Some Identities on the 𝑞-Integral Representation of the Product of Several 𝑞-Bernstein-Type Polynomials

Abstract

The purpose of this paper is to give some properties of several 𝑞-Bernstein-type polynomials to express the 𝑞-integral on [0, 1] in terms of 𝑞-beta and 𝑞-gamma functions. Finally, we derive some identities on the 𝑞-integral of the product of several 𝑞-Bernstein-type polynomials.

1. Introduction

Let 𝑞∈ℝ with 0≤𝑞<1. We assume that 𝑞-number is defined by [𝑥]_𝑞=(1−𝑞^𝑥)/(1−𝑞) and [0]_𝑞=0. Note that lim_𝑞→1[𝑥]_𝑞=𝑥. The 𝑞-derivative of a map 𝑓∶ℝ→ℝ at 𝑥∈ℝ⧵{0} is given by𝐷_𝑞(𝑓)=𝑑^𝑞𝑓(𝑥)𝑑_𝑞𝑥=𝑓((𝑞𝑞𝑥−)−1)𝑓𝑥(𝑥)(1.1) (see [1–6]). For 𝑛∈ℕ, by (1.1), we get 𝐷^𝑛𝑞(𝑥^𝑛)=[𝑛]_𝑞[𝑛−1]_𝑞⋯[2]_𝑞[1]_𝑞=[𝑛]₁!. The 𝑞-binomial formula is given by(𝑎+𝑏)^𝑛𝑞=^𝑛−1_𝑖=0𝑎+𝑏𝑞^𝑖=^𝑛_𝑙=0⎜⎝⎛⎜𝑛𝑙⎞⎟⎟⎠_𝑞𝑞^𝑙2𝑎^𝑛−𝑙𝑏^𝑙(1.2) (see [2, 5, 7–11]), where (^𝑛𝑘)_𝑞=[𝑛]_𝑞!/[𝑘]_𝑞![𝑛−𝑘]_𝑞!=[𝑛]_𝑞[𝑛−1]_𝑞⋯[𝑛−𝑘+1]_𝑞/[𝑘]_𝑞!.

