On the 𝑞-Genocchi Numbers and Polynomials with Weight 𝛼 and Weak Weight 𝛽

Abstract

We construct a new type of 𝑞-Genocchi numbers and polynomials with weight 𝛼 and weak weight 𝛽∶𝐺^{(𝛼,𝛽)𝑛,𝑞},𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥), respectively. Some interesting results and relationships are obtained.

1. Introduction

The Genocchi numbers and polynomials possess many interesting properties and are arising in many areas of mathematics and physics. Recently, many mathematicians have studied in the area of the 𝑞-Genocchi numbers and polynomials (see [1–13]). In this paper, we construct a new type of 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽)𝑛,𝑞} and polynomials 𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽.

Throughout this paper, we use the following notations. By ℤ_𝑝, we denote the ring of 𝑝-adic rational integers, ℚ_𝑝 denotes the field of 𝑝-adic rational numbers, ℂ_𝑝 denotes the completion of algebraic closure of ℚ_𝑝, ℕ denotes the set of natural numbers, ℤ denotes the ring of rational integers, ℚ denotes the field of rational numbers, ℂ denotes the set of complex numbers, and ℤ⁺=ℕ∪{0}. Let 𝜈_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{−𝜈𝑝(𝑝)}=𝑝⁻¹. When one talks of 𝑞-extension, 𝑞 is considered in many ways such as an indeterminate, a complex number 𝑞∈ℂ, or 𝑝-adic number 𝑞∈ℂ_𝑝. If 𝑞∈ℂ, one normally assume that |𝑞|<1. If 𝑞∈ℂ_𝑝, we normally assumes that |𝑞−1|_𝑝<𝑝^{−(1/𝑝−1)} so that 𝑞^𝑥=exp(𝑥log𝑞) for |𝑥|_𝑝≤1. Throughout this paper, we use the notation [𝑥]_𝑞=1−𝑞^𝑥1−𝑞,[𝑥]^−𝑞=1−(−𝑞)^𝑥1+𝑞,(1.1) cf. [1–13].

Hence, lim_𝑞→1[𝑥]=𝑥 for any 𝑥 with |𝑥|_𝑝≤1 in the present 𝑝-adic case. For 𝑓∈𝑈𝐷ℤ_𝑝=𝑓∣𝑓∶ℤ_𝑝⟶ℂ_𝑝isuniformlydifferentiablefunction,(1.2) 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𝑝^𝑁_𝑝−𝑞^𝑁−1_𝑥=0𝑓(𝑥)(−𝑞)^𝑥,(1.3) cf. [3–6].

If we take 𝑓₁(𝑥)=𝑓(𝑥+1) in (1.1), then we easily see that 𝑞𝐼_−𝑞𝑓₁+𝐼_−𝑞(𝑓)=[2]_𝑞𝑓(0).(1.4) From (1.4), we obtain 𝑞^𝑛𝐼_−𝑞𝑓_𝑛+(−1)^𝑛−1𝐼_−𝑞(𝑓)=[2]_𝑞𝑛−1_𝑙=0(−1)^{𝑛−1−𝑙}𝑞^𝑙𝑓(𝑙),(1.5) where 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛) (cf. [3–6]).

As-well-known definition, the Genocchi polynomials are defined by 𝐹(𝑡)=2𝑡𝑒^𝑡+1=𝑒^𝐺𝑡=^∞_𝑛=0𝐺_𝑛𝑡^𝑛𝐹(𝑡,𝑥)=2𝑡𝑛!,𝑒^𝑡+1𝑒^𝑥𝑡=𝑒^{𝐺(𝑥)𝑡}=^∞_𝑛=0𝐺_𝑛(𝑥)𝑡^𝑛𝑛!.(1.6) with the usual convention of replacing 𝐺^𝑛(𝑥) by 𝐺_𝑛(𝑥). In the special case, 𝑥=0, 𝐺_𝑛(0)=𝐺_𝑛 are called the 𝑛-th Genocchi numbers (cf. [1–11]).

