Some Identities on the Twisted (ℎ,𝑞)-Genocchi Numbers and Polynomials Associated with 𝑞-Bernstein Polynomials

Abstract

We give some interesting identities on the twisted (ℎ,𝑞)-Genocchi numbers and polynomials associated with 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 integer, ℚ_𝑝 denotes the ring 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={𝜁∣𝜁^𝑝𝑛=1} be the cyclic group of order 𝑝^𝑛 and letT_𝑝=_𝑛≥1𝐶_𝑝𝑛=lim_𝑛→∞𝐶_𝑝𝑛=𝐶_𝑝∞. (1.1) The 𝑝-adic absolute value is defined by |𝑥|=1/𝑝^𝑟, where 𝑥=𝑝^𝑟(𝑠/𝑡) (𝑟∈ℚ and 𝑠,𝑡∈ℤ with (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 [1–15]). Note that lim_𝑞→1[𝑥]_𝑞=𝑥. Let 𝑈𝐷(ℤ_𝑝) be the space of uniformly differentiable function on ℤ_𝑝. For 𝑓∈𝑈𝐷(ℤ_𝑝), Kim defined the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 as follows:𝐼_−𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁𝑝_−𝑞^𝑁−1_𝑥=0𝑓(𝑥)−𝑞^𝑥 (1.3) (see [2–6, 8–15]). From (1.3), we note that𝑞^𝑛𝐼_−𝑞𝑓_𝑛=(−1)^𝑛𝐼_−𝑞𝑓+[2]_𝑞^𝑛−1_ℓ=0(−1)^{𝑛−1−ℓ}𝑞^ℓ𝑓(ℓ) (1.4) (see [4–6, 8–12]), where 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛) for 𝑛∈ℕ. For 𝑘,𝑛∈ℤ₊ and 𝑥∈[0,1], Kim defined the 𝑞-Bernstein polynomials of the degree 𝑛 as follows:𝐵_𝑘,𝑛=𝑛𝑘𝑥,𝑞[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘₋₁, (1.5) (see [13–15]). For ℎ∈ℤ and 𝜁∈𝑇_𝑝, let us consider the twisted (ℎ,𝑞)-Genocchi polynomials as follows:𝑡_ℤ𝑝𝑒^{[𝑥+𝑦]𝑞𝑡}𝜁^𝑦𝑞^{(ℎ−1)𝑦}𝑑𝜇_−𝑞𝑦=^∞_𝑛=0𝐺^(ℎ)_{𝑛,𝑞,𝜁}(𝑥)𝑡^𝑛.𝑛! (1.6) Then, 𝐺^(ℎ)_{𝑛,𝑞,𝜁}(𝑥) is called 𝑛th twisted (ℎ,𝑞)-Genocchi polynomials.

In the special case, 𝑥=0 and 𝐺^(ℎ)_{𝑛,𝑞,𝜁}(0)=𝐺^(ℎ)_{𝑛,𝑞,𝜁} are called the 𝑛th twisted (ℎ,𝑞)-Genocchi numbers.

In this paper, we give the fermionic 𝑝-adic integral representation of 𝑞-Bernstein polynomial, which are defined by Kim [13], associated with twisted (ℎ,𝑞)-Genocchi numbers and polynomials. And we construct some interesting properties of 𝑞-Bernstein polynomials associated with twisted (ℎ,𝑞)-Genocchi numbers and polynomials.

