−𝑘+1)/𝑘!. Here 𝐵_𝑛(𝑓∶𝑥) is called the Bernstein operator of order 𝑛 for 𝑓. For 𝑘,𝑛∈ℤ₊, the Bernstein polynomial of degree 𝑛 is defined by 𝐵_𝑘,𝑛𝑛𝑘(𝑥)=𝑥^𝑘(1−𝑥)^𝑛−𝑘, (1.2) where 𝑥∈[0,1]. For example, 𝐵_0,1(𝑥)=1−𝑥,𝐵_1,1𝐵(𝑥)=𝑥,_0,2(𝑥)=(1−𝑥)²,𝐵_1,2(𝑥)=2𝑥(1−𝑥),𝐵_2,2(𝑥)=𝑥²,…. (1.3) Also, 𝐵_𝑘,𝑛(𝑥)=0, for 𝑘>𝑛, because (^𝑛_𝑘)=0.

Some people have studied the Bernstein polynomials in the area of approximation theory (see [2] through [3]). Note that for 𝑘∈ℤ₊ and 𝑥∈[0,1], 𝑡^𝑘𝑒^{(1−𝑥)𝑡}𝑥^𝑘=𝑥𝑘!𝑘^𝑘!(𝑡^𝑘∞_𝑛=0(1−𝑥)^𝑛𝑡^𝑛)=𝑥𝑛!_𝑘𝑘!^∞_n=0(1−𝑥)^𝑛(𝑛+1)⋯(𝑛+𝑘)𝑡(𝑛+𝑘)!^𝑛+𝑘=^∞_𝑛=𝑘(𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘)𝑡^𝑛=𝑛!_∞𝑛=𝑘𝐵_𝑘,𝑛𝑡(𝑥)^𝑛.𝑛! (1.4) Because 𝐵_𝑘,0(𝑥)=𝐵_𝑘,1(𝑥)=⋯=𝐵_{𝑘,𝑘−1}(𝑥)=0, we obtain the generating function for 𝐵_𝑘,𝑛(𝑥) as follows:𝐹^(𝑘)𝑡(𝑡,𝑥)∶=^𝑘𝑒^{(1−𝑥)𝑡}𝑥^𝑘=𝑘!^∞_𝑛=0𝐵_𝑘,𝑛𝑡(𝑥)^𝑛𝑛! (1.5) (see [4, 5]), where 𝑘∈ℤ₊ and 𝑥∈[0,1]. Notice that 𝐵_𝑘,𝑛𝑛𝑘𝑥(𝑥)=^𝑘(1−𝑥)^𝑛−𝑘if0𝑛≥𝑘,if𝑛<𝑘, (1.6) for 𝑛,𝑘∈ℤ₊ (see [2]).

Let 0<𝑞<1. Define the 𝑞-number of 𝑥 by [𝑥]_𝑞∶=1−𝑞^𝑥.1−𝑞 (1.7) See [2] through [3] for details and related facts. Note that lim_𝑞→1[𝑥]_𝑞=𝑥. In [6], Phillips proposed a generalization of the classical Bernstein polynomials based on 𝑞-integers. In the last decade some new generalizations of well-known positive linear operators, based on 𝑞-integers were introduced and studied by several authors (see [1–13]). Recently, Simsek and Acikgoz have also studied the 𝑞-extension of Bernstein-type polynomials [5]. Their 𝑞-Bernstein-type polynomials are given by 𝑌_𝑛𝑛𝑘(𝑘;𝑥∶𝑞)=(−1)^𝑘𝑘!(1−𝑞)^{∞𝑛−𝑘}_𝑚,𝑙=0^𝑛−𝑘_𝑗=0𝑙𝑘𝑘+𝑙−1𝑛−𝑘×((−1)^𝑗𝑞^{𝑙+𝑗(1−𝑥)}𝑆(𝑚,𝑘)(𝑥ln𝑞)^𝑚𝑚!), (1.8) where 𝑆(𝑚,𝑘) are the second-kind stirling number. In [5], we can find some interesting formulae related to 𝑞-extension of Bernstein polynomials which are different 𝑞-Bernstein polynomials of Phillips. In the conference of Jangjeon Mathematical Society which was held in IRAN (on Feb.2010), Acikgoz and Arci has introduced several-type Bernstein polynomials (see [2]). The Acikgoz paper [2] announced in the conference is actually what motivated us to write this paper. In this paper, we considered the 𝑞-extension of Bernstein polynomials which were introduced by Acikgoz at the conference of Jangjeon Mathematical Society on Feb. 2010. First, we consider the 𝑞-extension of the generating function of Bernstein polynomials in (1.5). Indeed, this generating function is also treated by Simsek and Acikgoz in a previous paper (see [5]). From this 𝑞-extension of the generating function for the Bernstein polynomials, we propose the modified 𝑞-Bernstein polynomials of degree 𝑛 which are different 𝑞-Bernstein polynomials of Phillips. By using the properties of the modified 𝑞-Bernstein polynomials, we obtain some recurrence formulae for the modified 𝑞-Bernstein polynomials of degree 𝑛.