These numbers and polynomials are interpolated by the Genocchi zeta function and Hurwitz-type Genocchi zeta function, respectively. 𝜁_𝐺(𝑠)=2^∞_𝑛=1(−1)^𝑛𝑛^𝑠𝜁_𝐺,(𝑠,𝑥)=2^∞_𝑛=0(−1)^𝑛(𝑛+𝑥)^𝑠.(1.7) Our aim in this paper is to define 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽)𝑛,𝑞} and polynomials 𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽. We investigate some properties which are related to 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽)𝑛,𝑞} and polynomials 𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽. We also derive the existence of a specific interpolation function which interpolates 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽)𝑛,𝑞} and polynomials 𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽 at negative integers.

2. 𝑞-Genocchi Numbers and Polynomials with Weight 𝛼 and Weak Weight 𝛽

Our primary goal of this section is to define 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽)𝑛,𝑞} and polynomials 𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽. We also find generating functions of 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽)𝑛,𝑞} and polynomials 𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽.

For 𝛼∈ℤ and 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝≤1, 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽)𝑛,𝑞} are defined by 𝐺^{(𝛼,𝛽)𝑛,𝑞}=𝑛_ℤ𝑝[𝑥]^𝑞𝑛−1_𝛼𝑑𝜇_−𝑞𝛽(𝑥).(2.1) By using 𝑝-adic 𝑞-integral on ℤ_𝑝, we obtain 𝑛_ℤ𝑝[𝑥]^𝑞𝑛−1_𝛼𝑑𝜇_−𝑞𝛽(𝑥)=𝑛lim_𝑁→∞1𝑝^𝑁_−𝑞𝛽^𝑝𝑁−1_𝑥=0[𝑥]^𝑞𝑛−1_𝛼−𝑞^𝛽^𝑥=𝑛[2]_𝑞𝛽1−1𝑞^𝛼^{𝑛−1𝑛−1}_𝑙=0⎛⎜⎜⎝𝑛−𝑙1⎞⎟⎟⎠(−1)_𝑙11+𝑞^{𝛼𝑙+𝛽}=𝑛[2]_𝑞𝛽^∞_𝑚₌₀(−1)^𝑚𝑞^𝛽𝑚[𝑚]^𝑞𝑛−1_𝛼.(2.2) By (2.1), we have 𝐺^{(𝛼,𝛽)𝑛,𝑞}=𝑛[2]_𝑞𝛽^∞_𝑚₌₀(−1)^𝑚𝑞^𝛽𝑚[𝑚]^𝑞𝑛−1_𝛼.(2.3) From the above, we can easily obtain that 𝐹^{𝑞(𝛼,𝛽)}(𝑡)=^∞_𝑛=0𝐺^{(𝛼,𝛽)𝑛,𝑞}𝑡^𝑛=𝑡[2]_𝑞𝛽∞𝑛!_𝑚₌₀(−1)^𝑚𝑞^𝛽𝑚𝑒^{[𝑚]𝑞𝛼𝑡}.(2.4) Thus, 𝑞-Genocchi numbers 𝐺^{(𝛼,𝛽)𝑛,𝑞} with weight 𝛼 and weak weight 𝛽 are defined by means of the generating function 𝐹^{𝑞(𝛼,𝛽)}(𝑡)=𝑡[2]_𝑞𝛽^∞_𝑚₌₀(−1)^𝑚𝑞^𝛽𝑚𝑒^{[𝑚]𝑞𝛼𝑡}.(2.5)

Using similar method as above, we introduce 𝑞-Genocchi polynomials 𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥) with weight 𝛼 and weak weight 𝛽.

𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥) are defined by 𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥)=𝑛_ℤ𝑝[𝑥+𝑦]^𝑞𝑛−1_𝛼𝑑𝜇_−𝑞𝛽(𝑦).(2.6) By using 𝑝-adic 𝑞-integral, we have 𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥)=𝑛[2]_𝑞𝛽1−1𝑞^𝛼^{𝑛−1𝑛−1}_𝑙=0⎛⎜⎜⎝𝑛−𝑙1⎞⎟⎟⎠(−1)_𝑙𝑞^𝛼𝑥𝑙11+𝑞^{𝛼𝑙+𝛽}.(2.7) By using (2.6) and (2.7), we obtain 𝐹^{𝑞(𝛼,𝛽)}(𝑡,𝑥)=^∞_𝑛=0𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥)𝑡^𝑛𝑛!=𝑡[2]^𝑞𝛽∞_𝑚=0(−1)^𝑚𝑞^𝛽𝑚𝑒^{[𝑚+𝑥]𝑞𝛼𝑡}.(2.8)