2. On the Twisted (ℎ,𝑞)-Genocchi Numbers and Polynomials

From (1.6), we note that𝐺^(ℎ)_{𝑛+1,𝑞,𝜁}(𝑥)=𝑛+1_ℤ𝑝𝑥+𝑦^𝑛_𝑞𝜁^𝑦𝑞^{(ℎ−1)𝑦}𝑑𝜇_−𝑞𝑦=_ℤ𝑝[𝑥]_𝑞+𝑞^𝑥𝑦_𝑞^𝑛𝜁^𝑦𝑞^{(ℎ−1)𝑦}𝑑𝜇_−𝑞𝑦=^𝑛_ℓ=0𝑛ℓ[𝑥]_𝑞^𝑛−ℓ𝑞^ℓ𝑥_ℤ𝑝𝑦^ℓ_𝑞𝜁^𝑦𝑞^{(ℎ−1)𝑦}𝑑𝜇_−𝑞𝑦=^𝑛_ℓ=0𝑛ℓ[𝑥]_𝑞^𝑛−ℓ𝑞^ℓ𝑥𝐺^(ℎ)_{ℓ+1,𝑞,𝜁}.ℓ+1 (2.1) We also have𝐺^(ℎ)_{𝑛,𝑞,𝜁}(𝑥)=𝑞^𝑛−𝑥_ℓ=0𝑛ℓ[𝑥]_𝑞^𝑛−ℓ𝑞^ℓ𝑥𝐺^(ℎ)_{ℓ,𝑞,𝜁}. (2.2) Therefore, we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊ and 𝜁∈T_𝑝, one has 𝐺^(ℎ)_{𝑛,𝑞,𝜁}(𝑥)=𝑞^−𝑥[𝑥]_𝑞+𝑞^𝑥𝐺^(ℎ)_𝑞,𝜁^𝑛 (2.3) with usual convention about replacing (𝐺^(ℎ)_𝑞,𝜁)^𝑛 by 𝐺^ℎ_{𝑛,𝑞,𝜁}.
By (1.6) and (2.1) one gets 𝐺^(ℎ)_{𝑛+1,𝑞−1,𝜁−1}(1−𝑥)=𝑛+1_ℤ𝑝1−𝑥+𝑦^𝑛_𝑞−1𝜁^−𝑦𝑞^{−(ℎ−1)𝑦}𝑑𝜇_−𝑞−1𝑦=[2]_𝑞1−𝑞^−1𝑛𝑛_ℓ=0𝑛ℓ(−1)^𝑛𝑞^ℎ−1𝜁𝑞^ℓ𝑥1+𝑞^ℎ+ℓ𝜁=(−1)^𝑛𝑞^𝑛+ℎ−1𝜁[2]_𝑞1−𝑞^𝑛𝑛_ℓ=0𝑛ℓ(−1)^ℓ𝑞^ℓ𝑥1+𝑞^ℎ+ℓ𝜁=(−1)^𝑛𝜁𝑞^𝑛+ℎ−1𝐺^(ℎ)_{𝑛+1,𝑞,𝜁}(𝑥).𝑛+1 (2.4)

Therefore, we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℤ₊ and 𝜁∈T_𝑝, one has 𝐺^(ℎ)_{𝑛,𝑞−1,𝜁−1}(1−𝑥)=(−1)^𝑛−1𝜁𝑞^𝑛+ℎ−2𝐺^(ℎ)_{𝑛,𝑞,𝜁}. (2.5)
From (1.5), one gets the following recurrence formula: 𝑞^ℎ𝜁𝐺^(ℎ)_{𝑛,𝑞,𝜁}(1)+𝐺^(ℎ)_{𝑛,𝑞,𝜁}=[2]_𝑞𝑖𝑓𝑛=1,0𝑖𝑓𝑛>1. (2.6)

Therefore, we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℤ₊ and 𝜁∈T_𝑝, one has 𝐺_0,𝑞,𝜁=0,𝑞^ℎ−1𝜁𝑞𝐺^(ℎ)_𝑞,𝜁+1^𝑛+𝐺^(ℎ)_{𝑛,𝑞,𝜁}=[2]_𝑞𝑖𝑓𝑛=1,0𝑖𝑓𝑛>1 (2.7) with usual convention about replacing (𝐺^(ℎ)_𝑞,𝜁)^𝑛 by 𝐺^ℎ_{𝑛,𝑞,𝜁}.