2. The Modified 𝑞-Bernstein Polynomials

For 0<𝑞<1, consider the 𝑞-extension of (1.5) as follows:𝐹_𝑞^(𝑘)𝑡(𝑡,𝑥)∶=^𝑘𝑒^{[1−𝑥]𝑞𝑡}[𝑥]^𝑘_𝑞=𝑘![𝑥]^𝑘_𝑞𝑘!^∞_𝑛=0[1−𝑥]^𝑛_𝑞𝑡𝑛!^𝑛+𝑘=^∞_𝑛=𝑘𝑛𝑘[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘𝑡^𝑛,𝑛! (2.1) where 𝑘,𝑛∈ℤ₊ and 𝑥∈[0,1]. Note that lim_𝑞→1𝐹_𝑞^(𝑘)(𝑡,𝑥)=𝐹^(𝑘)(𝑡,𝑥). We define the modified 𝑞-Bernstein polynomials as follows:𝐹_𝑞^(𝑘)𝑡(𝑡,𝑥)=^𝑘𝑒^{[1−𝑥]𝑞𝑡}[𝑥]^𝑘_𝑞=𝑘!^∞_𝑛=0𝐵_𝑘,𝑛𝑡(𝑥,𝑞)^𝑛,𝑛! (2.2) where 𝑘,𝑛∈ℤ₊ and 𝑥∈[0,1].

Remark 2. This generating function is also introduced by Simsek and Acikgoz in a previous paper (see [5]).

By comparing the coefficients of (2.1) and (2.2), we obtain the following theorem.

Theorem 2.1. For 𝑘,𝑛∈ℤ₊ and 𝑥∈[0,1], 𝐵_𝑘,𝑛𝑛(𝑥,𝑞)=𝑘[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘,if𝑛≥𝑘0,if𝑛<𝑘. (2.3)

For 0≤𝑘≤𝑛, we have [1−𝑥]_𝑞𝐵_{𝑘,𝑛−1}(𝑥,𝑞)+[𝑥]_𝑞𝐵_{𝑘−1,𝑛−1}(𝑥,𝑞)=[1−𝑥]_𝑞𝑘𝑛−1[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^{𝑛−1−𝑘}+[𝑥]_𝑞𝑛−1𝑘−1[𝑥]_𝑞^𝑘−1[1−𝑥]_𝑞^𝑛−𝑘=𝑘𝑛−1[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘+𝑛−1𝑘−1[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘=𝑛𝑘[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘, (2.4) and the derivatives of the modified 𝑞-Bernstein polynomials of degree 𝑛 are also polynomials of degree 𝑛−1, that is, 𝑑𝐵𝑑𝑥^𝑘,𝑛𝑛𝑘(𝑥,𝑞)=𝑘[𝑥]_𝑞^𝑘−1[1−𝑥]_𝑞^𝑛−𝑘ln𝑞𝑞𝑞−1^𝑥+𝑛𝑘[𝑥]^𝑘_𝑞(𝑛−𝑘)[1−𝑥]_𝑞^{𝑛−𝑘−1}(−ln𝑞𝑞−1)𝑞^1−𝑥=ln𝑞{𝑛𝑞−1𝑘𝑘[𝑥]_𝑞^𝑘−1[1−𝑥]_𝑞^𝑛−𝑘𝑞^𝑥−𝑛𝑘[𝑥]^𝑘_𝑞(𝑛−𝑘)[1−𝑥]_𝑞^{𝑛−𝑘−1}𝑞^1−𝑥}=𝑛(𝑞^𝑥𝐵_{𝑘−1,𝑛−1}(𝑥,𝑞)−𝑞^1−𝑥𝐵_{𝑘,𝑛−1}(𝑥,𝑞))ln𝑞.𝑞−1 (2.5) Therefore, we obtain the following recurrence formulae.

