A de Casteljau Algorithm for 𝑞-Bernstein-Stancu Polynomials

Abstract

This paper is concerned with a generalization of the 𝑞-Bernstein polynomials and Stancu operators, where the function is evaluated at intervals which are in geometric progression. It is shown that these polynomials can be generated by a de Casteljau algorithm, which is a generalization of that relating to the classical case and 𝑞-Bernstein case.

1. Introduction

Let 𝑞>0. For any fixed real number 𝑞>0 and for 𝑛∈𝑍={0,±1,±2,…}, the 𝑞-integers of the number [𝑛] are defined by[𝑛]=(1−𝑞^𝑛),(1−𝑞)for𝑞≠1,[𝑛]=𝑛,for𝑞=1. (1.1) The 𝑞-factorial [𝑛]!, for 𝑛∈𝑁₀={0,1,2,…}, is defined by[𝑛]!=[1][2]⋯[𝑛](𝑛=1,2,…),[0]!=1. (1.2) For the integers 𝑛, 𝑘, (𝑛≥𝑘≥0), the 𝑞-binomial or the Gaussian coefficients are defined by (see [1, page 12])𝑛𝑘=[𝑛]!.[𝑘]![𝑛−𝑘]! (1.3) For 𝑓∈𝐶[0;1], 𝑞>0, 𝛼≥0 and each positive integer 𝑛, we introduce (see [2]) the following generalized 𝑞-Bernstein operators:𝐵_𝑛^𝑞,𝛼(𝑓;𝑥)=^𝑛_𝑘=0𝑝^𝑞,𝛼_𝑛,𝑘(𝑥)𝑓([𝑘][𝑛]), (1.4) where𝑝^𝑞,𝛼_𝑛,𝑘𝑛𝑘∏(𝑥)=^{𝑘−1𝑖=0}(𝑥+𝛼[𝑖])∏^{𝑛−1−𝑘}_𝑠=0(1−𝑞^𝑠𝑥+𝛼[𝑠])∏^𝑛−1_𝑖=0.(1+𝛼[𝑖]) (1.5) Note, that an empty product in (1.5) denotes 1. In the case where 𝛼=0, 𝐵_𝑛^𝑞,𝛼(𝑓;𝑥) reduces to the well-known 𝑞-Bernstein polynomials introduced by Phillips [3, 4] in 1997𝐵_𝑛,𝑞(𝑓;𝑥)=^𝑛_𝑘=0𝑛𝑘𝑥^{𝑘𝑛−𝑘−1}_𝑖=0(1−𝑞^𝑖𝑥)𝑓([𝑘][𝑛]). (1.6) In the case where 𝑞=1, 𝐵_𝑛^𝑞,𝛼(𝑓;𝑥) reduces to Bernstein-Stancu polynomials, introduced by Stancu [5] in 1968𝑆_𝑛(𝑓;𝑥)=^𝑛_𝑘=0𝑛𝑘∏^𝑘−1_𝑖=0(𝑥+𝛼𝑖)∏^{𝑛−𝑘−1}_𝑠=0(1−𝑥+𝑠𝛼)∏^𝑛−1_𝑖=0𝑘(1+𝑖𝛼)𝑓(𝑛). (1.7) When 𝑞=1 and 𝛼=0, we obtain the classical Bernstein polynomial defined by𝐵_𝑛(𝑓;𝑥)=^𝑛_𝑘=0𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘𝑘𝑓(𝑛). (1.8) Basic facts on Bernstein polynomials, their generalizations, and applications can be found for example in [6–8]. In recent years, the 𝑞-Bernstein polynomials have attracted much interest, and a great number of interesting results related to the 𝐵_𝑛,𝑞(𝑓) polynomials have been obtained (see [3, 4, 9–12]). Some approximation properties of the Stancu operators are presented in [5, 13–15].

