Some Relations between Twisted (ℎ,𝑞)-Euler Numbers with Weight α and 𝑞-Bernstein Polynomials with Weight α

Abstract

By using fermionic p-adic q-integral on ℤ_𝑝, we give some interesting relationship between the twisted (h, q)-Euler numbers with weight α and the q-Bernstein polynomials.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper, we always make use of the following notations: ℤ denotes the ring of rational integers, ℤ_𝑝 denotes the ring of 𝑝-adic rational integers, ℚ_𝑝 denotes the field of 𝑝-adic rational numbers, and ℂ_𝑝 denotes the completion of algebraic closure of ℚ_𝑝, respectively. Let ℕ be the set of natural numbers and ℤ₊=ℕ∪{0}. Let 𝐶_𝑝𝑛={𝑤∣𝑤^𝑝𝑛=1} be the cyclic group of order 𝑝^𝑛, and let 𝑇_𝑝=lim_𝑛→∞𝐶_𝑝𝑛=𝐶_𝑝∞=∪_𝑛≥0𝐶_𝑝𝑛,(1.1) (see [1–22]), be the locally constant space. For 𝑤∈𝑇_𝑝, we denote by 𝜙_𝑤∶ℤ_𝑝→ℂ_𝑝 the locally constant function 𝑥↦𝑤^𝑥. The 𝑝-adic absolute value is defined by |𝑥|_𝑝=1/𝑝^𝑟, where 𝑥=𝑝^𝑟𝑠/𝑡(𝑟∈ℚand𝑠,𝑡∈ℤwith(𝑠,𝑡)=(𝑝,𝑠)=(𝑝,𝑡)=1). In this paper, we assume that 𝑞∈ℂ_𝑝 with |𝑞−1|_𝑝<1 as an indeterminate. The 𝑞-number is defined by [𝑥]_𝑞=1−𝑞^𝑥1−𝑞,(1.2) (see [1–22]). Note that lim_𝑞→1[𝑥]_𝑞=𝑥. For 𝑓∈𝑈𝐷ℤ_𝑝=𝑓∣𝑓∶ℤ_𝑝⟶ℂ_𝑝isuniformlydifferentiablefunction,(1.3) the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 is defined by Kim as follows: 𝐼_−𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1+𝑞1+𝑞_𝑝^{𝑁𝑝𝑁−1}_𝑥=0𝑓(𝑥)(−𝑞)^𝑥,(1.4) (see [1–7]). From (1.4), we note that 𝑞^𝑛𝐼_−𝑞𝑓_𝑛=(−1)^𝑛𝐼_−𝑞(𝑓)+[2]^𝑞𝑛−1_𝑙=0(−1)^{𝑛−1−𝑙}𝑞^𝑙𝑓(𝑙),(1.5) where 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛) for 𝑛∈ℕ.

For 𝑘,𝑛∈ℤ₊ and 𝑥∈[0,1], Kim defined 𝑞-Bernstein polynomials, which are different 𝑞-Bernstein polynomials of Phillips, as follows: 𝐵_𝑘,𝑛(𝑥,𝑞)=⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠[𝑥]^𝑘_𝑞[1−𝑥]^𝑛−𝑘_𝑞−1,(1.6) (see [5]). In [9], the 𝑝-adic extension of (1.6) is given by 𝐵_𝑘,𝑛(𝑥,𝑞)=⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠[𝑥]^𝑘_𝑞[1−𝑥]^𝑛−𝑘_𝑞−1,where𝑥∈ℤ_𝑝,𝑛,𝑘∈ℤ₊.(1.7) For 𝛼∈ℤ, ℎ∈ℤ, 𝑤∈𝑇_𝑝, and 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝≤1, twisted (ℎ,𝑞)-Euler numbers 𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤} with weight 𝛼 are defined by 𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}=_ℤ𝑝𝜙_𝑤(𝑥)𝑞^{𝑥(ℎ−1)}[𝑥]^𝑛_𝑞𝛼𝑑𝜇_−𝑞(𝑥).(1.8) In the special case, 𝑥=0, 𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}(0)=𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤} are called the 𝑛-th twisted (ℎ,𝑞)-Euler numbers with weight 𝛼.

