Some New Identities on the 𝑞-Genocchi Numbers and Polynomials with Weight 𝜶

Abstract

We construct a new type of 𝑞-Genocchi numbers and polynomials with weight 𝛼. From these 𝑞-Genocchi numbers and polynomials with weight 𝛼, we establish some interesting identities and relations.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper, ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 will, respectively denote the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, and the completion of the algebraic closure of ℚ_𝑝. Let ℕ be the set of natural numbers and ℤ₊=ℕ⋃{0}. Let 𝑣_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{−𝑣𝑝(𝑝)}=1/𝑝. When one talks of 𝑞-extension, 𝑞 is variously considered as an indeterminate, a complex 𝑞∈ℂ, or a 𝑝-adic number 𝑞. In this paper, we assume that 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1. As a definition of 𝑞-numbers, we use the notation of 𝑞-number of [𝑥]_𝑞=1−𝑞^𝑥1−𝑞,[𝑥]_−𝑞=1−(−𝑞)^𝑥1+𝑞(1.1)(cf. [1–11]). Note that lim_𝑞→1[𝑥]_𝑞=𝑥. Let 𝐶(ℤ_𝑝) be the space of continuous functions on ℤ_𝑝. For 𝑓∈𝐶(ℤ_𝑝), the 𝑝-adic invariant integral on ℤ_𝑝 is defined by Kim [1, 3], 𝐼_−𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁_−𝑞^𝑝𝑁−1_𝑥=0𝑓(𝑥)(−𝑞)^𝑥.(1.2) From (1.2), we have the well-known integral equation 𝑞^𝑛𝐼₋₁𝑓_𝑛+(−1)^𝑛−1𝐼_𝑞(𝑓)=[2]^𝑞𝑛−1_𝑙=0(−1)^𝑙𝑞^𝑙𝑓(𝑙)(1.3)(see [1, 3]), where 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛), (𝑛∈ℕ).

For 𝛼∈ℕ, in [11], the 𝑞-Genocchi polynomials with weight 𝛼 are introduced by 𝑡_ℤ𝑝𝑒^{[𝑥+𝑦]𝑞𝛼𝑡}𝑑𝜇_−𝑞(𝑦)=^∞_𝑛=0𝐺^(𝛼)_𝑛,𝑞(𝑥)𝑡^𝑛𝑛!.(1.4) By comparing the coefficients of both sides of (1.4), we have 𝐺^(𝛼)_0,𝑞(𝑥)=0,𝐺^(𝛼)_𝑛+1,𝑞(𝑥)(𝑛+1)=_ℤ𝑝[𝑥+𝑦]^𝑛_𝑞𝛼𝑑𝜇_−𝑞(𝑦),for𝑛∈ℕ.(1.5)

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

2. 𝑞-Genocchi Numbers and Polynomials with Weight 𝛼

In this section, we show some new identities on the 𝑞-Genocchi numbers and polynomials with weight 𝛼. And we establish the distribution relation for 𝑞-Genocchi polynomials with weight 𝛼.

From (1.5), we can easily see that 𝐺^(𝛼)_𝑛+1,𝑞(𝑥)𝑛+1=[2]_𝑞[𝛼]^𝑛_𝑞(1−𝑞)^𝑛𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)^𝑙𝑞^𝛼𝑙𝑥1+𝑞^𝛼𝑙+1.(2.1) From (1.5) and (2.1), we note that 𝐺^(𝛼)_𝑛+1,𝑞(𝑥)𝑛+1=_ℤ𝑝[𝑥+𝑦]^𝑛_𝑞𝛼𝑑𝜇₋₁(𝑦)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠[𝑥]^𝑛−𝑙_𝑞𝛼𝑞^𝛼𝑙𝑥_ℤ𝑝[𝑦]^𝑙_𝑞𝛼𝑑𝜇_−𝑞(𝑦)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠[𝑥]^𝑛−𝑙_𝑞𝛼𝑞^𝛼𝑙𝑥𝐺^(𝛼)_𝑙+1,𝑞𝑙+1.(2.2) Note that (1/(𝑙+1))(^𝑛_𝑙)=(1/(𝑛+1))^𝑛+1_𝑙+1 and from (2.2), we have the relation of polynomials and numbers, 𝐺^(𝛼)_𝑛+1,𝑞(𝑥)𝑛+1=1(𝑛+1)^𝑛_𝑙=0⎛⎜⎜⎝𝑛+1𝑙+1⎞⎟⎟⎠[𝑥]^{𝑛+1−𝑙}_𝑞𝛼𝑞^𝛼𝑙𝑥𝐺^(𝛼)_𝑙+1,𝑞=1(𝑛+1)𝑞^{𝛼𝑥𝑛+1}_𝑙=0⎛⎜⎜⎝𝑛+1𝑙+1⎞⎟⎟⎠[𝑥]^{𝑛+1−𝑙}_𝑞𝛼𝑞^{𝛼(𝑙+1)𝑥}𝐺^(𝛼)_𝑙+1,𝑞=1(𝑛+1)𝑞^𝛼𝑥[𝑥]_𝑞𝛼+𝑞^𝛼𝑙𝑥𝐺^(𝛼)_𝑞^𝑛+1,(2.3) with the usual convention of replacing (𝐺^(𝛼)_𝑞)^𝑛 by (𝐺^(𝛼)_𝑛,𝑞).