For 𝑎,𝑏∈ℝ, the Jackson 𝑞-integral of 𝑓∶ℝ→ℝ is defined by^𝑏𝑎𝑓(𝑥)𝑑_𝑞𝑥=(1−𝑞)^∞_𝑛=0𝑞^𝑛(𝑏𝑓(𝑏𝑞^𝑛)−𝑎𝑓(𝑎𝑞^𝑛))(1.3) (see [1, 2, 5, 6, 9, 12, 13]). From (1.2), we note that⎛⎜⎜⎝𝑛+𝑘1⎞⎟⎟⎠_𝑞=⎛⎜⎜⎝𝑘−𝑛1⎞⎟⎟⎠_𝑞+𝑞^𝑘⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞=𝑞^𝑛−𝑘⎛⎜⎜⎝𝑘−𝑛1⎞⎟⎟⎠_𝑞+⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞.(1.4) By (1.2) and (1.4), we get(1−𝑏)^𝑛𝑞=(𝑏∶𝑞)_𝑛=^𝑛−1_𝑖=01−𝑞^𝑖𝑏=^𝑛_𝑖=0⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠_𝑞1𝑞^𝑖2(−𝑏)^𝑖,(1−𝑏)^𝑛𝑞=1(𝑏∶𝑞)_𝑛=1∏^{𝑛−1𝑖=0}(1−𝑞^𝑖𝑏)=^∞_𝑖=0⎛⎜⎜⎝𝑛+𝑖𝑖−1⎞⎟⎟⎠_𝑞𝑏^𝑖.(1.5) 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 operators (see [1, 4, 9, 11, 14]):𝔹_𝑛(𝑓∣𝑥)=^𝑛_𝑘=0𝑓𝑘𝑛⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝑥^𝑘(1−𝑥)^𝑛−𝑘=^𝑛_𝑘=0𝑓𝑘𝑛𝐵^𝑘,𝑛(𝑥).(1.6) Here 𝔹_𝑛(𝑓∣𝑥) is called Bernstein operator of order 𝑛 for 𝑓. For 𝑘,𝑛∈ℤ₊(=ℕ∪{0}), the Bernstein polynomials of degree 𝑛 are defined by𝐵_𝑘,𝑛(𝑥)=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝑥^𝑘(1−𝑥)^𝑛−𝑘(1.7) (see [1, 3, 4, 11–14]). By the definition of Bernstein polynomials (see (1.6) and (1.7)), we can see that Bernstein basis is the probability mass function of binomial distribution. A Bernoulli trial involves performing an experiment once and noting whether a particular event 𝐴 occurs. The outcome of Bernoulli trial is said to be “success” if 𝐴 occurs and a “failure” otherwise. Let 𝑘 be the number of successes in 𝑛 independent Bernoulli trials, the probabilities of 𝑘 are given by the binomial probability law:𝑝_𝑛(𝑘)=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝑝^𝑘(1−𝑝)^𝑛−𝑘,for𝑘=0,1,…,𝑛,(1.8) where 𝑝_𝑛(𝑘) is the probability of 𝑘 successes in 𝑛 trials. For example, a communication system transmits binary information over channel that introduces random bit errors with probability 𝜉=10⁻³. The transmitter transmits each information bit three times, an a decoder takes a majority vote of the received bits to decide on what the transmitted bit was. The receiver can correct a single error, but it will make the wrong decision if the channel introduces two or more errors. If we view each transmission as a Bernoulli trial in which a “success” corresponds to the introduction of an error, then the probability of two or more errors in three Bernoulli trials is𝑝(𝑘≥2)=⎛⎜⎜⎝32⎞⎟⎟⎠(0.001)²(0.999)+⎛⎜⎜⎝33⎞⎟⎟⎠(0.001)³≈310⁻⁶,(1.9) see [9]. Based on the 𝑞-integers Phillips introduced the 𝑞-analogue of well-known Bernstein polynomials (see [4, 5, 9, 11, 15]). For 𝑓∈𝐶([0,1]), Phillips introduced the 𝑞-extension of (1.6) as follows:𝔹_𝑛,𝑞(𝑓∣𝑥)=^𝑛_𝑛=0𝑓[𝑘]^𝑞[𝑛]_𝑞⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞(1−𝑥)^{𝑞𝑛−𝑘}=^𝑛_𝑛=0𝑓[𝑘]^𝑞[𝑛]_𝑞𝐵^𝑘,𝑛(𝑥,𝑞),for𝑘,𝑛∈ℤ₊(1.10) (see [4, 5, 9, 11, 15]). Here 𝔹_𝑛,𝑞(𝑓∣𝑥) is called the 𝑞-Bernstein operator of order 𝑛 for 𝑓. For 𝑘,𝑛∈ℤ₊, the 𝑞-Bernstein polynomial of degree 𝑛 is defined by𝐵_𝑘,𝑛(𝑥,𝑞)=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞𝑥^𝑘(1−𝑥)^{𝑞𝑛−𝑘},where𝑥∈[0,1].(1.11) Note that (1.11) is the 𝑞-extension of (1.7). That is, lim_𝑞→1𝐵_𝑘,𝑛(𝑥,𝑞)=𝐵_𝑘,𝑛(𝑥). For example, 𝐵_0,1(𝑥,𝑞)=1−𝑥, 𝐵_1,1(𝑥,𝑞)=𝑥, and 𝐵_0,2(𝑥,𝑞)=1−[2]_𝑞𝑥+𝑞𝑥²,…. Also 𝐵_𝑘,𝑛(𝑥,𝑞)=0 for 𝑘>𝑛, because (^𝑛𝑘)_𝑞=0. For 𝑛,𝑘∈ℤ₊, its probabilities are given by𝑝(𝑥=𝑘)=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞𝑥^𝑘(1−𝑥)^{𝑞𝑛−𝑘},where𝑥∈[0,1].(1.12) This distributions are studied by several authors and they have applications in physics as well as in approximation theory due to the 𝑞-Bernstein polynomials and the 𝑞-Bernstein operators (see [1–16]). By the definition of the 𝑞-Bernstein polynomials, we easily see that the 𝑞-Bernstein basis is the probability mass function of 𝑞-binomial distribution. In this paper we use the two 𝑞-analogues of exponential function as follows:𝐸_𝑞(𝑥)=((1−𝑞)𝑥∶𝑞)_∞=(1+(1−𝑞)𝑥)^∞𝑞=^∞_𝑛=0𝑞^𝑛2𝑥^𝑛[𝑛]_𝑞!,(1.13)𝑒_𝑞(𝑥)=1((1−𝑞)𝑥∶𝑞)_∞=1(1+(1−𝑞)𝑥)^∞𝑞=^∞_𝑛=0𝑥^𝑛[𝑛]_𝑞!,(1.14) (see [2–4, 6, 10]). From (1.3), the improper 𝑞-integral is given by^0∞/𝐴𝑓(𝑥)𝑑_𝑞𝑥=(1−𝑞)_𝑛∈ℤ𝑞^𝑛𝐴𝑓𝑞𝐴^𝑛(1.15) (see [6]), where the improper 𝑞-integral depends on 𝐴. The purpose of this paper is to give some properties of several 𝑞-Bernstein type polynomials to express the 𝑞-integral on [0,1] in terms of 𝑞-beta and 𝑞-gamma functions. Finally, we derive some identities on the 𝑞-integral of the product of several 𝑞-Bernstein type polynomials.