Let Δ⁰_𝑞𝑓_𝑗=𝑓_𝑗, for 𝑗=0,1,…,𝑛, and recursively,Δ_𝑞^𝑘+1𝑓_𝑗=Δ^𝑘_𝑞𝑓_𝑗+1−𝑞^𝑘Δ^𝑘_𝑞𝑓_𝑗, (1.9) for 𝑘=0,1,…,𝑛−𝑗−1 and 𝑓_𝑗=𝑓([𝑗]/[𝑛]). It is easily established by induction that 𝑞-differences satisfy the relationΔ^𝑘_𝑞𝑓_𝑗=^𝑘_𝑖=0(−1)^𝑘𝑞^{𝑖(𝑖−1)/2}𝑘𝑖𝑓_{𝑗+𝑘−𝑖}. (1.10) In [2], we prove that the operators 𝐵_𝑛^𝑞,𝛼(𝑓;𝑥) defined by (1.4) can be expressed in terms of 𝑞-differences𝐵_𝑛^𝑞,𝛼𝑓(𝑥)=^𝑛_𝑘=0𝑛𝑘Δ^𝑘_𝑞𝑓₀^𝑘−1_𝑖=0𝑥+𝛼[𝑖],1+𝛼[𝑖] (1.11) which generalized the well-known result [3, 4] for the 𝑞-Bernstein polynomial. In this paper, we show that polynomials defined by (1.4) can be generated by a de Castljau algorithm, which is a generalization of that relating to the classical case [16] and 𝑞-Bernstein case [4, 11].

2. Auxiliary Results

We note that 𝐵_𝑛^𝑞,𝛼(𝑓;𝑥) defined by (1.4), is a monotone linear operator for any 0<𝑞≤1 and 𝛼≥0. These operators reproduces linear functions [2], that is,𝐵_𝑛^𝑞,𝛼(𝑎𝑥+𝑏;𝑥)=𝑎𝑥+𝑏,𝑎,𝑏∈𝑅. (2.1) They also satisfy the end point interpolation conditions 𝐵_𝑛^𝑞,𝛼(𝑓;0)=𝑓(0) and 𝐵_𝑛^𝑞,𝛼(𝑓;1)=𝑓(1). These properties are significant in designing curves and surfaces.

Moreover, the following holds.

Lemma 2.1. Let 0<𝑞≤1, 𝛼≥0. Then, ^𝑚−1_𝑢=0(𝑞^𝑟−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟]))=^𝑚_𝑠=0(−1)^𝑠𝑞^{𝑠(𝑠−1)/2+(𝑚−𝑠)𝑟}𝑚𝑠^{𝑠−1𝑖=0}(𝑥+𝛼[𝑖])^{𝑚−𝑟−1}_{𝑗=𝑠−𝑟}(1+𝛼[𝑗]), (2.2) for all 𝑚∈𝑁, 𝑟∈𝑁₀=𝑁∪{0} and 𝑥∈[0;1].

