Some Properties of Multiple Generalized q-Genocchi Polynomials with Weight 𝜶 and Weak Weight 𝜷

Abstract

The present paper deals with the various q-Genocchi numbers and polynomials. We define a new type of multiple generalized q-Genocchi numbers and polynomials with weight α and weak weight β by applying the method of p-adic q-integral. We will find a link between their numbers and polynomials with weight α and weak weight β. Also we will obtain the interesting properties of their numbers and polynomials with weight α and weak weight β. Moreover, we construct a Hurwitz-type zeta function which interpolates multiple generalized q-Genocchi polynomials with weight α and weak weight β and find some combinatorial relations.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper ℤ_𝑝,ℚ_𝑝,ℂ, and ℂ_𝑝 denote the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, the complex number field, and the completion of the algebraic closure of ℚ_𝑝, respectively. 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 (see [1–21]). When one talks of 𝑞-extension, 𝑞 is variously considered as an indeterminate, a complex 𝑞∈ℂ, or a 𝑝-adic number 𝑞∈ℂ_𝑝. If 𝑞∈ℂ, then one normally assumes |𝑞|<1. If 𝑞∈ℂ_𝑝, then we assume that |𝑞−1|_𝑝<1.

Throughout this paper, we use the following notation: [𝑥]_𝑞=1−𝑞^𝑥,[𝑥]1−𝑞^−𝑞=1−−𝑞^𝑥.1+𝑞 (1.1) Hence lim_𝑞→1[𝑥]_𝑞=𝑥 for all 𝑥∈ℤ_𝑝 (see [1–14, 16, 18, 20, 21]).

We say that 𝑔∶ℤ_𝑝→ℂ_𝑝 is uniformly differentiable function at a point 𝑎∈ℤ_𝑝 and we write 𝑔∈UD(ℤ_𝑝) if the difference quotients Φ_𝑔∶ℤ_𝑝×ℤ_𝑝→ℂ_𝑝 such that Φ_𝑔=𝑔(𝑥)𝑦𝑥,𝑦−𝑔𝑥−𝑦 (1.2) have a limit 𝑔^′(𝑎) as (𝑥,𝑦)→(𝑎,𝑎).

Let 𝑑 be a fixed integer, and let 𝑝 be a fixed prime number. For any positive integer 𝑁, we set 𝑋=𝑋_𝑑=lim_←𝑁ℤ𝑑𝑝^𝑁ℤ,𝑋₁=ℤ_𝑝,𝑋^∗=_{0<𝑎<𝑑𝑝𝑎,𝑝=1}𝑎+𝑑𝑝ℤ_𝑝,𝑎+𝑑𝑝^𝑁ℤ_𝑝=𝑥∈𝑋∣𝑥≡𝑎mod𝑑𝑝^𝑁, (1.3) where 𝑎∈ℤ lies in 0≤𝑎<𝑑𝑝^𝑁.

For any positive integer 𝑁, 𝜇_𝑞𝑎+𝑑𝑝^𝑁ℤ_𝑝=𝑞^𝑎𝑑𝑝^𝑁_𝑞 (1.4) is known to be a distribution on 𝑋.

For 𝑔∈UD(ℤ_𝑝), Kim defined the 𝑞-deformed fermionic 𝑝-adic integral on ℤ_𝑝: 𝐼_−𝑞𝑔=_ℤ𝑝𝑔(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁𝑝_−𝑞^𝑁−1_𝑥=0𝑔(𝑥)−𝑞^𝑥. (1.5)(see [1–13]), and note that _ℤ𝑝𝑔(𝑥)𝑑𝜇_−𝑞(𝑥)=_𝑋𝑔(𝑥)𝑑𝜇_−𝑞(𝑥). (1.6) We consider the case 𝑞∈(−1,0) corresponding to 𝑞-deformed fermionic certain and annihilation operators and the literature given there in [9, 13, 14].

In [9, 12, 14, 19], we introduced multiple generalized Genocchi number and polynomials. Let 𝜒 be a primitive Dirichlet character of conductor 𝑓∈ℕ. We assume that 𝑓 is odd. Then the multiple generalized Genocchi numbers, 𝐺^(𝑟)_𝑛,𝜒, and the multiple generalized Genocchi polynomials, 𝐺^(𝑟)_𝑛,𝜒(𝑥), associated with 𝜒, are defined by 𝐹_𝜒^(𝑟)(𝑡)=2𝑡∑^𝑓−1_𝑎=0𝜒(𝑎)(−1)^𝑎𝑒^𝑎𝑡𝑒^𝑓𝑡+1^𝑟=^∞_𝑛=0𝐺^(𝑟)_𝑛,𝜒𝑡^𝑛,𝐹𝑛!_𝜒(𝑟)()=∑𝑡,𝑥2𝑡^𝑓−1_𝑎=0𝜒(𝑎)(−1)^𝑎𝑒^𝑎𝑡𝑒^𝑓𝑡𝑒+1^𝑡𝑥𝑟=^∞_𝑛=0𝐺^(𝑟)_𝑛,𝜒(𝑥)𝑡^𝑛.𝑛! (1.7) In the special case 𝑥=0, 𝐺^(𝑟)_𝑛,𝜒=𝐺^(𝑟)_𝑛,𝜒(0) are called the 𝑛th multiple generalized Genocchi numbers attached to 𝜒.

