A Study on the Fermionic 𝑝-Adic 𝑞-Integral Representation on ℤ_𝑝 Associated with Weighted 𝑞-Bernstein and 𝑞-Genocchi Polynomials

Abstract

We consider weighted 𝑞-Genocchi numbers and polynomials. We investigated some interesting properties of the weighted 𝑞-Genocchi numbers related to weighted 𝑞-Bernstein polynomials by using fermionic 𝑝-adic integrals on ℤ_𝑝.

1. Introduction, Definitions, and Notations

The main motivation of this paper is [1] by Kim, in which he introduced and studied properties of 𝑞-Bernoulli numbers and polynomials with weight 𝛼. Recently, many mathematicians have studied weighted special polynomials (see [1–5]).

This numbers and polynomials are used in not only number theory, complex analysis, and the other branch of mathematics, but also in other parts of the 𝑝-adic analysis and mathematical physics. Kurt Hensel (1861–1941) invented the so-called 𝑝-adic numbers around the end of the nineteenth century. In spite of their being already one hundred years old, these numbers are still today enveloped in an aura of mystery within scientific community [6] although they have penetrated several mathematical fields such as number theory, algebraic geometry, algebraic topology, analysis, and mathematical physics (see, for details, [6–8]).

The 𝑝-adic 𝑞-integral (or 𝑞-Volkenborn integral) are originally constructed by Kim [9]. The 𝑞-Volkenborn integral is used in mathematical physics, for example, the functional equation of the 𝑞-zeta function, the 𝑞-Stirling numbers, and 𝑞-Mahler theory of integration with respect to the ring ℤ_𝑝 together with Iwasawa's 𝑝-adic 𝑞-𝐿 function.

Let 𝑝 be a fixed odd prime number. Throughout this paper, we use the following notations. By ℤ_𝑝, we denote the ring of 𝑝-adic rational integers, ℚ denotes the field of rational numbers, ℚ_𝑝 denotes the field of 𝑝-adic rational numbers, and ℂ_𝑝 denotes the completion of algebraic closure of ℚ_𝑝. Let ℕ be the set of natural numbers and ℕ^∗=ℕ∪{0}. The 𝑝-adic absolute value is defined by |𝑝|_𝑝=1/𝑝. In this paper, we assume |𝑞−1|_𝑝<1 as an indeterminate. In [10–12], let UD (ℤ_𝑝) be the space of uniformly differentiable functions on ℤ_𝑝. For 𝑓∈ UD (ℤ_𝑝), the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 is defined by Kim𝐼_−𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁𝑝_−𝑞^𝑁−1_𝑥=0𝑞^𝑥𝑓(𝑥)(−1)^𝑥. (1.1)

For 𝛼,𝑘,𝑛∈ℕ^∗ and 𝑥∈[0,1], Kim et al. defined weighted 𝑞-Bernstein polynomials as follows:𝐵^(𝛼)_𝑘,𝑛=𝑛𝑘𝑥,𝑞[𝑥]^𝑘_𝑞𝛼[1−𝑥]_𝑞^𝑛−𝑘_−𝛼, (1.2) (see [13, 14]). When we put 𝑞→1 and 𝛼=1 in (1.2), [𝑥]^𝑘_𝑞𝛼→𝑥^𝑘,[1−𝑥]_𝑞^𝑛−𝑘_−𝛼→(1−𝑥)^𝑛−𝑘, and we obtain the classical Bernstein polynomials (see [13, 14]), where [𝑥]_𝑞 is a 𝑞-extension of 𝑥 which is defined by[𝑥]_𝑞=1−𝑞^𝑥,1−𝑞 (1.3) (see [1–4, 7, 9–12, 14–26]). Note that lim_𝑞→1[𝑥]_𝑞=𝑥.