Theorem 2.2 (recurrence formulae for 𝐵_𝑘,𝑛(𝑥,𝑞)). For 𝑘,𝑛∈ℤ₊ and for 𝑥∈[0,1], [1−𝑥]_𝑞𝐵_{𝑘,𝑛−1}(𝑥,𝑞)+[𝑥]_𝑞𝐵_{𝑘−1,𝑛−1}(𝑥,𝑞)=𝐵_𝑘,𝑛𝑑(𝑥,𝑞),𝐵𝑑𝑥^𝑘,𝑛(𝑥,𝑞)=𝑛(𝑞^𝑥𝐵_{𝑘−1,𝑛−1}(𝑥,𝑞)−𝑞^1−𝑥𝐵_k,𝑛−1(𝑥,𝑞))ln𝑞.𝑞−1 (2.6)

Let 𝑓 be a continuous function on [0,1]. Then the modified 𝑞-Bernstein operator of order 𝑛 for 𝑓 is defined by𝐵_𝑛,𝑞(𝑓∶𝑥)∶=^𝑛_𝑘=0𝑘𝑓(𝑛)𝐵_𝑘,𝑛(𝑥,𝑞), (2.7) where 0≤𝑥≤1, 𝑛∈ℤ₊. We get from Theorem 2.1 and (2.7) that for 𝑓(𝑥)=𝑥,𝐵_𝑛,𝑞(𝑓∶𝑥)=^𝑛_𝑘=0𝑘𝑓(𝑛)𝑛𝑘[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘=[𝑥]_𝑞(1−[1−𝑥]_𝑞[𝑥]_𝑞(𝑞−1))^𝑛−1=𝑓([𝑥]_𝑞)(1+(1−𝑞)[𝑥]_𝑞[1−𝑥]_𝑞)^𝑛−1. (2.8) We also see from Theorem 2.1 that 𝐵_𝑛,𝑞(1∶𝑥)=^𝑛_𝑘=0𝐵_𝑘,𝑛=(𝑥,𝑞)_𝑛𝑘=0𝑛𝑘[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘=^𝑛_𝑘=0𝑛𝑘[𝑥]^𝑘_𝑞(1−𝑞^1−𝑥[𝑥]_𝑞)^𝑛−𝑘=(1+(1−𝑞)[𝑥]_𝑞[1−𝑥]_𝑞)^𝑛. (2.9)

The modified 𝑞-Bernstein polynomials are symmetric polynomials in the following sense: 𝐵_{𝑛−𝑘,𝑛}𝑛(1−𝑥,𝑞)=𝑛−𝑘[1−𝑥]_𝑞^𝑛−𝑘[𝑥]^𝑘_𝑞=𝐵_𝑘,𝑛(𝑥,𝑞). (2.10) Therefore, we get the following theorem.