Proof. We use induction on 𝑚. First, we see from equality [−𝑟]=−𝑞^−𝑟[𝑟], (𝑟∈𝑁), that (2.2) is evident for 𝑚=1. Let us assume that (2.2) holds for a given 𝑚∈𝑁. Then, using (2.2), we obtain ^𝑚_𝑢=0(𝑞^𝑟−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟]))=(𝑞^𝑟−𝑞^𝑚𝑥+𝛼([𝑚]−[𝑟]))^𝑚_𝑠=0(−1)^𝑠𝑞^{𝑠(𝑠−1)/2+(𝑚−𝑠)𝑟}𝑚𝑠⋅^𝑠−1_𝑖=0(𝑥+𝛼[𝑖])^{𝑚−𝑟−1}_{𝑗=𝑠−𝑟}=(1+𝛼[𝑗])_𝑚𝑠=0(−1)^𝑠𝑞^{𝑠(𝑠−1)/2+(𝑚−𝑠)𝑟}(𝑞^𝑟+𝛼[𝑚]−𝛼[𝑟]+𝛼𝑞^𝑚𝑚𝑠⋅[𝑠])_{𝑠−1𝑖=0}(𝑥+𝛼[𝑖])^{𝑚−𝑟−1}_{𝑗=𝑠−𝑟}+(1+𝛼[𝑗])_𝑚+1𝑠=1(−1)^𝑠𝑞^{(𝑠−1)(𝑠−2)/2+(𝑚−𝑠+1)𝑟+𝑚}𝑚⋅(1+𝛼[𝑠−𝑟−1])𝑠−1_{𝑠−1𝑖=0}(𝑥+𝛼[𝑖])^{𝑚−𝑟−1}_{𝑗=𝑠−𝑟}(1+𝛼[𝑗])=𝑞^𝑚𝑟(𝑞^𝑟+𝛼[𝑚]−𝛼[𝑟])^{𝑚−𝑟−1}_𝑗=−𝑟(1+𝛼[𝑗])+(−1)^𝑚+1𝑞^{𝑚(𝑚−1)/2+𝑚}(1+𝛼[𝑚−𝑟])^𝑚_𝑖=0+(𝑥+𝛼[𝑖])_𝑚𝑠=1(−1)^𝑠𝑞^{𝑠(𝑠−1)/2+(𝑚+1−𝑠)𝑟}𝑈_𝑠^𝑠−1_𝑖=0(𝑥+𝛼[𝑖])^{𝑚−𝑟−1}_{𝑗=𝑠−𝑟}(1+𝛼[𝑗]), (2.3) where 𝑈_𝑠=𝑚𝑠(𝑞^𝑟+𝛼[𝑚]−𝛼[𝑟]+𝛼𝑞^𝑚[𝑠])𝑞^−𝑟+𝑞^{𝑚−𝑠+1}𝑚𝑠−1(1+𝛼[𝑠−𝑟−1]). (2.4) Using the obvious equalities (𝑞^𝑟+𝛼[𝑚]−𝛼[𝑟])𝑞^−𝑟𝑚𝑠𝑚=1+𝛼[𝑚−𝑟],[𝑠]=𝑠−1[𝑚−𝑠+1], (2.5) we have 𝑈_𝑠=𝑚𝑠+𝑚𝑞(1+𝛼[𝑚−𝑟])𝑠−1^{𝑚−𝑠+1}(1+𝛼([𝑚−𝑠+1]𝑞^{𝑠−𝑟−1}+[𝑠−𝑟−1])). (2.6) It is easy to see that 𝑞^{𝑠−𝑟−1}𝑚𝑠+𝑚𝑞+[𝑠−𝑟−1]=[𝑚−𝑟],𝑠−1^{𝑚−𝑠+1}=𝑠.𝑚+1 (2.7) Therefore, 𝑈_𝑠𝑠=(1+𝛼[𝑚−𝑟]).𝑚+1 (2.8) From last equality and (2.3), we obtain ^𝑚_𝑢=0(𝑞^𝑟−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟]))=𝑞^𝑚𝑟(𝑞^𝑟+𝛼[𝑚]−𝛼[𝑟])^{𝑚−𝑟−1}_𝑗=−𝑟(1+𝛼[𝑗])+(−1)^𝑚+1𝑞^{𝑚(𝑚−1)/2+𝑚}(1+𝛼[𝑚−𝑟])^𝑚_𝑖=0+(𝑥+𝛼[𝑖])_𝑚𝑠=1(−1)^𝑠𝑞^{𝑠(𝑠−1)/2+(𝑚+1−𝑠)𝑟}𝑠𝑚+1(1+𝛼[𝑚−𝑟])^𝑠−1_𝑖=0(𝑥+𝛼[𝑖])^{𝑚−𝑟−1}_{𝑗=𝑠−𝑟}=(1+𝛼[𝑗])_𝑚+1𝑠=0(−1)^𝑠𝑞^{𝑠(𝑠−1)/2+(𝑚+1−𝑠)𝑟}𝑠𝑚+1^𝑠−1_𝑖=0(𝑥+𝛼[𝑖])^𝑚−𝑟_{𝑗=𝑠−𝑟}(1+𝛼[𝑗]). (2.9) This completes the proof of the lemma.