In [3], For 𝑛∈ℕ^∗, S. Araci et al. defined weighted 𝑞-Genocchi polynomials as follows:𝐺^(𝛼)_𝑛+1,𝑞(𝑥)=𝑛+1_ℤ𝑝𝑥+𝑦^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝑦=[2]_𝑞1−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼ℓ𝑥11+𝑞^𝛼ℓ+1=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝑞^𝑚[𝑚+𝑥]^𝑛_𝑞𝛼. (1.4)

In the special case, 𝑥=0, 𝐺^(𝛼)_𝑛,𝑞(0)=𝐺^(𝛼)_𝑛,𝑞  are called the 𝑞-Genocchi numbers with weight 𝛼.

In [3], For 𝛼∈ℕ^∗ and 𝑛∈ℕ, S. Araci et al. defined 𝑞-Genocchi numbers with weight 𝛼 as follows:𝐺^(𝛼)_0,𝑞=0,𝑞𝐺^(𝛼)_𝑛,𝑞(1)+𝐺^(𝛼)_𝑛,𝑞=[2]_𝑞,if𝑛=0,0,if𝑛≠0. (1.5)

In this paper we obtained some relations between the weighted 𝑞-Bernstein polynomials and the 𝑞-Genocchi numbers. From these relations, we derive some interesting identities on the 𝑞-Genocchi numbers and polynomials with weight 𝛼.

2. On the Weighted 𝑞-Genocchi Numbers and Polynomials

By the definition of 𝑞-Genocchi polynomials with weight 𝛼, we easily get 𝐺^(𝛼)_𝑛+1,𝑞(𝑥)=𝑛+1_ℤ𝑝𝑥+𝑦^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝑦=_ℤ𝑝[𝑥]_𝑞𝛼+𝑞^𝛼𝑥𝑦_𝑞𝛼^𝑛𝑑𝜇_−𝑞𝑦=^𝑛_𝑘=0𝑛𝑘[𝑥]_𝑞^𝑛−𝑘_𝛼𝑞^𝛼𝑘𝑥_ℤ𝑝𝑦^𝑘_𝑞𝛼𝑑𝜇_−𝑞𝑦=^𝑛_𝑘=0𝑛𝑘[𝑥]_𝑞^𝑛−𝑘_𝛼𝑞^𝛼𝑘𝑥𝐺^(𝛼)_𝑘+1,𝑞.𝑘+1 (2.1)

Therefore, we obtain the following theorem.

Theorem 2.1. For 𝑛,𝛼∈ℕ^∗, one has 𝐺^(𝛼)_𝑛,𝑞(𝑥)=𝑞^{𝑛−𝛼𝑥}_𝑘=0𝑛𝑘𝑞^𝛼𝑘𝑥𝐺^(𝛼)_𝑘,𝑞[𝑥]_𝑞^𝑛−𝑘_𝛼, (2.2) with usual convention about replacing (𝐺_𝑞^(𝛼))^𝑛 by 𝐺^(𝛼)_𝑛,𝑞.

By Theorem 2.1, we have𝐺^(𝛼)_𝑛,𝑞(𝑥)=𝑞^−𝛼𝑥𝑞^𝛼𝑥𝐺_𝑞^(𝛼)+[𝑥]_𝑞𝛼^𝑛. (2.3)

By (1.4), we get 𝐺^(𝛼)_{𝑛+1,𝑞−1}(1−𝑥)=𝑛+1_ℤ𝑝1−𝑥+𝑦^𝑛_𝑞−𝛼𝑑𝜇_−𝑞𝑦=[2]_𝑞−11−𝑞^{−𝛼𝑛𝑛}_𝑙=0𝑛𝑙𝑞^{−𝛼𝑙(1−𝑥)}(−1)^𝑙11+𝑞^{−𝛼𝑙−1}=(−1)^𝑛𝑞^𝛼𝑛[2]_𝑞1−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥11+𝑞^𝛼𝑙+1=(−1)^𝑛𝑞^𝛼𝑛𝐺^(𝛼)_𝑛+1,𝑞(𝑥).𝑛+1 (2.4)

Therefore, we obtain the following theorem.