From Theorem 2.3, we note that𝑞^2ℎ𝜁²𝐺^(ℎ)_{𝑛,𝑞,𝜁}(2)−𝑞^ℎ𝜁𝑛[2]_𝑞=−𝑞^ℎ−1𝜁^𝑛_ℓ=0𝑛ℓ𝑞^ℓ𝐺^(ℎ)_{ℓ,𝑞,𝜁}=−𝑞^ℎ−1𝜁𝑞𝐺^(ℎ)_𝑞,𝜁+1^𝑛=𝐺^(ℎ)_{𝑛,𝑞,𝜁}if𝑛>1. (2.8)

Therefore, we obtain the following theorem.

Theorem 2.4. For 𝑛∈ℤ₊ and 𝜁∈T_𝑝, one has 𝑞^2ℎ𝜁²𝐺^(ℎ)_{𝑛,𝑞,𝜁}(2)=𝐺^(ℎ)_{𝑛,𝑞,𝜁}+𝑛𝑞^ℎ𝜁[2]_𝑞. (2.9)

Remark 2.5. We note that Theorem 2.4 also can be proved by using fermionic integral equation (1.4) in case of 𝑛=2.

By (2.4) and Theorem 2.2, we get𝐺^(ℎ)_{𝑛+1,𝑞−1,𝜁−1}(2)𝑛+1=(−1)^𝑛𝑞^𝑛+ℎ−1𝜁𝐺^(ℎ)_{𝑛+1,𝑞,𝜁}(−1)𝑛+1=(−1)^𝑛𝑞^𝑛+ℎ−1𝜁_ℤ𝑝[𝑥−1]^𝑛_𝑞𝜁^𝑥𝑞^{(ℎ−1)𝑥}𝑑𝜇_−𝑞(𝑥)=𝑞^ℎ−1𝜁_ℤ𝑝[1−𝑥]^𝑛_𝑞−1𝜁^𝑥𝑞^{(ℎ−1)𝑥}𝑑𝜇_−𝑞(𝑥). (2.10)

Therefore, we obtain the following theorem.

Theorem 2.6. For 𝑛∈ℤ₊ and 𝜁∈T_𝑝, one has (𝑛+1)𝑞^ℎ−1𝜁_ℤ𝑝[1−𝑥]^𝑛_𝑞−1𝜁^𝑥𝑞^{(ℎ−1)𝑥}𝑑𝜇_−𝑞(𝑥)=𝐺^(ℎ)_{𝑛+1,𝑞−1,𝜁−1}(2). (2.11)

Let 𝑛∈ℕ. By Theorems 2.4 and 2.6, we get(𝑛+1)𝑞^ℎ−1𝜁_ℤ𝑝[1−𝑥]^𝑛_𝑞−1𝜁^𝑥𝑞^{(ℎ−1)𝑥}𝑑𝜇_−𝑞(𝑥)=𝑞^2ℎ𝜁²𝐺^(ℎ)_{𝑛+1,𝑞−1,𝜁−1}+(𝑛+1)𝑞^ℎ−1𝜁[2]_𝑞. (2.12) Therefore, we obtain the following corollary.

Corollary 2.7. For 𝑛∈ℤ₊ and 𝜁∈T_𝑝, one has _ℤ𝑝[1−𝑥]^𝑛_𝑞−1𝜁^𝑥𝑞^{(ℎ−1)𝑥}𝑑𝜇_−𝑞(𝑥)=𝑞^ℎ+1𝜁𝐺^(ℎ)_{𝑛+1,𝑞−1,𝜁−1}+[2]𝑛+1^𝑞. (2.13)
By (1.5), we get the symmetry of 𝑞-Bernstein polynomials as follows: 𝐵_𝑘,𝑛𝑥,𝑞=𝐵_{𝑛−𝑘,𝑛}1−𝑥,𝑞⁻¹ (2.14) (see [11]).