3. Main Result

The generalized 𝑞-Bernstein polynomials, defined by (1.4), may be evaluated by Algorithm 1.

input:  𝑞;𝑓([0]/[𝑛]),𝑓([1]/[𝑛]),…,𝑓([𝑛]/[𝑛]) for  𝑟=0 to 𝑛                  𝑓𝑟[0]∶=𝑓([𝑟])[𝑛] next   𝑟 for  𝑚=1 to 𝑛 for  𝑟=0 to 𝑛−𝑚         𝑓𝑟[𝑚]∶={(𝑞𝑟−𝑞𝑚−1𝑥+𝛼([𝑚−1]−[𝑟]))𝑓𝑟[𝑚−1]+(𝑥+𝛼[𝑟])𝑓[𝑚−1]𝑟+1}1+𝛼[𝑚−1] next 𝑟 next 𝑚

Algorithm 1 

De Casteljau type algorithm.

In the case, where 𝛼=0, this is the de Casteljau algorithm for evaluating the 𝑞-Bernstein polynomial [3, 4]. Note that with 𝑞=1 and 𝛼=0, we recover the original classical de Casteljau algorithm (see Hoschek and Lasser [16]). The algorithm is justifed by the following theorem.

Theorem 3.1. Each intermediate point 𝑓_𝑟^[𝑚] of the algorithm can be expressed as 𝑓_𝑟^[𝑚]=(^𝑚−1_𝑖=0(1+𝛼[𝑖]))⁻¹⋅^𝑚_𝑡=0𝑓_𝑟+𝑡𝑚𝑡^{𝑡−1𝑠=0}(𝑥+𝛼[𝑟+𝑠])^{𝑚−𝑡−1}_𝑢=0(𝑞^𝑟−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟])), (3.1) and, in particular 𝑓₀^[𝑛]=𝐵_𝑛^𝑞,𝛼(𝑓;𝑥). (3.2)