Theorem 2.3. For 𝑘,𝑛∈ℤ₊ and 𝑥∈[0,1], 𝐵_{𝑛−𝑘,𝑛}(1−𝑥,𝑞)=𝐵_𝑘,𝑛𝐵(𝑥,𝑞),_𝑛,𝑞(1∶𝑥)=(1+(1−𝑞)[𝑥]_𝑞[1−𝑥]_𝑞)^𝑛. (2.11)

For 𝜁∈ℂ, 𝑥∈[0,1] and for 𝑛∈ℤ₊, consider𝑛!2𝜋𝑖_𝐶([𝑥]_𝑞𝜁)^𝑘𝑒𝑘!^{([1−𝑥]𝑞𝜁)}𝑑𝜁𝜁^𝑛+1, (2.12) where 𝐶 is a circle around the origin and integration is in the positive direction. We see from the definition of the modified 𝑞-Bernstein polynomials and the basic theory of complex analysis including Laurent series that _𝐶([𝑥]_𝑞𝜁)^𝑘𝑒𝑘!^{[1−𝑥]𝑞𝜁}𝑑𝜁𝜁^𝑛+1=^∞_𝑚=0𝐶𝐵_𝑘,𝑚(𝑥,𝑞)𝜁^𝑚𝑚!𝑑𝜁𝜁^𝑛+1𝐵=2𝜋𝑖(^𝑘,𝑛(𝑥,𝑞)𝑛!). (2.13) We get from (2.12) and (2.13) that𝑛!2𝜋𝑖_𝐶([𝑥]_𝑞𝜁)^𝑘𝑒𝑘!^{[1−𝑥]𝑞𝜁}𝑑𝜁𝜁^𝑛+1=𝐵_𝑘,𝑛(𝑥,𝑞),_𝐶([𝑥]_𝑞𝜁)^𝑘𝑒𝑘!^{[1−𝑥]𝑞𝜁}𝑑𝜁𝜁^𝑛+1=[𝑥]^𝑘_𝑞𝑘!^∞_𝑚=0([1−𝑥]^𝑚_𝑞𝑚!_𝐶𝜁^{𝑚−𝑛−1+𝑘}𝑑𝜁)=2𝜋𝑖([𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘𝑘!(𝑛−𝑘)!). (2.14)(2.15) We also get from (2.12) and (2.15) that𝑛!2𝜋𝑖_𝐶([𝑥]_𝑞𝜁)^𝑘𝑒𝑘!^{([1−𝑥]𝑞𝜁)}𝑑𝜁𝜁^𝑛+1=𝑛𝑘[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘. (2.16) Therefore, we see from (2.14) and (2.16) that𝐵_𝑘,𝑛𝑛𝑘(𝑥,𝑞)=[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘. (2.17) Note that (𝑛−𝑘𝑛)𝐵_𝑘,𝑛(𝑥,𝑞)+(𝑘+1𝑛)𝐵_𝑘+1,𝑛=(𝑥,𝑞)(𝑛−1)!𝑘!(𝑛−𝑘−1)![𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘+(𝑛−1)!𝑘!(𝑛−𝑘−1)![𝑥]_𝑞^𝑘+1[1−𝑥]_𝑞^{𝑛−𝑘−1}=([1−𝑥]_𝑞+[𝑥]_𝑞)𝐵_{𝑘,𝑛−1}(𝑥,𝑞)=(1+[𝑥]_𝑞(1−𝑞^1−𝑥))𝐵_{𝑘,𝑛−1}(𝑥,𝑞)=(1+(1−𝑞)[𝑥]_𝑞[1−𝑥]_𝑞)𝐵_{𝑘,𝑛−1}(𝑥,𝑞). (2.18) Therefore, we can write the modified 𝑞-Bernstein polynomials as a linear combination of polynomials of higher order as follows.

Theorem 2.4. For 𝑘,𝑛∈ℤ₊ and 𝑥∈[0,1], (𝑛+1−𝑘𝑛+1)𝐵_𝑘,𝑛+1(𝑥,𝑞)+(𝑘+1𝑛+1)𝐵_{𝑘+1,𝑛+1}(𝑥,𝑞)=(1+(1−𝑞)[𝑥]_𝑞[1−𝑥]_𝑞)𝐵_𝑘,𝑛(𝑥,𝑞). (2.19)

We easily see from (2.17) that for 𝑛,𝑘∈ℕ, (𝑛−𝑘+1𝑘)([𝑥]_𝑞[1−𝑥]_𝑞)𝐵_{𝑘−1,𝑛}(𝑥,𝑞)=(𝑛−𝑘+1𝑘)([𝑥]_𝑞[1−𝑥]_𝑞)𝑛𝑘−1[𝑥]_𝑞^𝑘−1[1−𝑥]_𝑞^{𝑛−𝑘+1}=𝑛!𝑘!(𝑛−𝑘)![𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘=𝐵_𝑘,𝑛(𝑥,𝑞). (2.20) Thus, the following corollary holds.