Thus, by Corollary 2.7 and (2.14), we get_ℤ𝑝𝐵_𝑘,𝑛𝑥,𝑞𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇_−𝑞(𝑥)=_ℤ𝑝𝐵_{𝑛−𝑘,𝑛}1−𝑥,𝑞⁻¹𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇_−𝑞(𝑥)=𝑛𝑘^𝑘ℓ=0𝑘ℓ(−1)^𝑘−ℓ_ℤ𝑝[1−𝑥]_𝑞^𝑛−ℓ₋₁𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇₋₁(𝑥)=𝑛𝑘^𝑘ℓ=0𝑘ℓ(−1)^𝑘−ℓ𝑞^ℎ+1𝜁𝐺^(ℎ)_{𝑛−ℓ+1,𝑞−1,𝜁−1}+[2]𝑛−ℓ+1^𝑞=𝑞^ℎ+1𝜁𝐺^(ℎ)_{𝑛+1,𝑞−1,𝜁−1}+[2]𝑛+1^𝑞if𝑞𝑘=0,_ℎ+1𝜁𝑛𝑘^𝑘_ℓ=0𝑘ℓ(−1)^𝑘−ℓ𝐺^(ℎ)_{𝑛−ℓ+1,𝑞−1,𝜁−1}𝑛−ℓ+1if𝑘>0. (2.15)

From (2.15), we have the following theorem.

Theorem 2.8. For 𝑛∈ℤ₊ and 𝜁∈T_𝑝, one has _ℤ𝑝𝐵_𝑘,𝑛𝑥,𝑞𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇_−𝑞(𝑥)=𝑞^ℎ+1𝜁𝐺^(ℎ)_{𝑛+1,𝑞−1,𝜁−1}+[2]𝑛+1^𝑞if𝑞𝑘=0,_ℎ+1𝜁𝑛𝑘^𝑘_ℓ=0𝑘ℓ(−1)^𝑘−ℓ𝐺^(ℎ)_{𝑛−ℓ+1,𝑞−1,𝜁−1}𝑛−ℓ+1if𝑘>0. (2.16)
For 𝑛,𝑘∈ℤ₊ with 𝑛>𝑘, fermionic 𝑝-adic invariant integral for multiplication of two 𝑞-Bernstein polynomials on ℤ_𝑝 can be given by the following: _ℤ𝑝𝐵_𝑘,𝑛𝑥,𝑞𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇_−𝑞(𝑥)=_ℤ𝑝𝑛𝑘[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘₋₁𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇_−𝑞(𝑥)=_ℤ𝑝𝑛𝑘[𝑥]^𝑘_𝑞1−[𝑥]_𝑞^𝑛−𝑘𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇₋₁(𝑥)=𝑛𝑘^{𝑛−𝑘ℓ=0}ℓ()𝑛−𝑘−1^ℓ_ℤ𝑝[𝑥]_𝑞^𝑘+ℓ𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇₋₁(𝑥). (2.17)

From Theorem 2.8 and (2.17), we have the following corollary.

Corollary 2.9. For 𝑛∈ℤ₊ and 𝜁∈T_𝑝, one has ^{𝑛−𝑘ℓ=0}ℓ()𝑛−𝑘−1^ℓ𝐺^(ℎ)_{𝑘+ℓ+1,𝑞,𝜁}=𝑞𝑘+ℓ+1^ℎ+1𝜁𝐺^(ℎ)_{𝑛+1,𝑞−1,𝜁−1}+[2]𝑛+1^𝑞if𝑞𝑘=0,_ℎ+1𝜁^𝑘_ℓ=0𝑘ℓ(−1)^𝑘−ℓ𝐺^(ℎ)_{𝑛−ℓ+1,𝑞−1,𝜁−1}𝑛−ℓ+1if𝑘>0. (2.18)
Let 𝑛₁,𝑛₂,𝑘∈ℤ₊ with 𝑛₁+𝑛₂>2𝑘. Then we get _ℤ𝑝𝐵_𝑘,𝑛1𝑥,𝑞𝐵_𝑘,𝑛2𝑥,𝑞𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇_−𝑞(𝑥)=𝑛₁𝑘𝑛₂𝑘^{ℤ𝑝2𝑘ℓ=0}ℓ()2𝑘−1^2𝑘−ℓ[1−𝑥]^𝑛1+𝑛2_𝑞^−ℓ₋₁𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇_−𝑞(𝑥)=𝑛₁𝑘𝑛₂𝑘^2𝑘ℓ=0ℓ()2𝑘−1^2𝑘−ℓ𝐺_𝑛^(ℎ)_{1+𝑛2−ℓ+1,𝑞−1,𝜁−1}𝑛₁+𝑛₂𝑞−ℓ+1^ℎ+1𝜁+[2]_𝑞. (2.19)