Theorem 2.2. For 𝑛,𝛼∈ℕ^∗, one has 𝐺^(𝛼)_{𝑛,𝑞−1}(1−𝑥)=(−1)^𝑛−1𝑞^{𝛼(𝑛−1)}𝐺^(𝛼)_𝑛,𝑞(𝑥). (2.5)

From (1.5) and Theorem 2.1, we have the following theorem.

Theorem 2.3. For 𝑛,𝛼∈ℕ^∗, one has 𝐺^(𝛼)_0,𝑞=0,𝑞^1−𝛼𝑞^𝛼𝐺_𝑞^(𝛼)+1^𝑛+𝐺^(𝛼)_𝑛,𝑞=[2]_𝑞,if𝑛=0,0,if𝑛≠0, (2.6) with usual convention about replacing (𝐺_𝑞^(𝛼))^𝑛 by 𝐺^(𝛼)_𝑛,𝑞.

For 𝑛,𝛼∈ℕ, by Theorem 2.3, we note that 𝑞^2𝛼𝐺^(𝛼)_𝑛,𝑞(2)=𝑞^𝛼𝑞^𝛼𝐺_𝑞^(𝛼)+1+1^𝑛=^𝑛_𝑘=0𝑛𝑘𝑞^𝑘𝛼𝑞^𝛼𝐺_𝑞+1^𝑘=𝑛𝑞^2𝛼−1[2]_𝑞−𝐺^(𝛼)_1,𝑞−𝑞^𝑛𝛼−1_𝑘=2𝑛𝑘𝑞^𝑘𝛼𝐺^(𝛼)_𝑘,𝑞=𝑞^𝛼−1[2]_𝑞−𝑞^𝑛𝛼−1_𝑘=1𝑛𝑘𝑞^𝛼𝑘𝐺^(𝛼)_𝑘,𝑞=𝑞^𝛼−1[2]_𝑞+𝑞^2𝛼−2𝐺^(𝛼)_𝑛,𝑞if𝑛>1. (2.7)

Therefore, we have the following theorem.

Theorem 2.4. For 𝑛∈ℕ, one has 𝐺^(𝛼)_𝑛,𝑞(2)=[2]_𝑞𝑞^𝛼+1+1𝑞²𝐺^(𝛼)_𝑛,𝑞. (2.8)

From Theorem 2.2 and (2.5), we see that(𝑛+1)_ℤ𝑝[1−𝑥]^𝑛_𝑞−𝛼𝑑𝜇_−𝑞(𝑥)=(𝑛+1)(−1)^𝑛𝑞^𝑛𝛼_ℤ𝑝[𝑥−1]^𝑛_𝑞𝛼𝑑𝜇_−𝑞(𝑥)=(−1)^𝑛𝑞^𝑛𝛼𝐺^(𝛼)_𝑛+1,𝑞(−1)=𝐺^(𝛼)_{𝑛+1,𝑞−1}(2). (2.9)

Therefore, we get the following theorem.

Theorem 2.5. For 𝑛,𝛼∈ℕ^∗, one has (𝑛+1)_ℤ𝑝[1−𝑥]^𝑛_𝑞−𝛼𝑑𝜇_−𝑞(𝑥)=𝐺^(𝛼)_{𝑛+1,𝑞−1}(2). (2.10)

Let 𝑛,𝛼∈ℕ. By Theorems 2.4 and 2.5, we get(𝑛+1)_ℤ𝑝[1−𝑥]^𝑛_𝑞−𝛼𝑑𝜇_−𝑞(𝑥)=𝑞^𝛼[2]_𝑞+𝑞²𝐺^(𝛼)_{𝑛+1,𝑞−1}. (2.11)

From (2.11), we get the following corollary.