2. 𝑞-Integral Representation of 𝑞-Bernstein Polynomials

The gamma and beta functions are defined as the following definite integrals (𝛼>0,𝛽>0):Γ(𝛼)=^∞0𝑒^−𝑡𝑡^𝛼−1𝑑𝑡,(2.1) (see [1–11, 14–16])𝐵(𝛼,𝛽)=¹⁰𝑡^𝛼−1(1−𝑡)^𝛽−1𝑑𝑡=^∞0𝑡^𝛼−1(1+𝑡)^𝛼+𝛽𝑑𝑡.(2.2) From (2.1) and (2.2), we can derive the following equations:Γ(𝛼+1)=𝛼Γ(𝛼),𝐵(𝛼,𝛽)=Γ(𝛼)Γ(𝛽)Γ(𝛼+𝛽).(2.3) As the 𝑞-extensions of (2.1) and (2.2), the 𝑞-gamma and 𝑞-beta functions are defined as the following 𝑞-integrals (𝛼>0,𝛽>0):Γ_𝑞(𝛼)=^{01/(1−𝑞)}𝑥^𝛼−1𝐸_𝑞(−𝑞𝑥)𝑑_𝑞𝑥(2.4) (see [2–6, 10]),𝐵_𝑞(𝛼,𝛽)=¹⁰𝑥^𝛼−1(1−𝑞𝑥)^𝑞𝛽−1𝑑_𝑞𝑥(2.5) (see [2, 4, 6, 10]).

By (2.4) and (2.5), we obtain the following lemma.

Lemma 2.1 (see [2, 6]). (a)  Γ_𝑞 can be equivalently expressed as Γ_𝑞(𝛼)=(1−𝑞)^𝑞𝛼−1(1−𝑞)^𝛼−1,where𝛼>0.(2.6) In particular, one has Γ_𝑞(𝛼+1)=[𝛼]_𝑞Γ_𝑞(𝛼),for𝛼>0,Γ_𝑞(1)=1.(2.7)(b) The 𝑞-gamma and 𝑞-beta functions are related to each other by the following two equations: Γ_𝑞(𝛼)=𝐵^𝑞(𝛼,∞)(1−𝑞)^𝛼,𝐵_𝑞(𝛼,𝛽)=Γ^𝑞(𝛼)Γ_𝑞(𝛽)Γ_𝑞(𝛼+𝛽),where𝛼>0,𝛽>0.(2.8)

Now one takes the 𝑞-integral for one 𝑞-Bernstein polynomial as follows: for 𝑛,𝑘∈ℤ₊, 𝑞^−𝑘¹⁰𝐵_𝑘,𝑛(𝑞𝑥,𝑞)𝑑_𝑞𝑥=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞¹⁰=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_{𝑞𝑛−𝑘}𝑥^𝑘(1−𝑞𝑥)^{𝑞𝑛−𝑘}𝑑_𝑞𝑥_𝑙=0⎛⎜⎜⎝𝑛−𝑙𝑘⎞⎟⎟⎠_𝑞(−1)^𝑙𝑞^{2𝑙+1}¹⁰=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_{𝑞𝑛−𝑘}𝑥^𝑙+𝑘𝑑_𝑞𝑥_𝑙=0⎛⎜⎜⎝𝑛−𝑙𝑘⎞⎟⎟⎠_𝑞(−1)^{𝑛−𝑘−𝑙}𝑞^{2𝑛−𝑘−𝑙+1}1[𝑛−𝑙+1]_𝑞.(2.9) Therefore, by (2.9), one obtains the following proposition.