Now, having discussed the multiple generalized Genocchi numbers and polynomials, we were ready to multiple-generalize them to their 𝑞-analogues. In generalizing the generating functions of the Genocchi numbers and polynomials to their respective 𝑞-analogues; it is more useful than defining the generating function for the Genocchi numbers and polynomials (see [12]).

Our aim in this paper is to define multiple generalized 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞} and polynomials 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽. We investigate some properties which are related to multiple generalized 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞} and polynomials 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽. We also derive the existence of a specific interpolation function which interpolate multiple generalized 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞} and polynomials 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽 at negative integers.

2. The Generating Functions of Multiple Generalized 𝑞-Genocchi Numbers and Polynomials with Weight 𝛼 and Weak Weight 𝛽

Many mathematicians constructed various kinds of generating functions of the 𝑞-Gnocchi numbers and polynomials by using 𝑝-adic 𝑞-Vokenborn integral. First we introduce multiple generalized 𝑞-Genocchi numbers and polynomials with weight 𝛼 and weak weight 𝛽.

Let us define the generalized 𝑞-Genocchi numbers 𝐺^(𝛼,𝛽)_{𝑛,𝜒,𝑞} and polynomials 𝐺^(𝛼,𝛽)_{𝑛,𝜒,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽, respectively, 𝐹^(𝛼,𝛽)_𝜒,𝑞(𝑡)=^∞_𝑛=0𝐺^(𝛼,𝛽)_{𝑛,𝜒,𝑞}𝑡^𝑛=𝑛!_𝑋𝑡𝜒(𝑥)𝑒^{[𝑥]𝑞𝛼𝑡}𝑑𝜇_−𝑞𝛽(𝑥),𝐹^(𝛼,𝛽)_𝜒,𝑞(𝑡,𝑥)=^∞_𝑛=0𝐺^(𝛼,𝛽)_{𝑛,𝜒,𝑞}(𝑥)𝑡^𝑛𝑛!=_𝑋𝑡𝜒𝑦𝑒^{[𝑥+𝑦]𝑞𝛼𝑡}𝑑𝜇_−𝑞𝛽𝑦. (2.1) By using the Taylor expansion of 𝑒^{[𝑥]𝑞𝛼𝑡}, we have ^∞_𝑛=0𝑋𝜒(𝑥)[𝑥]^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝛽(𝑥)𝑡^𝑛=𝑛!^∞_𝑛=0𝐺^(𝛼,𝛽)_{𝑛,𝜒,𝑞}𝑡^𝑛−1𝑛!=𝐺^(𝛼,𝛽)_0,𝜒,𝑞+^∞_𝑛=0𝐺^(𝛼,𝛽)_{𝑛+1,𝜒,𝑞}𝑡𝑛+1^𝑛.𝑛! (2.2) By comparing the coefficient of both sides of 𝑡^𝑛/𝑛! in (2.2), we get 𝐺^(𝛼,𝛽)_{𝑛+1,𝜒,𝑞}=[2]𝑛+11−𝑞^{𝑞𝛽𝛼𝑛𝑓−1}_𝑎=0(−1)^𝑎𝑞^𝛽𝑎𝜒(𝑎)^𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑎𝑙11+𝑞^{𝑓(𝛼𝑙+𝛽)}. (2.3) From (2.2) and (2.3), we can easily obtain that ^∞_𝑛=0𝐺^(𝛼,𝛽)_{𝑛,𝜒,𝑞}𝑡^𝑛=𝑛!^∞_𝑛=0𝑡_𝑋𝜒(𝑥)[𝑥]^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝛽(𝑥)𝑡^𝑛=[2]𝑛!^𝑞𝛽𝑡^∞_𝑙=0(−1)^𝑙𝑞^𝛽𝑙𝜒(𝑙)𝑒^{[𝑙]𝑞𝛼𝑡}. (2.4) Therefore, we obtain 𝐹^(𝛼,𝛽)_𝜒,𝑞(𝑡)=[2]_𝑞𝛽𝑡^∞_𝑙=0(−1)^𝑙𝑞^𝛽𝑙𝜒(𝑙)𝑒^{[𝑙]𝑞𝛼𝑡}=^∞_𝑛=0𝐺^(𝛼,𝛽)_{𝑛,𝜒,𝑞}𝑡^𝑛.𝑛! (2.5)