Corollary 2.6. For 𝑛,𝛼∈ℕ^∗, one has _ℤ𝑝[1−𝑥]^𝑛_𝑞−𝛼𝑑𝜇_−𝑞(𝑥)=𝑞^𝛼[2]𝑛+1^𝑞+𝑞²𝐺^(𝛼)_{𝑛+1,𝑞−1}.𝑛+1 (2.12)

3. Novel Identities on the Weighted 𝑞-Genocchi Numbers

In this section, we derive concerning the some interesting properties of 𝑞-Genocchi numbers via the 𝑝-adic 𝑞-integral on ℤ_𝑝, in the sense of fermionic and weighted 𝑞-Bernstein polynomials.𝐵^(𝛼)_𝑘,𝑛=𝑛𝑘𝑥,𝑞[𝑥]^𝑘_𝑞𝛼[1−𝑥]_𝑞^𝑛−𝑘_−𝛼,where𝑛,𝑘,𝛼∈ℕ^∗. (3.1)

By (3.1), Kim et al. get the symmetry of 𝑞-Bernstein polynomials weighted 𝛼 as follows:𝐵^(𝛼)_𝑘,𝑛𝑥,𝑞=𝐵^(𝛼)_{𝑛−𝑘,𝑛}1−𝑥,𝑞⁻¹, (3.2) (see [4]). Thus, from Corollary 2.6, (3.1), and (3.2), we see that_ℤ𝑝𝐵^(𝛼)_𝑘,𝑛𝑥,𝑞𝑑𝜇_−𝑞(𝑥)=_ℤ𝑝𝐵^(𝛼)_{𝑛−𝑘,𝑛}1−𝑥,𝑞⁻¹𝑑𝜇_−𝑞(𝑥)=𝑛𝑘^𝑘𝑙=0𝑘𝑙(−1)^𝑘+𝑙_ℤ𝑝[1−𝑥]_𝑞^𝑛−𝑙_−𝛼𝑑𝜇_−𝑞(𝑥)=𝑛𝑘^𝑘𝑙=0𝑘𝑙(−1)^𝑘+𝑙𝑞^𝛼[2]_𝑞𝑛−𝑙+1+𝑞²𝐺^(𝛼)_{𝑛−𝑙+1,𝑞−1}.𝑛−𝑙+1 (3.3)

For 𝑛,𝑘∈ℕ^∗ and 𝛼∈ℕ with 𝑛>𝑘, we obtain_ℤ𝑝𝐵_𝑘,𝑛(𝑥)𝑑𝜇_−𝑞(𝑥)=𝑛𝑘^𝑘𝑙=0𝑘𝑙(−1)^𝑘+𝑙𝑞^𝛼[2]_𝑞𝑛−𝑙+1+𝑞²𝐺^(𝛼)_{𝑛−𝑙+1,𝑞−1}=𝑞𝑛−𝑙+1_𝛼[2]_𝑞𝑛+1+𝑞²𝐺^(𝛼)_{𝑛+1,𝑞−1},𝑛+1if𝑘=0,_𝑘𝑙=0𝑘𝑙(−1)^𝑘+𝑙𝑞^𝛼[2]_𝑞𝑛−𝑙+1+𝑞²𝐺^(𝛼)_{𝑛−𝑙+1,𝑞−1},𝑛−𝑙+1if𝑘≠0. (3.4)

Let us take the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 for the weighted 𝑞-Bernstein polynomials of degree 𝑛 as follows:_ℤ𝑝𝐵^(𝛼)_𝑘,𝑛𝑥,𝑞𝑑𝜇_−𝑞(𝑥)=𝑛𝑘^ℤ𝑝[𝑥]^𝑘_𝑞𝛼[1−𝑥]_𝑞^𝑛−𝑘_−𝛼𝑑𝜇_−𝑞(𝑥)=𝑛𝑘^{𝑛−𝑘𝑙=0}𝑙)𝑛−𝑘(−1^𝑙𝐺^(𝛼)_{𝑙+𝑘+1,𝑞}.𝑙+𝑘+1 (3.5)

Therefore, by (3.4) and (3.5), we obtain the following theorem.