Corollary 2.5. For 𝑛,𝑘∈ℕ and 𝑥∈[0,1], (𝑛−𝑘+1𝑘)([𝑥]_𝑞[1−𝑥]_𝑞)𝐵_{𝑘−1,𝑛}(𝑥,𝑞)=𝐵_𝑘,𝑛(𝑥,𝑞). (2.21)

Note from the definition of the modified 𝑞-Bernstein polynomials and the binomial theorem that for 𝑘,𝑛∈ℤ₊, 𝐵_𝑘,𝑛𝑛𝑘(𝑥,𝑞)=[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘=𝑛𝑘[𝑥]^𝑘_𝑞(1−𝑞^1−𝑥[𝑥]_𝑞)^𝑛−𝑘=𝑛𝑘[𝑥]^𝑘_𝑞^𝑛−𝑘_𝑙=0𝑙𝑛−𝑘(−1)^𝑙𝑞^{𝑙(1−𝑥)}[𝑥]^𝑙_𝑞=^𝑛−𝑘_𝑙=0𝑘𝑘+𝑙𝑛𝑘+𝑙(−1)^𝑙𝑞^{𝑙(1−𝑥)}[𝑥]_𝑞^𝑙+𝑘=^𝑛_𝑗=𝑘𝑛𝑘𝑛𝑗(−1)^𝑗−𝑘𝑞^{(1−𝑥)(𝑗−𝑘)}[𝑥]^𝑗_𝑞. (2.22) Therefore, we showed that the following theorem holds.

Theorem 2.6. For k,𝑛∈ℤ₊ and 𝑥∈[0,1], 𝐵_𝑘,𝑛(𝑥,𝑞)=^𝑛_𝑗=𝑘𝑗𝑘𝑛𝑗(−1)^𝑗−𝑘𝑞^{(1−𝑥)(𝑗−𝑘)}[𝑥]^𝑗_𝑞. (2.23)

It is possible to write [𝑥]^𝑘_𝑞 as a linear combination of the modified 𝑞-Bernstein polynomials by using the degree evaluation formulae and mathematical induction. We easily see from the property of the modified 𝑞-Bernstein polynomials that ^𝑛_𝑘=1(𝑘𝑛)𝐵_𝑘,𝑛(𝑥,𝑞)=^𝑛_𝑘=1𝑛−1𝑘−1[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘=^𝑛−1_𝑘=0𝑘𝑛−1[𝑥]_𝑞^𝑘+1[1−𝑥]_𝑞^{𝑛−1−𝑘}=[𝑥]_𝑞([𝑥]_𝑞+[1−𝑥]_𝑞)^𝑛−1, (2.24) and that ^𝑛_𝑘=2(^𝑘₂)(^𝑛₂)𝐵_𝑘,𝑛(𝑥,𝑞)=^𝑛_𝑘=2𝑛−2𝑘−2[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘=^𝑛−2_𝑘=0𝑘𝑛−2[𝑥]_𝑞^𝑘+2[1−𝑥]_𝑞^{𝑛−2−𝑘}=[𝑥]²_𝑞([𝑥]_𝑞+[1−𝑥]_𝑞)^𝑛−2. (2.25) Continuing this process, we obtain ^𝑛_𝑘=𝑗(^𝑘_𝑗)(^𝑛_𝑗)𝐵_𝑘,𝑛(𝑥,𝑞)=[𝑥]^𝑗_𝑞([𝑥]_𝑞+[1−𝑥]_𝑞)^𝑛−𝑗, (2.26) for 𝑗∈ℕ. Therefore, we obtain the following theorem.