In this paper, we investigate some relations between the 𝑞-Bernstein polynomials and the twisted (ℎ,𝑞)-Euler numbers with weight 𝛼. From these relations, we derive some interesting identities on the twisted (ℎ,𝑞)-Euler numbers and polynomials with weight 𝛼.

2. Twisted (ℎ,𝑞)-Euler Numbers and Polynomials with Weight 𝛼

By using 𝑝-adic 𝑞-integral and (1.8), we obtain _ℤ𝑝𝜙_𝑤(𝑥)𝑞^{𝑥(ℎ−1)}[𝑥]^𝑛_𝑞𝛼𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁_−𝑞^𝑝𝑁−1_𝑥=0[𝑥]^𝑛_𝑞𝛼𝑤^𝑥𝑞^{𝑥(ℎ−1)}(−𝑞)^𝑥=[2]_𝑞11−𝑞^𝛼^𝑛𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)^𝑙11+𝑤𝑞^𝛼𝑙+ℎ.(2.1) We set 𝐹^(ℎ,𝛼)_𝑞,𝑤(𝑡)=^∞_𝑛=0𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}𝑡^𝑛𝑛!.(2.2) By (2.1) and (2.2), we have 𝐹^(ℎ,𝛼)_𝑞,𝑤(𝑡)=^∞_𝑛=0𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}𝑡^𝑛𝑛!=[2]^𝑞∞_𝑛=0⎛⎜⎜⎝11−𝑞^𝛼^𝑛𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)^𝑙11+𝑤𝑞^𝛼𝑙+ℎ⎞⎟⎟⎠𝑡^𝑛𝑛!=[2]^𝑞∞_𝑚=0(−1)^𝑚𝑤^𝑚𝑞^ℎ𝑚𝑒^{[𝑚]𝑞𝛼𝑡}.(2.3) Since [𝑥+𝑦]_𝑞𝛼=[𝑥]_𝑞𝛼+𝑞^𝛼𝑥[𝑦]_𝑞𝛼, we obtain 𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}(𝑥)=_ℤ𝑝𝜙_𝑤(𝑦)𝑞^{𝑦(ℎ−1)}[𝑦+𝑥]_𝑞𝛼𝑡𝑑𝜇_−𝑞(𝑦)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝑞^𝛼𝑥𝑙[𝑥]^𝑛−𝑙_𝑞𝛼_ℤ𝑝𝜙_𝑤(𝑦)𝑞^{𝑦(ℎ−1)}[𝑦]^𝑙_𝑞𝛼𝑑𝜇_−𝑞(𝑦)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝑞^𝛼𝑥𝑙[𝑥]^𝑛−𝑙_𝑞𝛼𝐸^(ℎ,𝛼)_{𝑙,𝑞,𝑤}.(2.4)

Therefore, we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊ and 𝑤∈𝑇_𝑝, we have 𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}(𝑥)=[2]^𝑞∞_𝑚=0(−1)^𝑚𝑤^𝑚𝑞^ℎ𝑚[𝑥+𝑚]^𝑛_𝑞𝛼.(2.5) Furthermore, 𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}(𝑥)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝑞^𝛼𝑥𝑙[𝑥]^𝑛−𝑙_𝑞𝛼𝐸^(ℎ,𝛼)_{𝑙,𝑞,𝑤}=[𝑥]_𝑞𝛼+𝑞^𝛼𝑥𝐸^(ℎ,𝛼)_𝑞,𝑤^𝑛,(2.6) with usual convention about replacing (𝐸^(ℎ,𝛼)_𝑞,𝑤)^𝑛 with 𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}.
Let 𝐹^(ℎ,𝛼)_𝑞,𝑤(𝑡,𝑥)=∑^∞_𝑛=0𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}(𝑥)𝑡^𝑛/𝑛!. Then we see that 𝐹^(ℎ,𝛼)_𝑞,𝑤(𝑡,𝑥)=[2]^𝑞∞_𝑚=0(−1)^𝑚𝑤^𝑚𝑞^𝑚ℎ𝑒^{[𝑥+𝑚]𝑞𝛼𝑡}.(2.7) In the special case, 𝑥=0, let 𝐹^(ℎ,𝛼)_𝑞,𝑤(𝑡,0)=𝐹^(ℎ,𝛼)_𝑞,𝑤(𝑡).
By (2.1), we get 𝐸^(ℎ,𝛼)_{𝑛,𝑞−1,𝑤−1}(1−𝑥)=(−1)^𝑛𝑤𝑞^{𝛼𝑛+ℎ−1}𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}(𝑥).(2.8) From (2.3) and (2.7), we note that 𝑤𝑞^ℎ𝐹^(ℎ,𝛼)_𝑞,𝑤(𝑡,1)+𝐹^(ℎ,𝛼)_𝑞,𝑤(𝑡)=[2]_𝑞.(2.9) By (2.9), we get the following recurrence formula: 𝐸^(ℎ,𝛼)_0,𝑞,𝑤=[2]_𝑞1+𝑞^ℎ𝑤,𝑞^ℎ𝑤𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}(1)+𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}=0if𝑛>0.(2.10)