Thus, by (2.3), we have a theorem.

Theorem 2.1. For 𝛼∈ℕ and 𝑛∈ℤ₊, one has 𝑞^𝛼𝑥𝐺^(𝛼)_𝑛+1,𝑞(𝑥)=(𝑛+1)𝑞^𝛼𝑥[𝑥]_𝑞𝛼+𝑞^𝛼𝑥𝐺^(𝛼)_𝑞^𝑛+1=^𝑛+1_𝑙=0⎛⎜⎜⎝𝑛+1𝑙⎞⎟⎟⎠[𝑥]^{𝑛+1−𝑙}_𝑞𝛼𝑞^𝛼𝑥𝐺^(𝛼)_𝑙,𝑞.(2.4)
In (1.3), if we take 𝑛=1, 𝑞𝐼₋₁𝑓₁+𝐼₋₁(𝑓)=[2]_𝑞.(2.5)
We apply 𝑓(𝑥)=𝑒^{[𝑥]𝑞𝛼𝑡} with (1.5), and we have the following: [2]_𝑞=^∞_𝑛=0⎛⎜⎝𝑞_ℤ𝑝[𝑥+1]^𝑛_𝑞𝛼𝑑𝜇_−𝑞(𝑥)+_ℤ𝑝[𝑥]^𝑛_𝑞𝛼𝑑𝜇_−𝑞(𝑥)⎞⎟⎠𝑡^𝑛𝑛!=^∞_𝑛=0⎛⎜⎜⎝𝑞𝐺^(𝛼)_𝑛+1,𝑞(1)𝑛+1+𝐺^(𝛼)_𝑛+1,𝑞𝑛+1⎞⎟⎟⎠𝑡^𝑛𝑛!.(2.6) By comparing the coefficients on both the sides in (2.6), we get 𝑞𝐺^(𝛼)_𝑛+1,𝑞(1)𝑛+1+𝐺^(𝛼)_𝑛+1,𝑞𝑛+1=[2]_𝑞if𝑛=0,0if𝑛>0.(2.7) From (2.2) and (2.7), we can derive the following: 𝐺^(𝛼)_1,𝑞(1)=1,𝑞^1−𝛼𝑞^𝛼𝐺^(𝛼)_𝑞+1^𝑛+𝐺^(𝛼)_𝑛,𝑞=0if𝑛∈ℕ,(2.8) with the usual convention of replacing (𝐺^(𝛼)_𝑞)^𝑛 by 𝐺^(𝛼)_𝑛,𝑞.
For a fixed odd positive integer 𝑑 with (𝑝,𝑑)=1, we set 𝑋=𝑋_𝑑=lim^⟵_𝑁ℤ𝑑𝑝^𝑁ℤ,𝑋₁=ℤ_𝑝,𝑋^∗=_0<𝑎<_{𝑑𝑝,(𝑎,𝑝)=1}𝑎+𝑑𝑝ℤ_𝑝,𝑎+𝑑𝑝^𝑁ℤ_𝑝=𝑥∈𝑋∣𝑥≡𝑎mod𝑑𝑝^𝑁,(2.9) where 𝑎∈ℤ satisfies the condition 0≤𝑎<𝑑𝑝^𝑁. For the distribution relation for the 𝑞-Genocchi polynomials with weight 𝛼, we consider the following: _ℤ𝑝[𝑥+𝑦]^𝑛_𝑞𝛼𝑑𝜇_−𝑞(𝑦)=_ℤ𝑝[𝑛+𝑦]^𝑛_𝑞𝛼𝑥𝑑𝜇_−𝑞(𝑦)=[𝑑]^𝑛_𝑞𝛼[𝑑]_−𝑞^𝑑−1_𝑎=0(−1)^𝑎𝑞^𝑎_ℤ𝑝𝑥+𝑎𝑑+𝑦^𝑛_𝑞𝛼𝑑𝑑𝜇_−𝑞(𝑦).(2.10) By (1.5) and (2.10), we get a theorem.

Theorem 2.2. For 𝛼∈ℕ and 𝑛∈ℤ₊, 𝑑∈ℕ with 𝑑≡1(mod2), one has 𝐺^(𝛼)_𝑛+1,𝑞(𝑥)𝑛+1=[𝑑]^𝑛_𝑞𝛼[𝑑]_−𝑞^𝑑−1_𝑎=0(−1)^𝑎𝑞^𝑎𝐺^(𝛼)_{𝑛+1,𝑞𝑑}𝑥+𝑎𝑎.(2.11)

3. Higher-Order 𝑞-Genocchi Numbers and Polynomials with Weight 𝛼

In this section, we define higher-order 𝑞-Genocchi polynomials 𝐺^(𝛼)_𝑛+1,𝑞(ℎ,𝑘∣𝑥) and numbers 𝐺^(𝛼)_𝑛+1,𝑞(ℎ,𝑘) with weight 𝛼. We find an integral equation for higher-order 𝑞-Genocchi numbers with weight 𝛼. And we establish a combination property.