Theorem 2.7. For 𝑛,𝑗∈ℤ₊ and 𝑥∈[0,1], 1([1−𝑥]_𝑞+[𝑥]_𝑞)^{𝑛𝑛−𝑗}_𝑘=𝑗(^𝑘_𝑗)(^𝑛_𝑗)𝐵_𝑘,𝑛(𝑥,𝑞)=[𝑥]^𝑗_𝑞. (2.27)

For 𝑘∈ℕ, the Bernoulli polynomial of order 𝑘 is defined by (𝑡𝑒^𝑡)−1^𝑘𝑒^𝑥𝑡𝑡=(𝑒^𝑡𝑡−1)×⋯×(𝑒^𝑡)−1_𝑘-times𝑒^𝑥𝑡=^∞_𝑛=0𝐵_𝑛^(𝑘)𝑡(𝑥)^𝑛,𝑛! (2.28) and 𝐵_𝑛^(𝑘)=𝐵_𝑛^(𝑘)(0) are called the 𝑛th Bernoulli numbers of order 𝑘. It is well known that the second kind stirling number is defined by(𝑒^𝑡−1)^𝑘𝑘!∶=^∞_𝑛=0𝑡𝑆(𝑛,𝑘)^𝑛,𝑛! (2.29) for 𝑘∈ℕ. We note from (2.2) that([𝑥]_𝑞𝑡)^𝑘𝑒^{[1−𝑥]𝑞𝑡}=𝑘![𝑥]^𝑘_𝑞(𝑒^𝑡−1)^𝑘(𝑡𝑘!𝑒^𝑡)−1^𝑘𝑒^{[1−𝑥]𝑞𝑡}=[𝑥]^𝑘_𝑞(^∞_𝑚=0𝑡𝑆(𝑚,𝑘)^𝑚𝑚!)(^∞_𝑛=0𝐵_𝑛^(𝑘)([1−𝑥]_𝑞)𝑡^𝑛)𝑛!=[𝑥]^𝑘_𝑞^∞_𝑙=0(^𝑙_𝑛=0𝐵_𝑛^(𝑘)([1−𝑥]_𝑞)𝑆(𝑙−𝑛,𝑘)𝑙!)𝑡𝑛!(𝑙−𝑛)!^𝑙.𝑙! (2.30) We have from (2.2) and (2.30) that𝐵_𝑘,𝑙(𝑥,𝑞)=[𝑥]^𝑘_𝑞^𝑙_𝑛=0𝑙𝑛𝐵_𝑛^(𝑘)([1−𝑥]_𝑞)𝑆(𝑙−𝑛,𝑘), (2.31) and 𝐵_𝑘,0(𝑥,𝑞)=𝐵_𝑘,1(𝑥,𝑞)=⋯=𝐵_{𝑘,𝑘−1}(𝑥,𝑞)=0.

Remark 2. The Equations (2.30) and (2.31) are already known by Simsek and Acikgoz in a previous paper [5, page 7].