By (2.10) and Theorem 2.1, we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℤ₊ and 𝑤∈𝑇_𝑝, we have 𝐸^(ℎ,𝛼)_0,𝑞,𝑤=[2]_𝑞1+𝑞^ℎ𝑤,𝑞^ℎ𝑤𝑞^𝛼𝐸^(ℎ,𝛼)_𝑞,𝑤+1^𝑛+𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}=0if𝑛>0,(2.11) with usual convention about replacing (𝐸^(ℎ,𝛼)_𝑞,𝑤)^𝑛 with 𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}.
By (2.4), Theorem 2.1, and Theorem 2.2, we have 𝑞^2ℎ𝑤²𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}(2)−[2]_𝑞1+𝑞^ℎ𝑤𝑞^2ℎ𝑤²−[2]_𝑞1+𝑞^ℎ𝑤𝑞^ℎ𝑤=𝑞^2ℎ𝑤^2𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝑞^𝛼𝑙𝑞^𝛼𝐸^(ℎ,𝛼)_𝑞,𝑤+1^𝑙−[2]_𝑞1+𝑞^ℎ𝑤𝑞^2ℎ𝑤²−[2]_𝑞1+𝑞^ℎ𝑤𝑞^ℎ𝑤=𝑞^2ℎ𝑤^2𝑛_𝑙=1⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝑞^𝛼𝑙𝑞^𝛼𝐸^(ℎ,𝛼)_𝑞,𝑤+1^𝑙−[2]_𝑞1+𝑞^ℎ𝑤𝑞^ℎ𝑤=−𝑞^ℎ𝑤^𝑛_𝑙=1⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝑞^𝛼𝑙𝐸^(ℎ,𝛼)_{𝑙,𝑞,𝑤}−[2]_𝑞1+𝑞^ℎ𝑤𝑞^ℎ𝑤=−𝑞^ℎ𝑤^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝑞^𝛼𝑙𝐸^(ℎ,𝛼)_{𝑙,𝑞,𝑤}=−𝑞^ℎ𝑤𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}(1)=𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}if𝑛>0.(2.12)

Therefore, we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℕ, we have 𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}(2)=1𝑞^2ℎ𝑤²𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}+[2]_𝑞1+𝑞^ℎ𝑤+1𝑞^ℎ𝑤[2]_𝑞1+𝑞^ℎ𝑤.(2.13)
By (2.8), we see that 𝑞^ℎ−1𝑤_ℤ𝑝[1−𝑥]^𝑛_𝑞−𝛼𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=(−1)^𝑛𝑞^{𝛼𝑛+ℎ−1}𝑤_ℤ𝑝[𝑥−1]^𝑛_𝑞𝛼𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=(−1)^𝑛𝑞^{𝛼𝑛+ℎ−1}𝑤𝐸^(ℎ,𝛼)_{𝑛,𝑞,𝑤}(−1)=𝐸^(ℎ,𝛼)_{𝑛,𝑞−1,𝑤−1}(2).(2.14)

Therefore, we obtain the following theorem.