Proposition 2.2. For 𝑛,𝑘∈ℤ₊, one has ¹⁰𝐵_𝑘,𝑛(𝑞𝑥,𝑞)𝑑_𝑞𝑥=𝑞^𝑘⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_{𝑞𝑛−𝑘}_l=0⎛⎜⎜⎝𝑛−𝑙𝑘⎞⎟⎟⎠_𝑞(−1)^{𝑛−𝑘−𝑙}𝑞^{2𝑛−𝑘−𝑙+1}1[𝑛−𝑙+1]_𝑞.(2.10)

The Proposition 2.2 is closely related to the 𝑞-beta function which is given by𝐵_𝑞(𝑛,𝑚)=¹⁰𝑥^𝑛−1(1−𝑞𝑥)^𝑞𝑚−1𝑑_𝑞𝑥,(2.11)Γ_𝑞(𝑚)=^{01/(1−𝑞)}𝑥^𝑛−1𝐸_𝑞(−𝑞𝑥)𝑑_𝑞𝑥,(2.12) (see (2.5)). From Lemma 2.1, one has𝐵_𝑞(𝑛,𝑚)=Γ^𝑞(𝑚)Γ_𝑞(𝑛)Γ_𝑞(𝑛+𝑚),where𝑚,𝑛∈ℕ.(2.13) By (2.9) and (2.13), one gets𝑞^−𝑘¹⁰𝐵_𝑘,𝑛(𝑞𝑥,𝑞)𝑑_𝑞𝑥==⎛⎜⎜⎝⎛⎜⎜⎝𝑘𝑘𝑛𝑛⎞⎟⎟⎠⎞⎟⎟⎠_𝑞𝐵_𝑞(𝑘+1,𝑛−𝑘+1)_𝑞Γ_𝑞(𝑘+1)Γ_𝑞(𝑛−𝑘+1)Γ_𝑞(𝑛+2),where𝑘>−1,𝑛>𝑘−1.(2.14) Therefore, by (2.14), one obtains the following theorem.

Theorem 2.3. For 𝑛,𝑘∈ℤ₊ with 𝑘>−1 and 𝑛>𝑘−1, one has ¹⁰𝐵_𝑘,𝑛(𝑞𝑥,𝑞)𝑑_𝑞𝑥=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞[𝑘]_𝑞[𝑛−𝑘]_𝑞(𝑞−1)[𝑘]_𝑞+1Γ^𝑞(𝑘)Γ_𝑞(𝑛−𝑘)Γ_𝑞(𝑛+2).(2.15)

By comparing the coefficients on the both sides of Proposition 2.2 and Theorem 2.3, one obtains the following corollary.

Corollary 2.4. For 𝑛,𝑘∈ℤ₊ with 𝑘>−1 and 𝑛>𝑘−1, one has ^𝑛−𝑘_𝑙=0⎛⎜⎜⎝𝑛−𝑙𝑘⎞⎟⎟⎠_𝑞(−1)^{𝑛−𝑘−𝑙}𝑞^{2𝑛−𝑘−𝑙+1}[𝑛−𝑙+1]_𝑞=Γ^𝑞(𝑘+1)Γ_𝑞(𝑛−𝑘+1)Γ_𝑞(𝑛+2).(2.16)