Proof. We use induction on 𝑚. From the initial conditions in the algorithm, 𝑓_𝑟^[0]=𝑓([𝑟]/[𝑛])=𝑓_𝑟, 0≤𝑟≤𝑛, it is clear that (3.1) holds for 𝑚=0 and 0≤𝑟≤𝑛. Let us assume that (3.1) holds for some 𝑚 such that 0≤𝑚<𝑛, and for all 𝑟 such that 0≤𝑟≤𝑛−𝑚. Then, for 0≤𝑟≤𝑛−𝑚−1, it follows from the algorithm that 𝑓_𝑟^[𝑚+1]∶={(𝑞^𝑟−𝑞^𝑚𝑥+𝛼([𝑚]−[𝑟]))𝑓_𝑟^[𝑚]+(𝑥+𝛼[𝑟])𝑓^[𝑚]_𝑟+1}1,1+𝛼[𝑚] (3.3) and using (3.1), we obtain 𝑓_𝑟^[𝑚+1](^𝑚_𝑖=0(1+𝛼[𝑖]))∶=(𝑞^𝑟−𝑞^𝑚⋅𝑥+𝛼([𝑚]−[𝑟]))_𝑚𝑡=0𝑓_𝑟+𝑡𝑚𝑡^{𝑡−1𝑠=0}(𝑥+𝛼[𝑟+𝑠])⋅^{𝑚−𝑡−1}_𝑢=0(𝑞^𝑟−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟]))+(𝑥+𝛼[𝑟])⋅^𝑚_𝑡=0𝑓_𝑟+𝑡+1𝑚𝑡^{𝑡−1𝑠=0}⋅(𝑥+𝛼[𝑟+𝑠+1])_{𝑚−𝑡−1𝑢=0}(𝑞^𝑟+1−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟+1]))=(𝑞^𝑟−𝑞^𝑚𝑥+𝛼([𝑚]−[𝑟]))𝑓_𝑟^𝑚−1_𝑢=0(𝑞^𝑟−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟]))+(𝑞^𝑟−𝑞^𝑚𝑥+𝛼([𝑚]−[𝑟]))⋅^𝑚_𝑡=1𝑓_𝑟+𝑡𝑚𝑡⋅^𝑡−1_𝑠=0(𝑥+𝛼[𝑟+𝑠])⋅^{𝑚−𝑡−1}_𝑢=0(𝑞^𝑟−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟]))+(𝑥+𝛼[𝑟])⋅^𝑚_𝑡=1𝑓_𝑟+𝑡𝑚𝑡−1^{𝑡−2𝑠=0}⋅(𝑥+𝛼[𝑟+𝑠+1])_{𝑚−𝑡𝑢=0}(𝑞^𝑟+1−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟+1]))+(𝑥+𝛼[𝑟])𝑓_𝑟+𝑚+1^𝑚−1_𝑠=0(𝑥+𝛼[𝑟+𝑠+1])=𝑓_𝑟^𝑚_𝑢=0(𝑞^𝑟−𝑞^𝑢+𝑥+𝛼([𝑢]−[𝑟]))_𝑚𝑡=1{(𝑞^𝑟−𝑞^𝑚𝑚𝑡⋅𝑥+𝛼([𝑚]−[𝑟]))_{𝑡−1𝑠=0}(𝑥+𝛼[𝑟+𝑠])^{𝑚−𝑡−1}_𝑢=0(𝑞^𝑟−𝑞^𝑢𝑚𝑥+𝛼([𝑢]−[𝑟]))+(𝑥+𝛼[𝑟])𝑡−1^{𝑡−2𝑠=0}(𝑥+𝛼[𝑟+𝑠+1])_{𝑚−𝑡𝑢=0}(𝑞^𝑟+1−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟+1]))}𝑓_𝑟+𝑡+𝑓^𝑚_{𝑟+𝑚+1𝑠=0}(𝑥+𝛼[𝑟+𝑠]). (3.4) We