Similarly, we find the generating function of generalized 𝑞-Genocchi polynomials with weight 𝛼 and weak weight 𝛽: 𝐺^(𝛼,𝛽)_0,𝜒,𝑞(𝑥)𝐺=0,^{(𝛼,𝛽)𝑛+1,𝜒,𝑞}(𝑥)=𝑛+1_𝑋𝜒𝑦𝑥+𝑦^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝛽𝑦=[2]_𝑞𝛽^∞_𝑙=0(−1)^𝑙𝑞^𝛽𝑙𝜒(𝑙)[𝑥+𝑙]^𝑛_𝑞𝛼. (2.6) From (2.6), we have 𝐹^(𝛼,𝛽)_𝜒,𝑞(𝑡,𝑥)=[2]_𝑞𝛽𝑡^∞_𝑙=0(−1)^𝑙𝑞^𝛽𝑙𝜒(𝑙)𝑒^{[𝑥+𝑙]𝑞𝛼𝑡}=^∞_𝑛=0𝐺^(𝛼,𝛽)_{𝑛,𝜒,𝑞}(𝑥)𝑡^𝑛.𝑛! (2.7) Observe that 𝐹^(𝛼,𝛽)_𝜒,𝑞(𝑡)=𝐹^(𝛼,𝛽)_𝜒,𝑞(𝑡,0). Hence we have 𝐺^(𝛼,𝛽)_{𝑛,𝜒,𝑞}=𝐺^(𝛼,𝛽)_{𝑛,𝜒,𝑞}(0). If 𝑞→1 into (2.7), then we easily obtain 𝐹_𝜒(𝑡,𝑥).

First, we define the multiple generalized 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞} with weight 𝛼 and weak weight 𝛽: 𝐹^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑡)=[2]^𝑟_𝑞𝛽𝑡^𝑟∞_{𝑘1,…,𝑘𝑟=0}(−1)^{∑𝑟𝑖=1𝑘𝑖}𝑞^{𝛽∑𝑟𝑖=1𝑘𝑖𝑟}_𝑖=1𝜒𝑘_𝑖𝑒^{[∑𝑟𝑖=1𝑘𝑖]𝑞𝛼𝑡}=𝑡^𝑟_𝑋⋯_{𝑋𝑟-times}𝜒𝑥₁⋯𝜒𝑥_𝑟𝑒^{[𝑥1+⋯+𝑥𝑟]𝑞𝛼𝑡}𝑑𝜇_−𝑞𝛽𝑥₁⋯𝑑𝜇_−𝑞𝛽𝑥_𝑟=^∞_𝑛=0𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}𝑡^𝑛.𝑛! (2.8) Then we have ^∞_{𝑛=0𝑋𝑋𝑟-}⋯_times𝜒𝑥₁⋯𝜒𝑥_𝑟𝑥₁+⋯+𝑥_𝑟^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝛽𝑥₁⋯𝑑𝜇_−𝑞𝛽𝑥_𝑟𝑡^𝑛=𝑛!_∞𝑛=0𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}𝑡^𝑛−𝑟=𝑛!^𝑟−1_𝑛=0𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}𝑡^𝑛−𝑟+𝑛!^∞_𝑛=0𝐺^{(𝛼,𝛽,𝑟)}_{𝑛+𝑟,𝜒,𝑞}(_𝑟^𝑛+𝑟)𝑡𝑟!^𝑛,𝑛! (2.9) where (_𝑟^𝑛+𝑟)=(𝑛+𝑟)!/𝑛!𝑟!.

By comparing the coefficients on the both sides of (2.9), we obtain the following theorem.

Theorem 2.1. Let 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1 and 𝑛∈ℤ₊. Then one has 𝐺^{(𝛼,𝛽,𝑟)}_0,𝜒,𝑞=𝐺^{(𝛼,𝛽,𝑟)}_1,𝜒,𝑞=⋯=𝐺^{(𝛼,𝛽,𝑟)}_{𝑟−1,𝜒,𝑞}𝐺=0,_{(𝛼,𝛽,𝑟)𝑛+𝑟,𝜒,𝑞}(_𝑟^𝑛+𝑟)=𝑟!_{𝑋𝑋𝑟-}⋯_times𝜒𝑥₁⋯𝜒𝑥_𝑟𝑥₁+⋯+𝑥_𝑟^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝛽𝑥₁⋯𝑑𝜇_−𝑞𝛽𝑥_𝑟=[2]^𝑟_𝑞𝛽1−𝑞^{𝛼𝑛𝑓−1}_{𝑎1,…,𝑎𝑟}^𝑛_=0𝑙=0𝑛𝑙^𝑟𝑖=1𝜒𝑎_𝑖(−1)^{𝑙+∑𝑟𝑖=1𝑎𝑖}𝑞^{(𝛼𝑙+𝛽)∑𝑟𝑖=1𝑎𝑖}1+𝑞^{𝑓(𝛼𝑙+𝛽)𝑟}=[2]^𝑟_𝑞𝛽^∞_𝑚=0^𝑓−1_{𝑎1,…,𝑎𝑟=0}𝑚()𝑚+𝑟−1−1^{∑𝑟𝑖=1𝑎𝑖+𝑚}𝑞^{𝛽(∑𝑟𝑖=1𝑎𝑖+𝑓𝑚)}×^𝑟_𝑖=1𝜒𝑎_𝑖^𝑟_𝑖=1𝑎_𝑖+𝑓𝑚^𝑛_𝑞𝛼. (2.10)