According to this result one can say that the 𝑞-integral of 𝑞-Bernstein polynomials from 0 to 1 is symmetric. Now one considers the 𝑞-integral for the multiplication of two 𝑞-Bernstein polynomials which is given by the following relation:∫¹⁰𝐵_𝑘,𝑛(𝑞𝑥,𝑞)𝐵_{𝑘,𝑚(𝑞𝑛−𝑘+1𝑥,𝑞)𝑑𝑞𝑥}𝑞^{𝑛𝑘−𝑘2+2𝑘}=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠_𝑞¹⁰=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞𝑥²𝑘(1−𝑞𝑥)^{𝑞𝑛+𝑚−2𝑘}𝑑_q𝑥⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠_𝑞¹⁰𝑢^{𝑛+𝑚−2𝑘}(1−𝑞𝑢)^𝑞2𝑘𝑑_𝑞𝑢.(2.17) For 𝑛,𝑘,𝑚∈ℤ₊, one can derive the following equation (2.20) from (2.17):∫¹⁰𝐵_𝑘,𝑛(𝑞𝑥,𝑞)𝐵_𝑘,𝑚𝑞^{𝑛−𝑘+1}𝑥,𝑞𝑑_𝑞𝑥𝑞^{𝑛𝑘−𝑘2+2𝑘}=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠_𝑞2𝑘_𝑙=0^𝑙2𝑘_𝑞(−1)^𝑙𝑞^{2𝑙+1}[𝑛+𝑚+𝑙−2𝑘+1]_𝑞=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠_𝑞2𝑘_𝑙=0^𝑙2𝑘_𝑞(−1)^2𝑘−𝑙𝑞^{22𝑘−𝑙+1}[𝑛+𝑚−𝑙+1]_𝑞.(2.18) Therefore, one obtains the following theorem.

Theorem 2.5. For 𝑚,𝑛,𝑘∈ℤ₊, one has ¹⁰𝐵_𝑘,𝑛(𝑞𝑥,𝑞)𝐵_𝑘,𝑚𝑞^{𝑛−𝑘+1}𝑥,𝑞𝑑_𝑞𝑥=𝑞^{𝑛𝑘−𝑘2+2𝑘}⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠_𝑞2𝑘_𝑙=0^𝑙2𝑘_𝑞(−1)^2𝑘−𝑙𝑞^{22𝑘−𝑙+1}[𝑛+𝑚−𝑙+1]_𝑞.(2.19)

For 𝑚,𝑛,𝑘∈ℤ₊, by (2.5) and (2.9), one gets∫¹⁰𝐵_𝑘,𝑛(𝑞𝑥,𝑞)𝐵_𝑘,𝑚𝑞^{𝑛−𝑘+1}𝑥,𝑞𝑑_𝑞𝑥𝑞^{𝑛𝑘−𝑘2+2𝑘}=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠_𝑞𝐵_𝑞(𝑛+𝑚−2𝑘+1,2𝑘+1).(2.20) Therefore, by Theorem 2.5 and (2.20), one obtains the following corollary.

Corollary 2.6. For 𝑘>−1 and 𝑛+𝑚−2𝑘>−1, one has ^2𝑘_𝑙=0^𝑙2𝑘_𝑞(−1)^2𝑘−𝑙𝑞^{22𝑘−𝑙+1}[𝑛+𝑚−𝑙+1]_𝑞=Γ^𝑞(𝑛+𝑚−2𝑘+1)Γ_𝑞(2𝑘+1)Γ_𝑞(𝑛+𝑚+2).(2.21)

By the same method, the multiplication of three 𝑞-Bernstein polynomials is given by the following relation: for 𝑘,𝑛,𝑚,𝑠∈ℤ₊,∫¹⁰𝐵_𝑘,𝑛(𝑞𝑥,𝑞)𝐵_𝑘,𝑚𝑞^{𝑛−𝑘+1}𝑥,𝑞𝐵_𝑘,𝑠𝑞^{𝑛+𝑚−2𝑘+1}𝑥,𝑞𝑑_𝑞𝑥𝑞^{3𝑘+2𝑛𝑘−3𝑘2+𝑚𝑘}=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑘𝑠⎞⎟⎟⎠_𝑞¹⁰=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞𝑥^3𝑘(1−𝑞𝑥)^{𝑞𝑛+𝑚+𝑠−3𝑘}𝑑_𝑞𝑥⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑘𝑠⎞⎟⎟⎠_𝑞¹⁰=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞𝑢^{𝑛+𝑚+𝑠−3𝑘}(1−𝑞𝑢)^𝑞3𝑘𝑑_𝑞𝑢⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑘𝑠⎞⎟⎟⎠_𝑞3𝑘_𝑙=0⎛⎜⎜⎝3𝑙𝑘⎞⎟⎟⎠_𝑞𝑞^{2𝑙+1}(−1)^𝑙¹⁰=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞𝑢^{𝑛+𝑚+𝑠−3𝑘+𝑙}𝑑_𝑞𝑢⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑘𝑠⎞⎟⎟⎠_𝑞3𝑘_𝑙=0⎛⎜⎜⎝3𝑙𝑘⎞⎟⎟⎠_𝑞𝑞^{23𝑘−𝑙+1}(−1)^𝑙+3𝑘1[𝑛+𝑚+𝑠−𝑙+1]_𝑞.(2.22) Therefore, by (2.22), one obtains the following theorem.