Theorem 2.4. For 𝑛∈ℤ₊, we have 𝑞^ℎ−1𝑤_ℤ𝑝[1−𝑥]^𝑛_𝑞−𝛼𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=𝐸^(ℎ,𝛼)_{𝑛,𝑞−1,𝑤−1}(2).(2.15) Let 𝑛∈ℕ. By Theorems 2.3 and 2.4, we get 𝑞^ℎ−1𝑤_ℤ𝑝[1−𝑥]^𝑛_𝑞−𝛼𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=𝑞^2ℎ𝑤²𝐸^(ℎ,𝛼)_{𝑛,𝑞−1,𝑤−1}+𝑞^ℎ−1𝑤[2]_𝑞1+𝑞^ℎ𝑤+𝑞^2ℎ−1𝑤²[2]_𝑞1+𝑞^ℎ𝑤.(2.16)
From (2.16), we have _ℤ𝑝[1−𝑥]^𝑛_𝑞−1𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛,𝑞−1,𝑤−1}+[2]_𝑞1+𝑞^ℎ𝑤+𝑞^ℎ𝑤[2]_𝑞1+𝑞^ℎ𝑤.(2.17)

Therefore, we obtain the following corollary.

Corollary 2.5. For 𝑛∈ℕ, we have _ℤ𝑝[1−𝑥]^𝑛_𝑞−𝛼𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛,𝑞−1,𝑤−1}+[2]_𝑞.(2.18)
Kim defined the 𝑞-Bernstein polynomials with weight 𝛼 of degree 𝑛 as below.
For 𝑥∈ℤ_𝑝, the 𝑝-adic 𝑞-Bernstein polynomials with weight 𝛼 of degree 𝑛 are given by 𝐵^(𝛼)_𝑘,𝑛(𝑥,𝑞)=⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠[𝑥]^𝑘_𝑞𝛼[1−𝑥]^𝑛−𝑘_𝑞−𝛼,where𝑛,𝑘∈ℤ₊.(2.19) compare [5, 10, 22] By (2.19), we get the symmetry of 𝑞-Bernstein polynomials as follows: 𝐵^(𝛼)_𝑘,𝑛(𝑥,𝑞)=𝐵^(𝛼)_{𝑛−𝑘,𝑛}1−𝑥,𝑞⁻¹,(2.20) see [8]. Thus, by Corollary 2.5, (2.19), and (2.20), we see that _ℤ𝑝𝐵^(𝛼)_𝑘,𝑛(𝑥,𝑞)𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=_ℤ𝑝𝐵^(𝛼)_{𝑛−𝑘,𝑛}1−𝑥,𝑞⁻¹𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠^𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)^𝑘+𝑙_ℤ𝑝[1−𝑥]^𝑛−𝑙_𝑞−𝛼𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠^𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)^𝑘+𝑙𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛−𝑙,𝑞−1,𝑤−1}+[2]_𝑞.(2.21)
For 𝑛,𝑘∈ℤ₊ with 𝑛>𝑘, we have _ℤ𝑝𝐵^(𝛼)_𝑘,𝑛(𝑥,𝑞)𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠^𝑘_𝑙=0⎛⎜⎜⎝𝑘𝑙⎞⎟⎟⎠(−1)^𝑘+𝑙𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛−𝑙,𝑞−1,𝑤−1}+[2]_𝑞=⎧⎪⎨⎪⎩𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛,𝑞−1,𝑤−1}+[2]_𝑞if𝑘=0,𝑞^ℎ+1𝑤𝑛𝑘^𝑘_𝑙=0𝑘𝑙(−1)^𝑘+𝑙𝐸^(ℎ,𝛼)_{𝑛−𝑙,𝑞−1,𝑤−1}if𝑘>0.(2.22) Let us take the fermionic 𝑞-integral on ℤ_𝑝 for the 𝑞-Bernstein polynomials with weight 𝛼 of degree 𝑛 as follows: _ℤ𝑝𝐵^(𝛼)_𝑘,𝑛(𝑥,𝑞)𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠_ℤ𝑝[𝑥]^𝑘_𝑞[1−𝑥]^𝑛−𝑘_𝑞−𝛼𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠^𝑛−𝑘_𝑙=0⎛⎜⎜⎝𝑛−𝑘𝑙⎞⎟⎟⎠(−1)^𝑙𝐸^(ℎ,𝛼)_{𝑙+𝑘,𝑞,𝑤}.(2.23)

Therefore, by (2.22) and (2.23), we obtain the following theorem.