Let Δ be the shift difference operator defined by Δ𝑓(𝑥)=𝑓(𝑥+1)−𝑓(𝑥). We see from the iterative method thatΔ^𝑛𝑓(0)=^𝑛_𝑘=0𝑛𝑘(−1)^𝑛−𝑘𝑓(𝑘), (2.32) for 𝑛∈ℕ. We get from (2.29) and (2.32) that ^∞_𝑛=0𝑡𝑆(𝑛,𝑘)^𝑛=1𝑛!𝑘!^𝑘_𝑙=0𝑘𝑙(−1)^𝑘−𝑙𝑒^𝑙𝑡=^∞_𝑛=0{1𝑘!^𝑘_𝑙=0𝑘𝑙(−1)^𝑘−𝑙𝑙^𝑛}𝑡^𝑛=𝑛!_∞𝑛=0Δ^𝑘0^𝑛𝑡𝑘!^𝑛.𝑛! (2.33) By comparing the coefficients on both sides above, we haveΔ𝑆(𝑛,𝑘)=^𝑘0^𝑛,𝑘! (2.34) for 𝑛,𝑘∈ℤ₊. Thus, we get from (2.31) and (2.34) that 𝐵_𝑘,𝑙(𝑥,𝑞)=[𝑥]^𝑘_𝑞^𝑙_𝑛=0𝑙𝑛𝐵_𝑛^(𝑘)([1−𝑥]_𝑞)Δ^𝑘0^𝑙−𝑛.𝑘! (2.35) Let (𝐸ℎ)(𝑥)=ℎ(𝑥+1) be the shift operator. Then the 𝑞-difference operator is defined by Δ^𝑛_𝑞=Π^𝑛−1_𝑗=0(𝐸−𝑞^𝑗𝐼), (2.36) where 𝐼 is an identity operator(see [7] through [11]). For 𝑓∈𝐶[0,1] and 𝑛∈ℕ, we have Δ^𝑛_𝑞𝑓(0)=^𝑛_𝑘=0𝑛𝑘_𝑞(−1)^𝑘𝑞^(𝑛2)𝑓(𝑛−𝑘), (2.37) where (^𝑛_𝑘)_𝑞 is the Gaussian binomial coefficient defined by 𝑥𝑘_𝑞=[𝑥]_𝑞[𝑥−1]_𝑞⋯[𝑥−𝑘+1]_𝑞[𝑘]_𝑞!. (2.38)

Let 𝐹_𝑞(𝑡) be the generating function of the 𝑞-extension of the second kind stirling number as follows:𝐹_𝑞𝑞(𝑡)∶=^𝑘2)−([𝑘]_𝑞!^𝑘_𝑗=0(−1)^𝑘−𝑗𝑘𝑗_𝑞𝑞^{(2)𝑘−𝑗}𝑒^{[𝑗]𝑞𝑡}=^∞_𝑛=0𝑡𝑆(𝑛,𝑘∶𝑞)^𝑛.𝑛! (2.39) We have from (2.39) that 𝑞𝑆(𝑛,𝑘∶𝑞)=^𝑘2)−([𝑘]_𝑞!^𝑘_𝑗=0(−1)^𝑗𝑞^(𝑗2)𝑘𝑗_𝑞[𝑘−𝑗]^𝑛_𝑞=𝑞^𝑘2)−([𝑘]_𝑞!Δ^𝑘_𝑞0^𝑛, (2.40) where [𝑘]_𝑞!=[𝑘]_𝑞[𝑘−1]_𝑞⋯[2]_𝑞[1]_𝑞. It is not difficult to see that[𝑥]^𝑛_𝑞=^𝑛_𝑘=0𝑞^(𝑘2)𝑥𝑘_𝑞[𝑘]_𝑞!𝑆(𝑛,𝑘∶𝑞). (2.41) See also [7] through [11] for details and related facts for above. Then, we get from (2.41) and Theorem 2.7 that ^𝑗_𝑘=0𝑞^(𝑘2)𝑥𝑘_𝑞[𝑘]_𝑞1!𝑆(𝑗,𝑘∶𝑞)=([1−𝑥]_𝑞+[𝑥]_𝑞)^{𝑛𝑛−𝑗}_𝑘=𝑗(^𝑘_𝑗)(^𝑛_𝑗)𝐵_𝑘,𝑛(𝑥,𝑞). (2.42) Therefore, this completes the proof of the following theorem.

Theorem 2.8. For 𝑛,𝑗∈ℤ₊ and 𝑥∈[0,1], 1([1−𝑥]_𝑞+[𝑥]_𝑞)^{𝑛𝑛−𝑗}_𝑘=𝑗(^𝑘_𝑗)(^𝑛_𝑗)𝐵_𝑘,𝑛(𝑥,𝑞)=^𝑗_𝑘=0𝑞^(𝑘2)𝑥𝑘_𝑞[𝑘]_𝑞!𝑆(𝑗,𝑘∶𝑞). (2.43)

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 2010.