From now on, we define the multiple generalized 𝑞-Genocchi polynomials 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽. 𝐹^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑡,𝑥)=[2]^𝑟_𝑞𝛽𝑡^𝑟∞_{𝑘1,…,𝑘𝑟=0}(−1)^{∑𝑟𝑖=1𝑘𝑖}𝑞^{𝛽∑𝑟𝑖=1𝑘𝑖𝑟}_𝑖=1𝜒𝑘_𝑖𝑒^{[∑𝑟𝑖=1𝑘𝑖+𝑥]𝑞𝛼𝑡}=𝑡^𝑟_𝑋⋯_{𝑋𝑟-times}𝜒𝑦₁⋯𝜒𝑦_𝑟𝑒^{[𝑥+𝑦1+⋯+𝑦𝑟]𝑞𝛼𝑡}𝑑𝜇_−𝑞𝛽𝑦₁⋯𝑑𝜇_−𝑞𝛽𝑦_𝑟=^∞_𝑛=0𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥)𝑡^𝑛.𝑛! (2.11) Then we have ^∞_{𝑛=0𝑋𝑋𝑟-}⋯_times𝜒𝑦₁⋯𝜒𝑦_𝑟𝑥+𝑦₁+⋯+𝑦_𝑟^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝛽𝑦₁⋯𝑑𝜇_−𝑞𝛽𝑦_𝑟𝑡^𝑛=𝑛!_∞𝑛=0𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥)𝑡^𝑛−𝑟=𝑛!^𝑟−1_𝑛=0𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥)𝑡^𝑛−𝑟+𝑛!^∞_𝑛=0𝐺^{(𝛼,𝛽,𝑟)}_{𝑛+𝑟,𝜒,𝑞}(𝑥)(_𝑟^𝑛+𝑟)𝑡𝑟!^𝑛,𝑛! (2.12) where (_𝑟^𝑛+𝑟)=(𝑛+𝑟)!/𝑛!𝑟!.

By comparing the coefficients on the both sides of (2.12), we have the following theorem.

Theorem 2.2. Let 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1 and 𝑛∈ℤ₊. Then one has 𝐺^{(𝛼,𝛽,𝑟)}_0,𝜒,𝑞(𝑥)=𝐺^{(𝛼,𝛽,𝑟)}_1,𝜒,𝑞(𝑥)=⋯=𝐺^{(𝛼,𝛽,𝑟)}_{𝑟−1,𝜒,𝑞}(𝑥)𝐺=0,_{(𝛼,𝛽,𝑟)𝑛+𝑟,𝜒,𝑞}(𝑥)(_𝑟^𝑛+𝑟)=𝑟!_{𝑋𝑋𝑟-}⋯_times𝜒𝑦₁⋯𝜒𝑦_𝑟𝑥+𝑦₁+⋯+𝑦_𝑟^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝛽𝑦₁⋯𝑑𝜇_−𝑞𝛽𝑦_𝑟=[2]^𝑟_𝑞𝛽1−𝑞^{𝛼𝑛𝑓−1}_{𝑎1,…,𝑎𝑟}^𝑛_=0𝑙=0𝑛𝑙^𝑟𝑖=1𝜒𝑎_𝑖(−1)^{𝑙+∑𝑟𝑖=1𝑎𝑖}𝑞^{𝛼𝑙𝑥+(𝛼𝑙+𝛽)∑𝑟𝑖=1𝑎𝑖}1+𝑞^{𝑓(𝛼𝑙+𝛽)𝑟}=[2]^𝑟_𝑞𝛽^∞_𝑚=0^𝑓−1_{𝑎1,…,𝑎𝑟=0}𝑚()𝑚+𝑟−1−1^{∑𝑟𝑖=1𝑎𝑖+𝑚}𝑞^{𝛽(∑𝑟𝑖=1𝑎𝑖+𝑓𝑚)}×^𝑟_𝑖=1𝜒𝑎_𝑖^𝑟_𝑖=1𝑎_𝑖+𝑓𝑚+𝑥^𝑛_𝑞𝛼. (2.13)

In (2.11), we simply identify that lim_𝑞→1𝐹^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑡,𝑥)=2^𝑟𝑡^𝑟∞_{𝑘1,…,𝑘𝑟=0}(−1)^{∑𝑟𝑖=1𝑘𝑖𝑟}_𝑖=1𝜒𝑘_𝑖𝑒^{(∑𝑟𝑖=1𝑘𝑖+𝑥)𝑡}=2𝑡∑^𝑓−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^𝑎𝑡1+𝑒^𝑓𝑡𝑟𝑒^𝑡𝑥=𝐹_𝜒^(𝑟)(𝑡,𝑥). (2.14)

So far, we have studied the generating functions of the multiple generalized 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞} and polynomials 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽.

3. Modified Multiple Generalized 𝑞-Genocchi Polynomials with Weight 𝛼 and Weak Weight 𝛽

In this section, we will investigate about modified multiple generalized 𝑞-Genocchi numbers and polynomials with weight 𝛼 and weak weight 𝛽. Also, we will find their relations in multiple generalized 𝑞-Genocchi numbers and polynomials with weight 𝛼 and weak weight 𝛽.

Firstly, we modify generating functions of 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞} and 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥). We access some relations connected to these numbers and polynomials with weight 𝛼 and weak weight 𝛽. For this reason, we assign generating function of modified multiple generalized 𝑞-Genocchi numbers and polynomials with weight 𝛼 and weak weight 𝛽 which are implied by 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞} and 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥). We give relations between these numbers and polynomials with weight 𝛼 and weak weight 𝛽.

We modify (2.11) as follows: 𝔉^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑡,𝑥)=𝐹^{(𝛼,𝛽,𝑟)}_𝜒,𝑞𝑞^−𝛼𝑥𝑡,𝑥, (3.1) where 𝐹^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑡,𝑥) is defined in (2.11).