Theorem 2.7. For 𝑛,𝑚,𝑠,𝑘∈ℤ₊, one has ¹⁰𝐵=_𝑘,𝑛𝑞(𝑞𝑥,𝑞)𝐵_𝑘,𝑚_{3𝑘+2𝑛𝑘−3𝑘2+𝑚𝑘}𝑞^{𝑛−𝑘+1}𝑥,𝑞𝐵_𝑘,𝑠𝑞^{𝑛+𝑚−2𝑘+1}𝑥,𝑞𝑑_𝑞𝑥⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑘𝑠⎞⎟⎟⎠_𝑞3𝑘_𝑙=0⎛⎜⎜⎝3𝑙𝑘⎞⎟⎟⎠_𝑞𝑞^{23𝑘−𝑙+1}(−1)^𝑙+3𝑘[𝑛+𝑚+𝑠−𝑙+1]_𝑞.(2.23)

From (2.5) and (2.22), one has∫¹⁰𝐵_𝑘,𝑛(𝑞𝑥,𝑞)𝐵_𝑘,𝑚𝑞^{𝑛−𝑘+1}𝑥,𝑞𝐵_𝑘,𝑠𝑞^{𝑛+𝑚−2𝑘+1}𝑥,𝑞𝑑_𝑞𝑥𝑞^{3𝑘+2𝑛𝑘−3𝑘2+𝑚𝑘}=⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑘𝑠⎞⎟⎟⎠_𝑞𝐵_𝑞(𝑛+𝑚+𝑠−3𝑘+1,3𝑘+1).(2.24) Therefore, by Theorem 2.7 and (2.24), one obtains the following corollary.

Corollary 2.8. For 𝑘>−1/3 and 𝑛+𝑚+𝑠−3𝑘>−1, one has ^3𝑘_𝑘=0⎛⎜⎜⎝3𝑙𝑘⎞⎟⎟⎠_𝑞(−1)^𝑙+3𝑘𝑞^{23𝑘−𝑙+1}[𝑛+𝑚+𝑠−𝑙+1]_𝑞=Γ^𝑞(𝑛+𝑚+𝑠−3𝑘+1)Γ_𝑞(3𝑘+1)Γ_𝑞(𝑛+𝑚+𝑠+2).(2.25)

For 𝑠∈ℕ, let 𝑛₁,𝑛₂,…,𝑛_𝑠,𝑘∈ℤ₊. Then one has∫¹⁰𝐵_𝑘,𝑛1(𝑞𝑥,𝑞)∏^{𝑠−1𝑖=1}𝐵_{𝑘,𝑛𝑖+1}𝑞^{∑𝑖𝑙=1𝑛𝑙−𝑖𝑘+1}𝑥,𝑞𝑑_𝑞𝑥𝑞^{∑𝑠𝑘+𝑘𝑠−1𝑖=1𝑖𝑛𝑠−𝑖−𝑘2𝑠2}=⎛⎜⎜⎝𝑛¹𝑘⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑛₂𝑘⎞⎟⎟⎠_𝑞⋯⎛⎜⎜⎝𝑛^𝑠𝑘⎞⎟⎟⎠_𝑞¹⁰=⎛⎜⎜⎝𝑛₁𝑥^𝑠𝑘(1−𝑞𝑥)^{𝑛1+⋯+𝑛𝑠𝑞−𝑠𝑘}𝑑_𝑞𝑥𝑘⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑛₂𝑘⎞⎟⎟⎠_𝑞⋯⎛⎜⎜⎝𝑛^𝑠𝑘⎞⎟⎟⎠_𝑞𝑠𝑘_𝑙=0⎛⎜⎜⎝𝑠𝑙𝑘⎞⎟⎟⎠_𝑞(−1)^𝑙𝑞^{2𝑙+1}¹⁰=⎛⎜⎜⎝𝑛₁𝑥^{𝑛1+⋯+𝑛𝑠−𝑠𝑘+𝑙}𝑑_𝑞𝑥𝑘⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑛₂𝑘⎞⎟⎟⎠_𝑞⋯⎛⎜⎜⎝𝑛^𝑠𝑘⎞⎟⎟⎠_𝑞𝑠𝑘_𝑙=0⎛⎜⎜⎝𝑠𝑙𝑘⎞⎟⎟⎠_𝑞(−1)^{𝑙+𝑠𝑘}𝑞^{2𝑠𝑘−𝑙+1}𝑛₁+⋯+𝑛_𝑠−𝑙+1_𝑞.(2.26) Therefore, by (2.26), one obtains the following theorem.