From (2.19), we have the following theorem.

Theorem 2.10. For 𝑛∈ℤ₊ and 𝜁∈T_𝑝, one has _ℤ𝑝𝐵_𝑘,𝑛1𝑥,𝑞𝐵_𝑘,𝑛2𝑥,𝑞𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇_−𝑞(𝑥)=𝑞^ℎ+1𝜁𝐺_𝑛^(ℎ)_{1+𝑛2+1,𝑞−1,𝜁−1}𝑛₁+𝑛₂+[2]+1^𝑞if𝑛𝑘=0,₁𝑘𝑛₂𝑘^2𝑘_ℓ=0ℓ()2𝑘−1^2𝑘−ℓ𝐺_𝑛^(ℎ)_{1+𝑛2−ℓ+1,𝑞−1,𝜁−1}𝑛₁+𝑛₂−ℓ+1if𝑘>0. (2.20)
Let 𝑛₁,𝑛₂,𝑘∈ℤ₊ with 𝑛₁+𝑛₂>2𝑘, fermionic 𝑝-adic invariant integral for multiplication of two 𝑞-Bernstein polynomials on ℤ_𝑝 can be given by the following: _ℤ𝑝𝐵_𝑘,𝑛1𝑥,𝑞𝐵_𝑘,𝑛2𝑥,𝑞𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇_−𝑞(𝑥)=𝑛₁𝑘𝑛₂𝑘^{ℤ𝑝𝑛1+𝑛2−2𝑘ℓ=0}(−1)^ℓ𝑛₁+𝑛₂ℓ[𝑥]−2𝑘_{𝑞2𝑘+ℓ}𝑞^{(ℎ−1)𝑥}𝜁^𝑥𝑑𝜇_−𝑞(𝑥)=𝑛₁𝑘𝑛₂𝑘^{𝑛1+𝑛2−2𝑘ℓ=0}(−1)^ℓ𝑛₁+𝑛₂ℓ−2𝑘𝐺^(ℎ)_{2𝑘+ℓ+1,𝑞,𝜁}.2𝑘+ℓ+1 (2.21)

From Theorem 2.10 and (2.21), we have the following corollary.

Corollary 2.11. For 𝑛₁,𝑛₂,𝑘∈ℤ₊ and 𝑛₁+𝑛₂>2𝑘, one has ^{𝑛1+𝑛2−2𝑘}_ℓ=0𝑛₁+𝑛₂ℓ()−2𝑘−1^ℓ𝐺^(ℎ)_{2𝑘+ℓ+1,𝑞,𝜁}=𝑞2𝑘+ℓ+1_ℎ+1𝜁𝐺_𝑛^(ℎ)_{1+𝑛2+1,𝑞−1,𝜁−1}𝑛₁+𝑛₂+[2]+1^𝑞if𝑘=0,_{𝑛1+𝑛2−2𝑘ℓ=0}𝑛₁+𝑛₂ℓ()−2𝑘−1^2𝑘−ℓ𝐺_𝑛^(ℎ)_{1+𝑛2−ℓ+1,𝑞−1,𝜁−1}𝑛₁+𝑛₂−ℓ+1if𝑘>0. (2.22)

Acknowledgment

The authors would like to thank the anonymous referee for his/her excellent detail comments and suggestion.