From the above we know that 𝔉^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑡,𝑥)=^∞_𝑛=0𝑞^{−(𝑛+𝑟)𝛼𝑥}𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥)𝑡^𝑛.𝑛! (3.2) After some elementary calculations, we attain 𝔉^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑡,𝑥)=𝑞^{−𝛼𝑟𝑥}𝑒^{(𝑞−𝛼𝑥[𝑥]𝑞𝛼𝑡)}𝐹^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑡), (3.3) where 𝐹^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑡) is defined in (2.8).

From the above, we can assign the modified multiple generalized 𝑞-Genocchi polynomials 𝜀^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽 as follows: 𝔉^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑡,𝑥)=^∞_𝑛=0𝜀^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥)𝑡^𝑛.𝑛! (3.4) Then we have 𝜀^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥)=𝑞^{−(𝑛+𝑟)𝛼𝑥}𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥). (3.5)

Theorem 3.1. For 𝑟∈ℕ and 𝑛∈ℤ₊, one has 𝜀^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥)=𝑞^{𝑛−(𝑛+𝑟)𝛼𝑥}_𝑖=0𝑛𝑖𝑞^𝛼𝑖𝑥[𝑥]_𝑞^𝑛−𝑖_𝛼𝐺^{(𝛼,𝛽,𝑟)}_{𝑖,𝜒,𝑞}. (3.6)

Corollary 3.2. For 𝑟∈ℕ and 𝑛∈ℤ₊, by using (3.7), one easily obtains 𝜀^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥)=𝑞^{∞−(𝑛+𝑟)𝛼𝑥𝑛}_𝑚=0𝑗=0^𝑛−𝑗_𝑙=0𝑛𝑚()𝑗,𝑙,𝑛−𝑗−𝑙𝑛−𝑗+𝑚−1−1^𝑙𝑞^{𝛼{(𝑗+𝑙)𝑥+𝑚}}𝐺^{(𝛼,𝛽,𝑟)}_{𝑗,𝜒,𝑞}. (3.7)

Secandly, by using generating function of the multiple generalized 𝑞-Genocchi polynomials with weight 𝛼 and weak weight 𝛽, which is defined by (2.11), we obtain the following identities.

By using (2.13), we find that 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛+𝑟,𝜒,𝑞}(𝑥)(_𝑟^𝑛+𝑟)=[2]𝑟!^{𝑟𝑞𝛽∞}_𝑚=0^𝑓−1_{𝑎1,…,𝑎𝑟=0}𝑚()𝑚+𝑟−1−1^{∑𝑟𝑖=1𝑎𝑖+𝑚}×𝑞^{𝛽(∑𝑟𝑖=1𝑎𝑖+𝑓𝑚)𝑟}_𝑖=1𝜒𝑎_𝑖^𝑟_𝑖=1𝑎_𝑖+𝑓𝑚+𝑥^𝑛_𝑞𝛼=[2]^𝑟_𝑞𝛽^𝑓−1_{𝑎1,…,𝑎𝑟}^𝑛₌₀^𝑙_𝑙=0𝑎=0𝑛()𝑎,𝑙−𝑎,𝑛−𝑙−1^{𝑎+∑𝑟𝑖=1𝑎𝑖}𝑞^{{𝛼(𝑎+𝑛−𝑙)+𝛽}∑𝑟𝑖=1𝑎𝑖}×^𝑟_𝑖=1𝜒𝑎_𝑖[𝑥]_𝑞^𝑛−𝑙_𝛼1−𝑞^𝛼𝑙1+𝑞^{𝑓{𝛼(𝑎+𝑛−𝑙)+𝛽}𝑟}. (3.8) Thus we have the following theorem.

Theorem 3.3. Let 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1 and 𝑟∈ℕ. Then one has 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛+𝑟,𝜒,𝑞}(𝑥)(_𝑟^𝑛+𝑟)=[2]𝑟!^{𝑟𝑞𝛽𝑓−1}_{𝑎1,…,𝑎𝑟}^𝑛₌₀^𝑙_𝑙=0𝑎=0𝑛()𝑎,𝑙−𝑎,𝑛−𝑙−1^{𝑎+∑𝑟𝑖=1𝑎𝑖}𝑞^{{𝛼(𝑎+𝑛−𝑙)+𝛽}∑𝑟𝑖=1𝑎𝑖}×^𝑟_𝑖=1𝜒𝑎_𝑖[𝑥]_𝑞^𝑛−𝑙_𝛼1−𝑞^𝛼𝑙1+𝑞^{𝑓{𝛼(𝑎+𝑛−𝑙)+𝛽}𝑟}. (3.9)