see that ^𝑚−𝑡_𝑢=0(𝑞^𝑟+1−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟+1]))=(𝑞^𝑟+1−𝑥−𝛼[𝑟+1])^{𝑚−𝑡−1}_𝑢=0(𝑞^𝑟+1−𝑞^𝑢+1𝑥+𝛼([𝑢+1]−[𝑟+1]))=(𝑞^𝑟+1−𝑥−𝛼[𝑟+1])^{𝑚−𝑡−1}_𝑢=0(𝑞^𝑟+1−𝑞^𝑢+1𝑥+𝛼𝑞([𝑢]−[𝑟]))=(𝑞^𝑟+1−𝑥−𝛼[𝑟+1])𝑞^{𝑚−𝑡𝑚−𝑡−1}_𝑢=0(𝑞^𝑟−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟])), (3.5) and hence, 𝑓_𝑟^[𝑚+1](^𝑚_𝑖=0(1+𝛼[𝑖]))∶=𝑓_𝑟^𝑚_𝑢=0(𝑞^𝑟−𝑞^𝑢+𝑥+𝛼([𝑢]−[𝑟]))_𝑚𝑡=1{𝑚𝑡(𝑞^𝑟−𝑞^𝑚𝑚𝑥+𝛼([𝑚]−[𝑟]))+𝑡−1(𝑞^𝑟+1−𝑥−𝛼[𝑟+1])𝑞^𝑚−𝑡}⋅𝑓_𝑟+𝑡^𝑡−1_𝑠=0(𝑥+𝛼[𝑟+𝑠])^{𝑚−𝑡−1}_𝑢=0(𝑞^𝑟−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟]))+𝑓^𝑚_{𝑟+𝑚+1𝑠=0}(𝑥+𝛼[𝑟+𝑠]). (3.6) It is easy to verify that 𝑚𝑡+𝑞^{𝑚−𝑡+1}𝑚=𝑡,𝑚𝑡−1𝑚+1𝑡−1+𝑞^𝑡𝑚𝑡=𝑡.𝑚+1 (3.7) Therefore, 𝑚𝑡(𝑞^𝑟−𝑞^𝑚𝑚𝑥+𝛼([𝑚]−[𝑟]))+𝑡−1(𝑞^𝑟+1−𝑥−𝛼[𝑟+1])𝑞^𝑚−𝑡=𝑞^𝑟(𝑚𝑡+𝑞^{𝑚−𝑡+1}𝑚𝑡−1)−𝑥𝑞^𝑚−𝑡(𝑚𝑡−1+𝑞^𝑡𝑚𝑡)+𝛼1−𝑞{𝑞^𝑟(𝑚𝑡+𝑞^{𝑚−𝑡+1}𝑚𝑡−1)−𝑞^𝑚−𝑡(𝑚𝑡−1+𝑞^𝑡𝑚𝑡=𝑡)}𝑚+1{(𝑞^𝑟−𝑥𝑞^𝑚−𝑡)+𝛼([𝑚−𝑡]−[𝑟])}. (3.8) Consequently, 𝑓_𝑟^[𝑚+1](^𝑚_𝑖=0(1+𝛼[𝑖]))∶=𝑓_𝑟^𝑚_𝑢=0(𝑞^𝑟−𝑞^𝑢+𝑥+𝛼([𝑢]−[𝑟]))_𝑚𝑡=1𝑡𝑚+1{(𝑞^𝑟−𝑥𝑞^𝑚−𝑡)+𝛼([𝑚−𝑡]−[𝑟])}⋅𝑓_𝑟+𝑡^𝑡−1_𝑠=0(𝑥+𝛼[𝑟+𝑠])^{𝑚−𝑡−1}_𝑢=0(𝑞^𝑟−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟]))+𝑓^𝑚_{𝑟+𝑚+1𝑠=0}=(𝑥+𝛼[𝑟+𝑠])_𝑚+1𝑡=0𝑡𝑚+1⋅𝑓_𝑟+𝑡^𝑡−1_𝑠=0(𝑥+𝛼[𝑟+𝑠])^𝑚−𝑡_𝑢=0(𝑞^𝑟−𝑞^𝑢𝑥+𝛼([𝑢]−[𝑟])). (3.9) Thus, one has the desired result.