Theorem 2.6. Let 𝑛,𝑘∈ℤ₊ with 𝑛>𝑘. Then we have _ℤ𝑝𝐵^(𝛼)_𝑘,𝑛(𝑥,𝑞)𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎧⎪⎨⎪⎩𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛,𝑞−1,𝑤−1}+[2]_𝑞if𝑘=0,𝑞^ℎ+1𝑤𝑛𝑘^𝑘_𝑙=0𝑘𝑙(−1)^𝑘+𝑙𝐸^(ℎ,𝛼)_{𝑛−𝑙,𝑞−1,𝑤−1}if𝑘>0.(2.24) Moreover, ^𝑛−𝑘_𝑙=0⎛⎜⎜⎝𝑛−𝑘𝑙⎞⎟⎟⎠(−1)^𝑙E^(ℎ,𝛼)_{𝑙+𝑘,𝑞,𝑤}=⎧⎪⎨⎪⎩𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛,𝑞−1,𝑤−1}+[2]_𝑞if𝑘=0,𝑞^ℎ+1𝑤^𝑘_𝑙=0𝑘𝑙(−1)^𝑘+𝑙𝐸^(ℎ,𝛼)_{𝑛−𝑙,𝑞−1,𝑤−1}if𝑘>0.(2.25)
Let 𝑛₁,𝑛₂,𝑘∈ℤ₊ with 𝑛₁+𝑛₂>2𝑘. Then we get _ℤ𝑝𝐵^(𝛼)_𝑘,𝑛1(𝑥,𝑞)𝐵^(𝛼)_𝑘,𝑛2(𝑥,𝑞)𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛₁𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑛₂𝑘⎞⎟⎟⎠^2𝑘_𝑙=0⎛⎜⎜⎝2𝑘𝑙⎞⎟⎟⎠(−1)^𝑙+2𝑘_ℤ𝑝[1−𝑥]^{𝑛1+𝑛2−𝑙}_𝑞−𝛼𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛₁𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑛₂𝑘⎞⎟⎟⎠^2𝑘_𝑙=0⎛⎜⎜⎝2𝑘𝑙⎞⎟⎟⎠(−1)^𝑙+2𝑘𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛1+𝑛2−𝑙,𝑞−1,𝑤−1}+[2]_𝑞=⎧⎪⎨⎪⎩𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛1+𝑛2,𝑞−1,𝑤−1}+[2]_𝑞if𝑘=0,𝑛₁𝑘𝑛₂𝑘^2𝑘_𝑙=02𝑘𝑙(−1)^2𝑘+𝑙𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛1+𝑛2−𝑙,𝑞−1,𝑤−1}+[2]_𝑞if𝑘≠0.(2.26)

Therefore, by (2.26), we obtain the following theorem.