Theorem 2.9. For 𝑠∈ℕ, let 𝑛₁,𝑛₂,…,𝑛_𝑠,𝑘∈ℤ₊. Then one has ¹⁰𝐵_𝑘,𝑛1(𝑞𝑥,𝑞)^𝑠−1_𝑖=1=𝑞𝐵_{𝑘,𝑛𝑖+1∑𝑠𝑘+𝑘𝑠−1𝑖=1𝑖𝑛𝑠−𝑖−𝑘2𝑠2}𝑞^{∑𝑖𝑙=1𝑛𝑙−𝑖𝑘+1}𝑥,𝑞𝑑_𝑞𝑥⎛⎜⎜⎝𝑛₁𝑘⎞⎟⎟⎠_𝑞⋯⎛⎜⎜⎝𝑛^𝑠𝑘⎞⎟⎟⎠_𝑞𝑠𝑘_𝑙=0^𝑙𝑠𝑘_𝑞(−1)^{𝑙+𝑠𝑘}𝑞^{2𝑠𝑘−𝑙+1}𝑛₁+⋯+𝑛_𝑠−𝑙+1_𝑞.(2.27)

By (2.5) and (2.26), we get∫¹⁰𝐵_𝑘,𝑛1(𝑞𝑥,𝑞)∏^{𝑠−1𝑖=1}𝐵_{𝑘,𝑛𝑖+1}𝑞^{∑𝑖𝑙=1𝑛𝑙−𝑖𝑘+1}𝑥,𝑞𝑑_𝑞𝑥𝑞^{∑𝑠𝑘+𝑘𝑠−1𝑖=1𝑖𝑛𝑠−𝑖−𝑘2𝑠2}=⎛⎜⎜⎝𝑛¹𝑘⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑛₂𝑘⎞⎟⎟⎠_𝑞⋯⎛⎜⎜⎝𝑛^𝑠𝑘⎞⎟⎟⎠_𝑞=⎛⎜⎜⎝𝑛𝐵_𝑞𝑠𝑘+1,𝑛₁+⋯+𝑛_𝑠−𝑠𝑘+1₁𝑘⎞⎟⎟⎠_𝑞⎛⎜⎜⎝𝑛₂𝑘⎞⎟⎟⎠_𝑞⋯⎛⎜⎜⎝𝑛^𝑠𝑘⎞⎟⎟⎠_𝑞Γ_𝑞(𝑠𝑘+1)Γ_𝑞𝑛₁+⋯+𝑛_𝑠−𝑠𝑘+1Γ_𝑞𝑛₁+⋯+𝑛_𝑠+2.(2.28) By comparing the coefficients on the both sides of Theorem 2.9 and (2.28), one obtains the following corollary.

Corollary 2.10. For 𝑠∈ℕ, let 𝑘>−1/𝑠 and 𝑛₁+⋯+𝑛_𝑠−𝑠𝑘>−1. Then one has ^𝑠𝑘_𝑙=0^𝑙𝑠𝑘_𝑞(−1)^{𝑙+𝑠𝑘}𝑞^{2𝑠𝑘−𝑙+1}𝑛₁+⋯+𝑛_𝑠−𝑙+1_𝑞=Γ^𝑞(𝑠𝑘+1)Γ_𝑞𝑛₁+⋯+𝑛_𝑠−𝑠𝑘+1Γ_𝑞𝑛₁+⋯+𝑛_𝑠+2.(2.29)