Theorem 3.1. For 𝑛,𝑘∈ℕ^∗ and 𝛼∈ℕ with 𝑛>𝑘, one has ^{𝑛−𝑘𝑙=0}𝑙()𝑛−𝑘−1^𝑙𝐺^(𝛼)_{𝑙+𝑘+1,𝑞}=𝑞𝑙+𝑘+1²[2]_𝑞𝑛+1+𝑞²𝐺^(𝛼)_{𝑛+1,𝑞−1},𝑛+1if𝑘=0,_𝑘𝑙=0𝑘𝑙(−1)^𝑘+𝑙𝑞^𝛼[2]_𝑞𝑛−𝑙+1+𝑞²𝐺^(𝛼)_{𝑛−𝑙+1,𝑞−1},𝑛−𝑙+1if𝑘≠0. (3.6)

Let 𝑛₁,𝑛₂,𝑘∈ℕ^∗ and 𝛼∈ℕ with 𝑛₁+𝑛₂>2𝑘.  Then, we get_ℤ𝑝𝐵^(𝛼)_𝑘,𝑛1𝑥,𝑞𝐵^(𝛼)_𝑘,𝑛2𝑥,𝑞𝑑𝜇_−𝑞(𝑥)=𝑛₁𝑘𝑛₂𝑘^2𝑘𝑙=0𝑙()2𝑘−1^2𝑘+𝑙_ℤ𝑝[1−𝑥]^𝑛1+𝑛2_𝑞^−𝑙_−𝛼𝑑𝜇_−𝑞(𝑥)=𝑛₁𝑘𝑛₂𝑘^2𝑘𝑙=0𝑙()2𝑘−1^2𝑘+𝑙𝑞^𝛼[2]_𝑞𝑛₁+𝑛₂−𝑙+1+𝑞²𝐺_𝑛^(𝛼)_{1+𝑛2−𝑙+1,𝑞−1}𝑛₁+𝑛₂=𝑞−𝑙+1_𝛼[2]_𝑞𝑛₁+𝑛₂+1+𝑞²𝐺_𝑛^(𝛼)_{1+𝑛2+1,𝑞−1}𝑛₁+𝑛₂,+1if𝑛𝑘=0,₁𝑘𝑛₂𝑘^2𝑘_𝑙=0𝑙()2𝑘−1^2𝑘+𝑙𝑞^𝛼[2]_𝑞𝑛₁+𝑛₂−𝑙+1+𝑞²𝐺_𝑛^(𝛼)_{1+𝑛2−𝑙+1,𝑞−1}𝑛₁+𝑛₂,−𝑙+1if𝑘≠0. (3.7)

Therefore, we obtain the following theorem.

Theorem 3.2. For 𝑛₁,𝑛₂,𝑘∈ℕ^∗ and 𝛼∈ℕ with 𝑛₁+𝑛₂>2𝑘, one has _ℤ𝑝𝐵^(𝛼)_𝑘,𝑛1𝑥,𝑞𝐵^(𝛼)_𝑘,𝑛2𝑥,𝑞𝑑𝜇_−𝑞(𝑥)=𝑞^𝛼[2]_𝑞𝑛₁+𝑛₂+1+𝑞²𝐺_𝑛^(𝛼)_{1+𝑛2+1,𝑞−1}𝑛₁+𝑛₂,+1if𝑛𝑘=0,₁𝑘𝑛₂𝑘^2𝑘_𝑙=0𝑙()2𝑘−1^2𝑘+𝑙𝑞^𝛼[2]_𝑞𝑛₁+𝑛₂−𝑙+1+𝑞²𝐺_𝑛^(𝛼)_{1+𝑛2−𝑙+1,𝑞−1}𝑛₁+𝑛₂,−𝑙+1if𝑘≠0. (3.8)

From the binomial theorem, we can derive_ℤ𝑝𝐵^(𝛼)_𝑘,𝑛1𝑥,𝑞𝐵^(𝛼)_𝑘,𝑛2𝑥,𝑞𝑑𝜇_−𝑞(𝑥)=²_𝑖=1𝑛_𝑖𝑘^{𝑛1+𝑛2−2𝑘𝑙=0}𝑛₁+𝑛₂𝑙()−2𝑘−1^𝑙_ℤ𝑝[𝑥]_𝑞^2𝑘+𝑙_𝛼𝑑𝜇_−𝑞(𝑥)=²_𝑖=1𝑛_𝑖𝑘^{𝑛1+𝑛2−2𝑘𝑙=0}𝑛₁+𝑛₂𝑙()−2𝑘−1^𝑙𝐺^(𝛼)_{𝑙+2𝑘+1,𝑞}.𝑙+2𝑘+1 (3.9)

Thus, for Theorem 3.4 and (3.13), we can obtain the following corollary.