Remark 2.1. In (2.8), we simply see that lim_𝑞→1𝐹^{𝑞(𝛼,𝛽)}(𝑡,𝑥)=2𝑡^∞_𝑚₌₀(−1)^𝑚𝑒^{(𝑚+𝑥)𝑡}=2𝑡1+𝑒^𝑡𝑒^𝑥𝑡=𝐹(𝑡,𝑥).(2.9) Since [𝑥+𝑦]_𝑞𝛼=[𝑥]_𝑞𝛼+𝑞^𝛼𝑥[𝑦]_𝑞𝛼, we easily obtain that 𝐺^{(𝛼,𝛽)𝑛+1,𝑞}(𝑥)==𝑞(𝑛+1)_{−𝛼𝑥𝑛+1}_ℤ𝑝[𝑥+𝑦]^𝑛𝑞_𝛼𝑑𝜇_−𝑞𝛽(𝑦)_𝑘=0⎛⎜⎜⎝𝑛+𝑘1⎞⎟⎟⎠[𝑥]_{𝑞𝑛+1−𝑘𝛼}𝑞^𝛼𝑥𝑘𝐺^{(𝛼,𝛽)𝑘,𝑞}=𝑞^−𝛼𝑥[𝑥]^𝑞𝛼+𝑞^𝛼𝑥𝐺^{𝑞(𝛼,𝛽)}^𝑛+1=(𝑛+1)[2]_𝑞𝛽^∞_𝑚₌₀(−1)^𝑚𝑞^𝛽𝑚[𝑥+𝑚]^𝑛𝑞_𝛼.(2.10) Observe that, if 𝑞→1, then 𝐺^{(𝛼,𝛽)𝑛,𝑞}→𝐺_𝑛 and 𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥)→𝐺_𝑛(𝑥).
By (2.7), we have the following complement relation.

Theorem 2.2. Property of complement 𝐺^{(𝛼,𝛽)𝑛,𝑞}₋₁(1−𝑥)=(−1)^𝑛−1𝑞^{𝛼(𝑛−1)}𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥).(2.11)

By (2.7), we have the following distribution relation.

Theorem 2.3. For any positive integer 𝑚 (=odd), one has 𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥)=[2]^𝑞𝛽[2]_𝑞𝛽𝑚[𝑚]^𝑞𝑛−1_𝛼^𝑚−1_𝑖=0(−1)^𝑖𝑞^𝛽𝑖𝐺^{(𝛼,𝛽)𝑛,𝑞}_𝑚𝑖+𝑥𝑚,𝑛∈ℤ⁺.(2.12) By (1.5), (2.1), and (2.6), one easily sees that 𝑚[2]_𝑞𝛽^𝑛−1_𝑙=0(−1)^{𝑛−1−𝑙}𝑞^𝛽𝑙[𝑙]^𝑞𝑚−1_𝛼=𝑞^𝛽𝑛𝐺^{(𝛼,𝛽)𝑚,𝑞}(𝑛)+(−1)^𝑛−1𝐺^{(𝛼,𝛽)𝑚,𝑞}.(2.13)

Hence, we have the following theorem.

Theorem 2.4. Let 𝑚∈ℤ⁺.
If 𝑛≡0(mod2), then 𝑞^𝛽𝑛𝐺^{(𝛼,𝛽)𝑚,𝑞}(𝑛)−𝐺^{(𝛼,𝛽)𝑚,𝑞}=𝑚[2]_𝑞𝛽^𝑛−1_𝑙=0(−1)^𝑙+1𝑞^𝛽𝑙[𝑙]^𝑞𝑚−1_𝛼.(2.14) If 𝑛≡1(mod2), then 𝑞^𝛽𝑛𝐺^{(𝛼,𝛽)𝑚,𝑞}(𝑛)+𝐺^{(𝛼,𝛽)𝑚,𝑞}=𝑚[2]_𝑞𝛽^𝑛−1_𝑙=0(−1)^𝑙𝑞^𝛽𝑙[𝑙]^𝑞𝑚−1_𝛼.(2.15) From (1.4), one notes that [2]_𝑞𝛽𝑡=𝑞^𝛽_ℤ𝑝=𝑡𝑒^{[𝑥+1]𝑞𝛼𝑡}𝑑𝜇_−𝑞𝛽(𝑥)+_ℤ𝑝𝑡𝑒^{[𝑥]𝑞𝛼𝑡}𝑑𝜇_−𝑞𝛽(𝑥)_∞_𝑛=0𝑞^𝛽_ℤ𝑝𝑛[𝑥+1]^𝑞𝑛−1_𝛼𝑑𝜇_−𝑞𝛽(𝑥)+_ℤ𝑝𝑛[𝑥]^𝑞𝑛−1_𝛼𝑑𝜇_−𝑞𝛽(𝑥)𝑡^𝑛=𝑛!_∞_𝑛=0𝑞^𝛽𝐺^{(𝛼,𝛽)𝑛,𝑞}(1)+𝐺^{(𝛼,𝛽)𝑛,𝑞}𝑡^𝑛𝑛!.(2.16)