Theorem 3.2. For 0≤𝑚≤𝑛 and 0≤𝑟≤𝑛−𝑚, we have 𝑓_𝑟^[𝑚]=^𝑚_𝑠=0𝑞^{(𝑚−𝑠)𝑟}𝑚𝑠Δ^𝑠_𝑞𝑓_𝑟∏^{𝑠+𝑟−1}_𝑖=𝑟(𝑥+𝛼[𝑖])∏^𝑠−1_𝑗=0,(1+𝛼[𝑗]) (3.10) for all 𝑥∈[0;1].

Proof. Using (2.2) and (3.1), we have 𝑓_𝑟^{[𝑚]𝑚−1}_𝑖=0(1+𝛼[𝑖])=^𝑚_𝑡=0𝑚𝑡𝑓_𝑟+𝑡𝑆_𝑡(𝑚), (3.11) where 𝑆_𝑡(𝑚)=^𝑚−𝑡_𝑢=0(−1)^𝑢𝑞^{𝑢(𝑢−1)/2+(𝑚−𝑡−𝑢)𝑟}𝑢×𝑚−𝑡_{𝑡+𝑟−1𝑠=𝑟}(𝑥+𝛼[𝑠])^𝑢−1_𝑖=0(𝑥+𝛼[𝑖])^{𝑚−𝑡−𝑟−1}_{𝑗=𝑢−𝑟}(1+𝛼[𝑗])(0≤𝑡≤𝑚). (3.12) First, we prove that 𝑆_𝑡(𝑚)=^𝑚−𝑡_𝑢=0(−1)^𝑢𝑞^{𝑢(𝑢−1)/2+(𝑚−𝑡−𝑢)𝑟}𝑢⋅𝑚−𝑡^{𝑡+𝑢+𝑟−1}_𝑖=𝑟(𝑥+𝛼[𝑖])^𝑚−1_{𝑗=𝑢+𝑡}(1+𝛼[𝑗]) (3.13) for all 𝑚∈𝑁₀={0,1,2,…}, 𝑡∈𝑁₀, and 𝑥∈[0;1]. Note that an empty sum denotes 0.
We use the induction on 𝑚. First, we see that (3.13) holds for 𝑚=0 and all 𝑡∈𝑁₀. Let us assume that (3.13) holds for a given 𝑚, and for all 𝑡∈𝑁₀. Then, from (3.12) and (3.13), we obtain 𝑆_𝑡(𝑚+1)=(𝑥+𝛼[𝑡+𝑟−1])^{𝑚+1−𝑡}_𝑢=0(−1)^𝑢𝑞^{𝑢(𝑢−1)/2+(𝑚−𝑡+1−𝑢)𝑟}𝑢⋅𝑚−𝑡+1_{𝑡+𝑢−2+𝑟𝑖=𝑟}(𝑥+𝛼[𝑖])^𝑚−1_{𝑗=𝑢+𝑡−1}=(1+𝛼[𝑗])_{𝑚+1−𝑡𝑢=0}(−1)^𝑢𝑞^{𝑢(𝑢−1)/2+(𝑚−𝑡+1−𝑢)𝑟}𝑢⋅𝑚−𝑡+1_{𝑡+𝑢+𝑟−1𝑖=𝑟}(𝑥+𝛼[𝑖])^𝑚−1_{𝑗=𝑢+𝑡−1}(1+𝛼[𝑗])+𝛼^{𝑚−𝑡+1}_𝑢=1(−1)^𝑢𝑞^{𝑢(𝑢−1)/2+(𝑚−𝑡+1−𝑢)𝑟}𝑢𝑚−𝑡+1⋅([𝑡+𝑟+1]−[𝑡+𝑢+𝑟+1])^{𝑡+𝑢−2+𝑟}_𝑖=𝑟(𝑥+𝛼[𝑖])^𝑚−1_{𝑗=𝑢+𝑡−1}(1+𝛼[𝑗]). (3.14) We see that 𝑢𝑚−𝑡+1([𝑡+𝑟−1]−[𝑡+𝑢+𝑟−1])=−𝑞^{𝑡+𝑟−1}[𝑚−𝑡−𝑢+2],𝑚−𝑡+1𝑢−1 (3.15) and hence, 