Corollary 3.3. For 𝑛₁,𝑛₂,𝑘∈ℕ^∗ and 𝛼∈ℕ with 𝑛₁+𝑛₂>2𝑘, one has ^{𝑛1+𝑛2−2𝑘}_𝑙=0𝑛₁+𝑛₂𝑙()−2𝑘−1^𝑙𝐺^(𝛼)_{𝑙+2𝑘+1,𝑞}=𝑞𝑙+2𝑘+1_𝛼[2]_𝑞𝑛₁+𝑛₂+1+𝑞²𝐺_𝑛^(𝛼)_{1+𝑛2+1,𝑞−1}𝑛₁+𝑛₂,+1if𝑘=0,_2𝑘𝑙=0𝑙()2𝑘−1^2𝑘+𝑙𝑞^𝛼[2]_𝑞𝑛₁+𝑛₂−𝑙+1+𝑞²𝐺_𝑛^(𝛼)_{1+𝑛2−𝑙+1,𝑞−1}𝑛₁+𝑛₂,−𝑙+1if𝑘≠0. (3.10)

For 𝑥∈ℤ_𝑝 and 𝑠∈ℕ with 𝑠≥2, let 𝑛₁,𝑛₂,…,𝑛_𝑠,𝑘∈ℕ^∗ and 𝛼∈ℕ with ∑^𝑠_𝑙=1𝑛_𝑙>𝑠𝑘. Then, we take the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 for the weighted 𝑞-Bernstein polynomials of degree 𝑛 as follows:_ℤ𝑝𝐵^(𝛼)_𝑘,𝑛1𝑥,𝑞𝐵^(𝛼)_𝑘,𝑛2𝑥,𝑞⋯𝐵^(𝛼)_{𝑘,𝑛𝑠}𝑥,𝑞_𝑠-times𝑑𝜇_−𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^ℤ𝑝[𝑥]_𝑞^𝑠𝑘_𝛼[1−𝑥]^{𝑛1+𝑛2+⋯+𝑛𝑠}_𝑞^−𝑠𝑘_−𝛼𝑑𝜇_−𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑠𝑘𝑙=0}𝑙()𝑠𝑘−1^{𝑙+𝑠𝑘}_ℤ𝑝[1−𝑥]^{𝑛1+𝑛2+⋯+𝑛𝑠}_𝑞^−𝑠𝑘_−𝛼𝑑𝜇_−𝑞(𝑥)=𝑞^𝛼[2]_𝑞𝑛₁+𝑛₂+⋯+𝑛_𝑠+1+𝑞²𝐺_𝑛^(𝛼)_{1+𝑛2+⋯+𝑛𝑠+1,𝑞−1}𝑛₁+𝑛₂+⋯+𝑛_𝑠,+1if𝑘=0,_𝑠𝑖=1𝑛_𝑖𝑘^𝑠𝑘_𝑙=0𝑙)𝑠𝑘(−1^{𝑠𝑘+𝑙}×𝑞^𝛼[2]_𝑞𝑛₁+𝑛₂+⋯+𝑛_𝑠−𝑙+1+𝑞²𝐺_𝑛^(𝛼)_{1+𝑛2+⋯+𝑛𝑠−𝑙+1,𝑞−1}𝑛₁+𝑛₂+⋯+𝑛_𝑠,−𝑙+1if𝑘≠0. (3.11)

Therefore, we obtain the following theorem.