Theorem 2.7. For 𝑛₁,𝑛₂,𝑘∈ℤ₊ with 𝑛₁+𝑛₂>2𝑘, we have _ℤ𝑝𝐵^(𝛼)_𝑘,𝑛1(𝑥,𝑞)𝐵^(𝛼)_𝑘,𝑛2(𝑥,𝑞)𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎧⎪⎨⎪⎩𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛1+𝑛2,𝑞−1,𝑤−1}+[2]_𝑞if𝑘=0,𝑞^ℎ+1𝑤𝑛₁𝑘𝑛₂𝑘^2𝑘_𝑙=02𝑘𝑙(−1)^2𝑘+𝑙𝐸^(ℎ,𝛼)_{𝑛1+𝑛2−𝑙,𝑞−1,𝑤−1}if𝑘≠0.(2.27)
From the binomial theorem, we can derive the following equation: _ℤ𝑝𝐵^(𝛼)_𝑘,𝑛1(𝑥,𝑞)𝐵^(𝛼)_𝑘,𝑛2(𝑥,𝑞)𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛₁𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑛₂𝑘⎞⎟⎟⎠^{𝑛1+𝑛2−2𝑘}_𝑙=0(−1)^𝑙⎛⎜⎜⎝𝑛₁+𝑛₂−2𝑘𝑙⎞⎟⎟⎠_ℤ𝑝[𝑥]^2𝑘+𝑙_𝑞𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛₁𝑘⎞⎟⎟⎠⎛⎜⎜⎝𝑛₂𝑘⎞⎟⎟⎠^{𝑛1+𝑛2−2𝑘}_𝑙=0(−1)^𝑙⎛⎜⎜⎝𝑛₁+𝑛₂−2𝑘𝑙⎞⎟⎟⎠𝐸^(ℎ,𝛼)_{2𝑘+𝑙,𝑞,𝑤}.(2.28)

Thus, by (2.28) and Theorem 2.7, we obtain the following corollary.

Corollary 2.8. Let 𝑛₁,𝑛₂,𝑘∈ℤ₊ with 𝑛₁+𝑛₂>2𝑘. Then we have ^𝑛1+^{𝑛2−2𝑘}_𝑙=0(−1)^𝑙⎛⎜⎜⎝𝑛₁+𝑛₂−2𝑘𝑙⎞⎟⎟⎠𝐸^(ℎ,𝛼)_{2𝑘+𝑙,𝑞}=⎧⎪⎨⎪⎩𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛1+𝑛2,𝑞−𝛼,𝑤−1}+[2]_𝑞if𝑘=0,𝑞^ℎ+1𝑤^2𝑘_𝑙=02𝑘𝑙(−1)^2𝑘+𝑙𝐸^(ℎ,𝛼)_{𝑛1+𝑛2−𝑙,𝑞−𝛼,𝑤−1}if𝑘>0.(2.29)
For 𝑥∈ℤ_𝑝 and 𝑠∈ℕ with 𝑠≥2, let 𝑛₁,𝑛₂,…,𝑛_𝑠,𝑘∈ℤ₊ with 𝑛₁+⋯+𝑛_𝑠>𝑠𝑘. Then we take the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 for the 𝑞-Bernstein polynomials with weight 𝛼 of degree 𝑛 as follows: _ℤ𝑝𝐵^(𝛼)_𝑘,𝑛1(𝑥,𝑞)⋯𝐵^(𝛼)_{𝑘,𝑛𝑠}(𝑥,𝑞)_{𝑠-times}𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛₁𝑘⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑛_𝑠𝑘⎞⎟⎟⎠_ℤ𝑝[𝑥]^𝑠𝑘_𝑞[1−𝑥]^{𝑛1+⋯+𝑛𝑠−𝑠𝑘}_𝑞−𝛼𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛₁𝑘⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑛_s𝑘⎞⎟⎟⎠^𝑠𝑘_𝑙=0⎛⎜⎜⎝𝑠𝑘𝑙⎞⎟⎟⎠(−1)^{𝑙+𝑠𝑘}_ℤ𝑝[1−𝑥]^{𝑛1+⋯+𝑛𝑠−𝑙}_𝑞−𝛼𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛₁𝑘⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑛_𝑠𝑘⎞⎟⎟⎠^𝑠𝑘_𝑙=0⎛⎜⎜⎝𝑠𝑘𝑙⎞⎟⎟⎠(−1)^{𝑙+𝑠𝑘}𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛1+⋯+𝑛𝑠−𝑙,𝑞−1,𝑤−1}+[2]_𝑞=⎧⎪⎨⎪⎩𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛1+⋯+𝑛𝑠,𝑞−1,𝑤−1}+[2]_𝑞if𝑘=0,𝑞^ℎ+1𝑤𝑛₁𝑘⋯𝑛_𝑠𝑘^𝑠𝑘_𝑙=0𝑠𝑘𝑙(−1)^{𝑙+𝑠𝑘}𝐸^(ℎ,𝛼)_{𝑛1+⋯+𝑛𝑠−𝑙,𝑞−1,𝑤−1}if𝑘>0.(2.30)

Therefore, by (2.30), we obtain the following theorem.