By using (2.13), we have 𝐹^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑡,𝑥)=[2]^𝑟_𝑞𝛽𝑡^𝑟∞𝑛_𝑛=0𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥1−𝑞^𝑛𝑓−1_{𝑎1,…,𝑎𝑟=0}(−1)^{∑𝑟𝑖=1𝑎𝑖}×𝑞^{(𝛼𝑙+𝛽)(∑𝑟𝑖=1𝑎𝑖)𝑟}_𝑖=1𝜒𝑎_𝑖^∞_𝑚=0𝑚𝑚+𝑟−1−𝑞^{𝑓(𝛼𝑙+𝛽)𝑚}𝑡^𝑛.𝑛! (3.10) Thus we have ^∞_𝑛=0𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥)𝑡^𝑛=𝑛!^∞_𝑛=0[2]^𝑟_𝑞𝛽𝑡^𝑟𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥1−𝑞^{𝛼−𝑛𝑓−1}_{𝑎1,…,𝑎𝑟=0}(−1)^{∑𝑟𝑖=1𝑎𝑖}×𝑞^{(𝛼𝑙+𝛽)(∑𝑟𝑖=1𝑎𝑖)𝑟}_𝑖=1𝜒𝑎_𝑖1+𝑞^{𝑓(𝛼𝑙+𝛽)−𝑟}𝑡^𝑛.𝑛! (3.11) By comparing the coefficients of both sides of (𝑛+𝑟)!/𝑡^𝑛+𝑟 in the above, we arrive at the following theorem.

Theorem 3.4. Let 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1, 𝑟∈ℕ. Then one has 𝐺^{(𝛼,𝛽,𝑟)}_{𝑛+𝑟,𝜒,𝑞}(𝑥)(_𝑟^𝑛+𝑟)=[2]𝑟!^{𝑟𝑞𝛽𝑛}_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥1−𝑞^{𝛼−𝑛𝑓−1}_{𝑎1,…,𝑎𝑟=0}(−1)^{∑𝑟𝑖=1𝑎𝑖}×𝑞^{(𝛼𝑙+𝛽)(∑𝑟𝑖=1𝑎𝑖)𝑟}_𝑖=1𝜒𝑎_𝑖1+𝑞^{𝑓(𝛼𝑙+𝛽)−𝑟}. (3.12)

From (2.12), we easily know that ^∞_𝑛=0𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥)𝑡^𝑛=𝑛!^∞_𝑛=0[2]^𝑟_𝑞𝛽𝑟𝑛+𝑟𝑟!^∞_{𝑘1,…,𝑘𝑟=0}(−1)^{∑𝑟𝑖=1𝑘𝑖}𝑞^{𝛽∑𝑟𝑖=1𝑘𝑖𝑟}_𝑖=1𝜒𝑘_𝑖×𝑥+^𝑟_𝑖=1𝑘_𝑖^𝑛_𝑞𝛼𝑡^𝑛+𝑟()!.𝑛+𝑟 (3.13) From the above, we get the following theorem.

Theorem 3.5. Let 𝑟∈ℕ, 𝑘∈ℤ₊. Then one has 𝐺^{(𝛼,𝛽,𝑟)}_0,𝜒,𝑞(𝑥)=𝐺^{(𝛼,𝛽,𝑟)}_1,𝜒,𝑞(𝑥)=⋯=𝐺^{(𝛼,𝛽,𝑟)}_{𝑟−1,𝜒,𝑞}(𝑥)𝐺=0,_{(𝛼,𝛽,𝑟)𝑙+𝑟,𝜒,𝑞}(𝑥)=[2]^𝑟_𝑞𝛽𝑟𝑙+𝑟𝑟!^∞_{𝑘1,…,𝑘𝑟=0}(−1)^{∑𝑟𝑖=1𝑘𝑖}𝑞^{𝛽∑𝑟𝑖=1𝑘𝑖𝑟}_𝑖=1𝜒𝑘_𝑖𝑥+^𝑟_𝑖=1𝑘_𝑖^𝑙_𝑞𝛼. (3.14)