Let 𝛼∈ℤ and ℎ,𝑘∈ℤ₊, for 𝑛∈ℤ₊, then we define higher-order 𝑞-Genocchi polynomials with weight 𝛼 as follows: 𝐺^(𝛼)_𝑛+1,𝑞(ℎ,𝑘∣𝑥)(𝑛+1)=_𝑍𝑝⋯_{𝑍𝑝}_{𝑘-times}𝑥₁+𝑥₂+⋯+𝑥_𝑘+𝑥^𝑛_𝑞𝛼𝑞^{(ℎ−1)𝑥1+⋯+(ℎ−𝑘)𝑥𝑘}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=[2]^𝑘_𝑞[𝛼]^𝑛_𝑞(1−𝑞)^𝑛𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)^𝑙𝑞^𝛼𝑙𝑥1+𝑞^𝛼𝑙+ℎ⋯1+𝑞^{𝛼𝑙+ℎ−𝑘+1}=[2]^𝑘_𝑞[𝛼]^𝑛_𝑞(1−𝑞)^𝑚𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠(−1)^𝑙𝑞^𝛼𝑙𝑥−𝑞^𝛼𝑙+ℎ∶𝑞⁻¹_𝑘,(3.1) where (𝑥∶𝑞)_ℎ=∏^ℎ−1_𝑖=0(1−𝑥𝑞^𝑖).

In the special case, 𝑥=0, 𝐺^(𝛼)_𝑛+1,𝑞(ℎ,𝑘|0)=𝐺^(𝛼)_𝑛+1,𝑞(ℎ,𝑘) are called the (𝑛+1)th (ℎ,𝑘)-Genocchi numbers with weight 𝛼.

In (3.1), apply the following identity: 𝑥₁+𝑥₂+⋯+𝑥_𝑘_𝑞𝛼(1−𝑞^𝛼)+𝑞^𝛼𝑥₁+𝑥₂+⋯+𝑥_𝑘=1,(3.2) and we have a theorem.

Theorem 3.1. For 𝛼∈ℕ and ℎ,𝑘∈ℤ₊, one has 𝐺^(𝛼)_𝑛+1,𝑞(ℎ,𝑘)𝑛+1=(1−𝑞^𝛼)𝐺^(𝛼)_𝑛+2,𝑞(ℎ,𝑘)𝑛+2+𝐺^(𝛼)_𝑛+1,𝑞(ℎ+𝛼,𝑘)𝑛+1.(3.3)

We consider, for 𝛼∈ℕ and ℎ,𝑘∈ℤ₊, ^𝑖_𝑗=0⎛⎜⎜⎝𝑖𝑗⎞⎟⎟⎠(𝑞^𝛼−1)^𝑗𝐺^(𝛼)_{𝑛+𝑗−𝑖+1}(ℎ−𝛼,𝑘)𝑛+𝑗−𝑖+1=^𝑖_𝑗=0⎛⎜⎜⎝𝑖𝑗⎞⎟⎟⎠(𝑞^𝛼−1)^𝑗_𝑍𝑝⋯_{𝑍𝑝}_{𝑘-times}⎡⎢⎣^𝑘_ℓ=1𝑥_ℓ⎤⎥⎦^{𝑛−𝑖−𝑗}_𝑞^𝛼𝑞^{∑𝑘ℓ=1(ℎ−𝛼−ℓ)𝑥ℓ}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=^𝑖_𝑗=0⎛⎜⎜⎝𝑖𝑗⎞⎟⎟⎠(𝑞^𝛼−1)^𝑗_𝑍𝑝⋯_{𝑍𝑝}_{𝑘-times}⎡⎢⎣^𝑘_ℓ=1𝑥_ℓ⎤⎥⎦^{𝑛−𝑖+𝑗}_𝑞^𝛼𝑞^{∑𝑘ℓ=1(ℎ−ℓ)𝑥ℓ}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=^𝑖_𝑗=0⎛⎜⎜⎝𝑖𝑗⎞⎟⎟⎠(𝑞^𝛼−1)^𝑗𝐺^(𝛼)_{𝑛+𝑗−𝑖+1}(ℎ,𝑘)𝑛+𝑗−𝑖+1.(3.4) Therefore, we obtain the following combinatorial property.

Theorem 3.2. For 𝛼∈ℕ and ℎ,𝑘∈ℤ₊, one has ^𝑖_𝑗=0⎛⎜⎜⎝𝑖𝑗⎞⎟⎟⎠(𝑞^𝛼−1)^𝑗𝐺^(𝛼)_{𝑛+𝑗−𝑖+1}(ℎ−𝛼,𝑘)𝑛+𝑗−𝑖+1=^𝑖−1_𝑗=0⎛⎜⎜⎝𝑖−1𝑗⎞⎟⎟⎠(𝑞^𝛼−1)^𝑗𝐺^(𝛼)_{𝑛+𝑗−𝑖+1}(ℎ,𝑘)𝑛+𝑗−𝑖+1.(3.5)