Theorem 2.9. For 𝑠∈ℕ with 𝑠≥2, let 𝑛₁,𝑛₂,…,𝑛_𝑠,𝑘∈ℤ₊ with 𝑛₁+⋯+𝑛_𝑠>𝑠𝑘. Then we get _ℤ𝑝𝐵^(𝛼)_𝑘,𝑛1(𝑥,𝑞)⋯𝐵^(𝛼)_{𝑘,𝑛𝑠}(𝑥,𝑞)_{𝑠-times}𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎧⎪⎨⎪⎩𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛1+⋯+𝑛𝑠,𝑞−1,𝑤−1}+[2]_𝑞if𝑘=0,𝑞^ℎ+1𝑤𝑛₁𝑘⋯𝑛_𝑠𝑘^𝑠𝑘_𝑙=0𝑠𝑘𝑙(−1)^{𝑙+𝑠𝑘}𝐸^(ℎ,𝛼)_{𝑛1+⋯+𝑛𝑠−𝑙,𝑞−1,𝑤−1}if𝑘>0.(2.31)
By the definition of 𝑞-Bernstein polynomials with weight 𝛼 and the binomial theorem, we easily get _ℤ𝑝𝐵^(𝛼)_𝑘,𝑛1(𝑥,𝑞)⋯𝐵^(𝛼)_{𝑘,𝑛𝑠}(𝑥,𝑞)_{𝑠-times}𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛₁𝑘⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑛_𝑠𝑘⎞⎟⎟⎠^{𝑛1+⋯+𝑛𝑠−𝑠𝑘}_𝑙=0(−1)^𝑙⎛⎜⎜⎝𝑛₁+⋯+𝑛_𝑠−𝑠𝑘𝑙⎞⎟⎟⎠_ℤ𝑝[𝑥]^{𝑠𝑘+𝑙}_𝑞𝑞^{(ℎ−1)𝑥}𝑤^𝑥𝑑𝜇_−𝑞(𝑥)=⎛⎜⎜⎝𝑛₁𝑘⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑛_𝑠𝑘⎞⎟⎟⎠^{𝑛1+⋯+𝑛𝑠−𝑠𝑘}_𝑙=0(−1)^𝑙⎛⎜⎜⎝𝑛₁+⋯+𝑛_𝑠−𝑠𝑘𝑙⎞⎟⎟⎠𝐸^(ℎ,𝛼)_{𝑠𝑘+𝑙,𝑞,𝑤}.(2.32)

Therefore, we have the following corollary.

Corollary 2.10. For 𝑤∈𝑇_𝑝,𝑠∈ℕ with 𝑠≥2, let 𝑛₁,𝑛₂,…,𝑛_𝑠,𝑘∈ℤ₊ with 𝑛₁+⋯+𝑛_𝑠>𝑠𝑘. Then we have ^𝑛1+⋯^{+n𝑠−𝑠𝑘}_𝑙=0(−1)^𝑙⎛⎜⎜⎝𝑛₁+⋯+𝑛_𝑠−𝑠𝑘𝑙⎞⎟⎟⎠𝐸^(ℎ,𝛼)_{𝑠𝑘+𝑙,𝑞,𝑤}=⎧⎪⎨⎪⎩𝑞^ℎ+1𝑤𝐸^(ℎ,𝛼)_{𝑛1+⋯+𝑛𝑠,𝑞−1,𝑤−1}+[2]_𝑞if𝑘=0,𝑞^ℎ+1𝑤^𝑠𝑘_𝑙=0𝑠𝑘𝑙(−1)^{𝑙+𝑠𝑘}𝐸^(ℎ,𝛼)_{𝑛1+⋯+𝑛𝑠−𝑙,𝑞−1,𝑤−1}if𝑘>0.(2.33)