For 𝑛∈ℤ₊, one gets∫¹⁰𝐵_0,𝑛(𝑞𝑥,𝑞)∏^𝑛𝑙=1𝐵_𝑙,𝑛𝑞^{𝑙2𝑛𝑙−+1}𝑥,𝑞𝑑_𝑞𝑥𝑞^{∑𝑛𝑙=1𝑙2(𝑛𝑙−+1)𝑙}=⎛⎜⎜⎜⎝^𝑛_𝑖=0⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠_𝑞⎞⎟⎟⎟⎠₁₀=⎛⎜⎜⎜⎝_𝑛𝑥^{2𝑛+1}(1−𝑞𝑥)^{2𝑞𝑛+1}𝑑_𝑞𝑥_𝑖=0=⎛⎜⎜⎝Γ𝑛𝑖⎞⎟⎟⎠_𝑞⎞⎟⎟⎟⎠𝐵_𝑞⎛⎜⎜⎝⎛⎜⎜⎝𝑛+21⎞⎟⎟⎠+1,⎛⎜⎜⎝𝑛+21⎞⎟⎟⎠+1_𝑞⎞⎟⎟⎠(𝑛+1)^𝑛+1∏^𝑛𝑖=1Γ_𝑞(𝑖+1)²Γ^𝑞(𝑛(𝑛+1)/2+1)²Γ_𝑞(𝑛(𝑛+1)+2).(2.30) Therefore, by (2.30), one obtains the following theorem.

Theorem 2.11. For 𝑛∈ℤ₊, one has ¹⁰𝐵_0,𝑛(𝑞𝑥,𝑞)^𝑛_𝑙=1𝐵_𝑙,𝑛𝑞^{𝑙2𝑛𝑙−+1}=𝑞_{∑𝑛𝑙=1𝑙2(𝑛𝑙−+1)𝑙}𝑥,𝑞𝑑_𝑞𝑥Γ^𝑞(𝑛+1)^𝑛+1∏^𝑛𝑖=1Γ_𝑞(𝑖+1)²Γ^𝑞(𝑛(𝑛+1)/2+1)²Γ_𝑞(𝑛(𝑛+1)+2).(2.31)

From (2.30), one can also derive the following equation:∫¹⁰𝐵_0,𝑛(𝑞𝑥,𝑞)∏^𝑛𝑙=1𝐵_𝑙,𝑛𝑞^{𝑙2𝑛𝑙−+1}𝑥,𝑞𝑑_𝑞𝑥𝑞^{∑𝑛𝑙=1𝑙2(𝑛𝑙−+1)𝑙}=⎛⎜⎜⎜⎝^𝑛_𝑖=0⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠_𝑞⎞⎟⎟⎟⎠₍^2𝑛+1₎_𝑙=0⎛⎜⎜⎜⎜⎝⎛⎜⎜⎝𝑛+2𝑙1⎞⎟⎟⎠⎞⎟⎟⎟⎟⎠_𝑞(−1)^𝑙𝑞^{2𝑙+1}¹⁰=⎛⎜⎜⎜⎝_𝑛𝑥^{2𝑛+1+𝑙}𝑑_𝑞𝑥_𝑖=0⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠_𝑞⎞⎟⎟⎟⎠₍^2𝑛+1₎_𝑙=0⎛⎜⎜⎜⎜⎝⎛⎜⎜⎝𝑛+2𝑙1⎞⎟⎟⎠⎞⎟⎟⎟⎟⎠_𝑞(−1)^𝑙𝑞^{2𝑙+1}1[𝑛(𝑛+1)/2+𝑙+1]_𝑞.(2.32) By comparing the coefficients on the both sides of Theorem 2.11 and (2.30), one can see that^𝑛(𝑛^+1)/2_𝑙=0^{𝑙𝑛(𝑛+1)/2}_𝑞(−1)^𝑙𝑞^{2𝑙+1}[𝑛(𝑛+1)/2+𝑙+1]_𝑞=𝐵_𝑞𝑛(𝑛2+1)+1,𝑛(𝑛2+1)+1.(2.33) Therefore, by (2.33), one obtains the following corollary.

Corollary 2.12. For 𝑛∈ℤ₊, one has ^𝑛(𝑛^+1)/2_𝑙=0^{𝑙𝑛(𝑛+1)/2}_𝑞(−1)^𝑙𝑞^{2𝑙+1}[𝑛(𝑛+1)/2+𝑙+1]_𝑞=Γ^𝑞(𝑛(𝑛+1)/2+1)²Γ_𝑞(𝑛(𝑛+1)+2).(2.34)