From (2.13), we have ^∞_𝑛=0𝐺^{(𝛼,𝛽,𝑟)}_{𝑛,𝜒,𝑞}(𝑥)𝑡^𝑛𝑛!^∞_𝑛=0𝐺^{(𝛼,𝛽,𝑠)}_{𝑛,𝜒,𝑞}(𝑥)𝑡^𝑛=[2]𝑛!_{𝑞𝑟+𝑠𝛽}𝑡^{𝑟+𝑠𝑓−1}_{𝑎1,…,𝑎𝑟}^∞_=0𝑚=0𝑚()𝑚+𝑟−1−1^{∑𝑟𝑖=1𝑎𝑖+𝑚}𝑞^{𝛽(∑𝑟𝑖=1𝑎𝑖+𝑓𝑚)}×^𝑟_𝑖=1𝜒𝑎_𝑖𝑒^{[∑𝑟𝑖=1𝑎𝑖+𝑓𝑚+𝑥]𝑞𝛼𝑡𝑓−1}_{𝑏1,…,𝑏𝑠}^∞_=0𝑘=0𝑘()𝑘+𝑠−1−1^{∑𝑠𝑖=1𝑏𝑖+𝑘}×𝑞^{𝛽(∑𝑠𝑖=1𝑏𝑖+𝑓𝑘)𝑠}_𝑖=1𝜒𝑏_𝑖𝑒^{[∑𝑟𝑖=1𝑏𝑖+𝑓𝑘+𝑥]𝑞𝛼𝑡}. (3.15) By using Cauchy product in (3.15), we obtain ^∞𝑛_𝑛=0𝑗=0𝑛𝑗𝐺^{(𝛼,𝛽,𝑟)}_{𝑗,𝜒,𝑞}(𝑥)𝐺^{(𝛼,𝛽,𝑠)}_{𝑛−𝑗,𝜒,𝑞}(𝑥)𝑡^𝑛=[2]𝑛!_{𝑞𝑟+𝑠𝛽}𝑡^{∞𝑟+𝑠𝑛}_𝑛=0𝑗=0^𝑓−1_{𝑎1,…,𝑎𝑟=0}^𝑓−1_{𝑏1,…,𝑏𝑠=0}𝑗×()𝑗+𝑟−1𝑛−𝑗+𝑠−1𝑛−𝑗−1^{∑𝑟𝑖=1𝑎𝑖+∑𝑠𝑖=1𝑏𝑖+𝑛}𝑞^{𝛽(∑𝑟𝑖=1𝑎𝑖+∑𝑠𝑖=1𝑏𝑖+𝑓𝑛)𝑟}_𝑖=1𝜒𝑎_𝑖^𝑠_𝑖=1𝜒𝑏_𝑖×𝑒^{[∑𝑟𝑖=1𝑎𝑖+𝑓𝑗+𝑥]𝑞𝛼𝑡}𝑒^{[∑𝑠𝑖=1𝑏𝑖+𝑓(𝑛−𝑗)+𝑥]𝑞𝛼𝑡}. (3.16) From (3.16), we have ^∞_𝑚=0^𝑚_𝑗=0𝑚𝑗𝐺^{(𝛼,𝛽,𝑟)}_{𝑗,𝜒,𝑞}(𝑥)𝐺^{(𝛼,𝛽,𝑠)}_{𝑚−𝑗,𝜒,𝑞}(𝑥)𝑡^𝑚=𝑚!_∞𝑚=0[2]_𝑞^𝑟+𝑠_𝛽𝑡^{∞𝑟+𝑠𝑛}_𝑛=0𝑗=0^𝑓−1_{𝑎1,…,𝑎𝑟=0}^𝑓−1_{𝑏1,…,𝑏𝑠=0}𝑗×()𝑗+𝑟−1𝑛−𝑗+𝑠−1𝑛−𝑗−1^{∑𝑟𝑖=1𝑎𝑖+∑𝑠𝑖=1𝑏𝑖+𝑛}𝑞^{𝛽(∑𝑟𝑖=1𝑎𝑖+∑𝑠𝑖=1𝑏𝑖+𝑓𝑛)𝑟}_𝑖=1𝜒𝑎_𝑖^𝑠_𝑖=1𝜒𝑏_𝑖×^𝑟_𝑖=1𝑎_𝑖+𝑓𝑗+𝑥_𝑞𝛼+^𝑠_𝑖=1𝑏_𝑖+𝑓(𝑛−𝑗)+𝑥_𝑞𝛼^𝑚𝑡^𝑚.𝑚! (3.17) By comparing the coefficients of both sides of 𝑡^{𝑚+𝑟+𝑠}/(𝑚+𝑟+𝑠)! in (3.17), we have the following theorem.

Theorem 3.6. Let 𝑟∈ℕand𝑠∈ℤ₊. Then one has ∑^{𝑙+𝑟+𝑠}_𝑗=0𝑗^{𝑙+𝑟+𝑠}𝐺^{(𝛼,𝛽,𝑟)}_{𝑗,𝜒,𝑞}(𝑥)𝐺^{(𝛼,𝛽,𝑠)}_{𝑙+𝑟+𝑠−𝑗,𝜒,𝑞}(𝑥)_{𝑙𝑙+𝑟+𝑠}()!=[2]𝑟+𝑠_{𝑞𝑟+𝑠𝛽∞𝑛𝑛=0𝑗=0𝑓−1𝑎1,…,𝑎𝑟=0𝑓−1𝑏1,…,𝑏𝑠=0}𝑗×()𝑗+𝑟−1𝑛−𝑗+𝑠−1𝑛−𝑗−1^{∑𝑟𝑖=1𝑎𝑖+∑𝑠𝑖=1𝑏𝑖+𝑛}𝑞^{𝛽(∑𝑟𝑖=1𝑎𝑖+∑𝑠𝑖=1𝑏𝑖+𝑓𝑛)𝑟}_𝑖=1𝜒𝑎_𝑖^𝑠_𝑖=1𝜒𝑏_𝑖×^𝑟_𝑖=1𝑎_𝑖+𝑓𝑗+𝑥_𝑞𝛼+^𝑠_𝑖=1𝑏_𝑖+𝑓𝑛−𝑗+𝑥_𝑞𝛼^𝑙. (3.18)

Corollary 3.7. In (3.18) setting 𝑠=1, one has ∑^𝑙+𝑟+1_𝑗=0𝑗^𝑙+𝑟+1𝐺^{(𝛼,𝛽,𝑟)}_{𝑗,𝜒,𝑞}(𝑥)𝐺^{(𝛼,𝛽,1)}_{𝑙+𝑟+1−𝑗,𝜒,𝑞}(𝑥)_{𝑙𝑙+𝑟+1}()!=[2]𝑟+1_{𝑞𝑟+1𝛽∞𝑛𝑛=0𝑗=0𝑓−1𝑎1,…,𝑎𝑟=0𝑓−1𝑏1=0}𝑗)𝑗+𝑟−1(−1^{∑𝑟𝑖=1𝑎𝑖+𝑏1+𝑛}𝑞^{𝛽(∑𝑟𝑖=1𝑎𝑖+𝑏1+𝑓𝑛)}×𝜒𝑏₁^𝑟_𝑖=1𝜒𝑎_𝑖^𝑟_𝑖=1𝑎_𝑖+𝑓𝑗+𝑥_𝑞𝛼+𝑏₁+𝑓𝑛−𝑗+𝑥_𝑞𝛼^𝑙. (3.19)

By using (2.13) we have the following theorem.