𝑆_𝑡(𝑚+1)=^{𝑚−𝑡+1}_𝑢=0(−1)^𝑢𝑞^{𝑢(𝑢−1)/2+(𝑚−𝑡+1−𝑢)𝑟}𝑢⋅𝑚−𝑡+1_{𝑡+𝑢+𝑟−1𝑖=𝑟}(𝑥+𝛼[𝑖])^𝑚−1_{𝑗=𝑢+𝑡−1}(1+𝛼[𝑗])+𝛼^𝑚−𝑡_𝑢=0(−1)^𝑢𝑞^{𝑢(𝑢−1)/2+(𝑚−𝑡+1−𝑢)𝑟}𝑞^{𝑢+𝑡−1}𝑢𝑚−𝑡+1⋅[𝑚−𝑡−𝑢+1]^{𝑡+𝑢+𝑟−1}_𝑖=𝑟(𝑥+𝛼[𝑖])^𝑚−1_{𝑗=𝑢+𝑡}(1+𝛼[𝑗])=(−1)^{𝑚−𝑡+1}𝑞^{(𝑚−𝑡+1)(𝑚−𝑡)/2𝑚+𝑟}_𝑖=𝑟+(𝑥+𝛼[𝑖])_{𝑚−𝑡𝑢=0}(−1)^𝑢𝑞^{𝑢(𝑢−1)/2+(𝑚−𝑡+1−𝑢)𝑟}⋅(1+𝛼[𝑢+𝑡−1]+𝛼𝑞^{𝑢+𝑡−1}𝑢⋅[𝑚−𝑡−𝑢+1])𝑚−𝑡+1_{𝑡+𝑢+𝑟−1𝑖=𝑟}(𝑥+𝛼[𝑖])^𝑚−1_{𝑗=𝑢+𝑡}(1+𝛼[𝑗]). (3.16) Next, in view of the equality (1+𝛼[𝑢+𝑡−1]+𝛼𝑞^{𝑢+𝑡−1}[𝑚−𝑡−𝑢+1])=1+𝛼[𝑚], (3.17) we obtain (3.13). Consequently, in view of (3.11) and (3.13), we get 𝑓_𝑟^{[𝑚]𝑚−1}_𝑖=0(1+𝛼[𝑖])=^𝑚_𝑡=0𝑚𝑡𝑓_𝑟+𝑡^𝑚−𝑡_𝑢=0(−1)^𝑢𝑞^{𝑢(𝑢−1)/2+(𝑚−𝑡−𝑢)𝑟}⋅𝑢𝑚−𝑡^{𝑡+𝑢+𝑟−1}_𝑖=𝑟(𝑥+𝛼[𝑖])^𝑚−1_{𝑗=𝑢+𝑡}=(1+𝛼[𝑗])_{𝑚𝑚𝑡=0𝑢=𝑡}𝑚𝑡𝑓_𝑟+𝑡(−1)^𝑢−𝑡𝑞^{(𝑢−𝑡)(𝑢−𝑡−1)/2+(𝑚−𝑢)𝑟}⋅𝑚−𝑡𝑢−𝑡^{𝑢+𝑟−1𝑖=𝑟}(𝑥+𝛼[𝑖])^𝑚−1_𝑗=𝑢(1+𝛼[𝑗]). (3.18) Next, in view of the equality 𝑚𝑡=𝑚𝑢𝑢𝑡,𝑚−𝑡𝑢−𝑡 (3.19) we obtain 𝑓_𝑟^{[𝑚]𝑚−1}_𝑖=0(1+𝛼[𝑖])=^𝑚𝑢_𝑢=0𝑡=0𝑚𝑢𝑓_𝑟+𝑡(−1)^𝑢−𝑡𝑞^{(𝑢−𝑡)(𝑢−𝑡−1)/2+(𝑚−𝑢)𝑟}⋅𝑢𝑡^{𝑢+𝑟−1𝑖=𝑟}(𝑥+𝛼[𝑖])^𝑚−1_𝑗=𝑢=(1+𝛼[𝑗])_𝑚𝑢=0𝑚𝑢𝑞^{(𝑚−𝑢)𝑟𝑢+𝑟−1}_𝑖=𝑟(𝑥+𝛼[𝑖])^𝑚−1_𝑗=𝑢⋅(1+𝛼[𝑗])_𝑢𝑡=0𝑢𝑡(−1)^𝑢−𝑡𝑞^{(𝑢−𝑡)(𝑢−𝑡−1)/2}𝑓_𝑟+𝑡. (3.20) The condition (1.10) completes the proof.

Theorems 3.1 and 3.2 are generalizations of Theorems 2.1 and 2.3 in [11].

Note that when 𝑚=𝑛 and 𝑟=0, (3.10) does indeed reduce to (1.11)