Theorem 3.4. For 𝑠∈ℕ with 𝑠≥2, let 𝑛₁,𝑛₂,…,𝑛_𝑠,𝑘∈ℕ^∗ and 𝛼∈ℕ with ∑^𝑠_𝑙=1𝑛_𝑙>𝑠𝑘. Then, one has _ℤ𝑝^s_𝑖=1𝐵^(𝛼)_{𝑘,𝑛𝑖}(𝑥)𝑑𝜇_−𝑞(𝑥)=𝑞^𝛼[2]_𝑞𝑛₁+𝑛₂+⋯+𝑛_𝑠𝑞+1²𝐺_𝑛^(𝛼)_{1+𝑛2+⋯+𝑛𝑠+1,𝑞−1}𝑛₁+𝑛₂+⋯+𝑛_𝑠,−𝑙+1if𝑘=0,_s𝑖=1𝑛_𝑖𝑘^𝑠𝑘_𝑙=0𝑙()𝑠𝑘−1^{𝑠𝑘+𝑙}𝑞^𝛼[2]_𝑞𝑛₁+𝑛₂−𝑙+1+𝑞²𝐺_𝑛^(𝛼)_{1+𝑛2−𝑙+1,𝑞−1}𝑛₁+𝑛₂,−𝑙+1if𝑘≠0. (3.12)

From the definition of weighted 𝑞-Bernstein polynomials and the binomial theorem, we easily get_ℤ𝑝𝐵^(𝛼)_𝑘,𝑛1𝑥,𝑞𝐵^(𝛼)_𝑘,𝑛2𝑥,𝑞⋯𝐵^(𝛼)_{𝑘,𝑛𝑠}𝑥,𝑞_𝑠-times𝑑𝜇_−𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑛1+⋯+𝑛𝑠−𝑠𝑘𝑙=0𝑠𝑑=1}𝑛_𝑑𝑙()−𝑘−1^𝑙_ℤ𝑝[𝑥]_𝑞^{𝑠𝑘+𝑙}_𝛼𝑑𝜇_−𝑞(𝑥)=^𝑠_𝑖=1𝑛_𝑖𝑘^{𝑛1+⋯+𝑛𝑠−𝑠𝑘𝑙=1𝑠𝑑=1}𝑛_𝑑𝑙()−𝑘−1^𝑙𝐺^(𝛼)_{𝑙+𝑠𝑘+1,𝑞}.𝑙+𝑠𝑘+1 (3.13)

Therefore, from (3.13) and Theorem 3.4, we have the following corollary.

Corollary 3.5. For 𝑠∈ℕ with 𝑠≥2, let 𝑛₁,𝑛₂,…,𝑛_𝑠,𝑘∈ℕ^∗ and 𝛼∈ℕ with ∑^𝑠_𝑙=1𝑛_𝑙>𝑠𝑘. One has ^𝑛₁^+⋯+𝑛_𝑠^−𝑠𝑘_𝑙=0^𝑠_𝑑=1𝑛_𝑑𝑙()−𝑘−1^𝑙𝐺^(𝛼)_{𝑙+𝑠𝑘+1,𝑞}=𝑞𝑙+𝑠𝑘+1_𝛼[2]_𝑞𝑛₁+𝑛₂+⋯+𝑛_𝑠+1+𝑞²𝐺_𝑛^(𝛼)_{1+𝑛2+⋯+𝑛𝑠+1,𝑞−1}𝑛₁+𝑛₂+⋯+𝑛_𝑠,+1if𝑘=0,_{𝑠𝑘𝑙=0}𝑙()𝑠𝑘−1^{𝑠𝑘+𝑙}𝑞^𝛼[2]_𝑞𝑛₁+𝑛₂+⋯+𝑛_𝑠−𝑙+1+𝑞²𝐺_𝑛^(𝛼)_{1+𝑛2+⋯+𝑛𝑠−𝑙+1,𝑞−1}𝑛₁+𝑛₂+⋯+𝑛_𝑠,−𝑙+1if𝑘≠0. (3.14)

Acknowledgments

The authors wish to express their sincere gratitude to the referees for their valuable suggestions and comments and Professor Toka Diagana for his cooperation and help.