Theorem 3.8. Distribution theorem is as follows:𝐺^{(𝛼,𝛽,𝑟)}_{𝑛+𝑟,𝜒,𝑞}=𝑓^𝑛_𝑞𝛼𝑓^𝑟_−𝑞𝛽^𝑓−1_{𝑎1,…,𝑎𝑟=0}(−1)^{∑𝑟𝑖=1𝑎𝑖}𝑞^{𝛽∑𝑟𝑖=1𝑎𝑖𝑟}_𝑖=1𝜒𝑎_𝑖𝐺^{(𝛼,𝛽,𝑟)}_{𝑛+𝑟,𝑞𝑓}𝑎₁+⋯+𝑎_𝑟𝑓,𝐺^{(𝛼,𝛽,𝑟)}_{𝑛+𝑟,𝜒,𝑞}(𝑥)=𝑓^𝑛_𝑞𝛼𝑓^𝑟_−𝑞𝛽^𝑓−1_{𝑎1,…,𝑎𝑟=0}(−1)^{∑𝑟𝑖=1𝑎𝑖}𝑞^{𝛽∑𝑟𝑖=1𝑎𝑖𝑟}_𝑖=1𝜒𝑎_𝑖𝐺^{(𝛼,𝛽,𝑟)}_{𝑛+𝑟,𝑞𝑓}𝑥+𝑎₁+⋯+𝑎_𝑟𝑓. (3.20)

4. Interpolation Function of Multiple Generalized 𝑞-Genocchi Polynomials with Weight 𝛼 and Weak Weight 𝛽

In this section, we see interpolation function of multiple generalized 𝑞-Genocchi polynomials with weak weight 𝛼 and find some relations.

Let us define interpolation function of the 𝐺^{(𝛼,𝛽,𝑟)}_{𝑘+𝑟,𝑞}(𝑥) as follows.

Definition 4.1. Let 𝑞,𝑠∈ℂ with |𝑞|<1 and 0<𝑥≤1. Then one defines 𝜁^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑠,𝑥)=[2]^𝑟_𝑞𝛽^∞_{𝑘1,…,𝑘𝑟=0}(−1)^{∑𝑟𝑖=1𝑘𝑖}𝑞^{𝛽∑𝑟𝑖=1𝑘𝑖}∏^𝑟_𝑖=1𝜒𝑘_𝑖𝑥+∑^𝑟_𝑖=1𝑘_𝑖^𝑠_𝑞𝛼. (4.1) We call 𝜁_𝑞^{(𝛼,𝛽,𝑟)}(𝑠,𝑥) the multiple generalized Hurwitz type 𝑞-zeta funtion.
In (4.1), setting 𝑟=1, we have 𝜁^{(𝛼,𝛽,1)}_𝜒,𝑞(𝑠,𝑥)=[2]_𝑞𝛽^∞_𝑙=0(−1)^𝑙𝑞^𝛽𝑙𝜒(𝑙)[𝑥+𝑙]^𝑠_𝑞𝛼=𝜁^(𝛼,𝛽)_𝜒,𝑞(𝑠,𝑥). (4.2)

Remark 4.2. It holds that lim_𝑞→1𝜁^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(𝑠,𝑥)=2^𝑟∞_{𝑘1,…,𝑘𝑟=0}(−1)^{∑𝑟𝑖=1𝑘𝑖}∏^𝑟_𝑖=1𝜒𝑘_𝑖𝑥+∑^𝑟_𝑖=1𝑘_𝑖^𝑠. (4.3) Substituting 𝑠=−𝑛,𝑛∈ℤ₊ into (4.1), then we have, 𝜁^{(𝛼,𝛽,𝑟)}_𝜒,𝑞(−𝑛,𝑥)=[2]^𝑟_𝑞𝛽^∞_{𝑘1,…,𝑘𝑟=0}(−1)^{∑𝑟𝑖=1𝑘𝑖}𝑞^{𝛽∑𝑟𝑖=1𝑘𝑖𝑟}_𝑖=1𝜒𝑘_𝑖𝑥+^𝑟_𝑖=1𝑘_𝑖^𝑛_𝑞𝛼. (4.4)

Setting (3.14) into the above, we easily get the following theorem.

Theorem 4.3. Let 𝑟∈ℕ, 𝑛∈ℤ₊. Then one has 𝜁^{(𝛼,𝛽,𝑟)}_𝜒,𝑞()=𝐺−𝑛,𝑥^{(𝛼,𝛽,𝑟)𝑛+𝑟,𝜒,𝑞}(𝑥)(_𝑟^𝑛+𝑟).𝑟! (4.5)