Therefore, we obtain the following theorem.

Theorem 2.5. For 𝑛∈ℤ⁺, one has 𝑞^𝛽𝐺^{(𝛼,𝛽)𝑛,𝑞}(1)+𝐺^{(𝛼,𝛽)𝑛,𝑞}=[2]^𝑞𝛽0,,if𝑛=1,if𝑛≠1.(2.17)

By Theorem 2.4 and (2.10), we have the following corollary.

Corollary 2.6. For 𝑛∈ℤ⁺, one has 𝑞^𝛽−𝛼𝑞^𝛼𝐺^{𝑞(𝛼,𝛽)}+1^𝑛+𝐺^{(𝛼,𝛽)𝑛,𝑞}=[2]^𝑞𝛽0,,if𝑛=1,if𝑛≠1.(2.18) with the usual convention of replacing (𝐺^{𝑞(𝛼,𝛽)})^𝑛 by 𝐺^{(𝛼,𝛽)𝑛,𝑞}.

3. The Analogue of the Genocchi Zeta Function

By using 𝑞-Genocchi numbers and polynomials with weight 𝛼 and weak weight 𝛽, 𝑞-Genocchi zeta function and Hurwitz 𝑞-Genocchi zeta functions are defined. These functions interpolate the 𝑞-Genocchi numbers and 𝑞-Genocchi polynomials with weight 𝛼 and weak weight 𝛽, respectively. In this section, we assume that 𝑞∈ℂ with |𝑞|<1. From (2.4), we note that 𝑑^𝑘+1𝑑𝑡^𝑘+1𝐹^{𝑞(𝛼,𝛽)}(𝑡)||||_𝑡=0=(𝑘+1)[2]_𝑞𝛽^∞_𝑚₌₀(−1)^𝑚𝑞^𝛽𝑚[𝑚]^𝑘𝑞_𝛼=𝐺^{(𝛼,𝛽)𝑘+1,𝑞},(𝑘∈ℕ).(3.1) By using the above equation, we are now ready to define 𝑞-Genocchi zeta functions.

Definition 3.1. Let 𝑠∈ℂ. We define 𝜁^{𝑞(𝛼,𝛽)}(𝑠)=[2]_𝑞𝛽^∞_𝑛=1(−1)^𝑛𝑞^𝛽𝑛[𝑛]^𝑠𝑞_𝛼.(3.2)

Note that 𝜁^{𝑞(𝛼,𝛽)}(𝑠) is a meromorphic function on ℂ. Note that, if 𝑞→1, then 𝜁^{𝑞(𝛼,𝛽)}(𝑠)=𝜁(𝑠) which is the Genocchi zeta functions. Relation between 𝜁^{𝑞(𝛼,𝛽)}(𝑠) and 𝐺^{(𝛼,𝛽)𝑘,𝑞} is given by the following theorem.

Theorem 3.2. For 𝑘∈ℕ, we have 𝜁^{𝑞(𝛼,𝛽)}(−𝑘)=𝐺^{(𝛼,𝛽)𝑘+1,𝑞}𝑘+1.(3.3)Observe that 𝜁^{𝑞(𝛼,𝛽)}(𝑠) function interpolates 𝐺^{(𝛼,𝛽)𝑘,𝑞} numbers at nonnegative integers. By using (2.3), one notes that 𝑑^𝑘+1𝑑𝑡^𝑘+1𝐹^{𝑞(𝛼,𝛽)}(𝑡,𝑥)||||_𝑡=0=(𝑘+1)[2]_𝑞𝛽^∞_𝑚₌₀(−1)^𝑚𝑞^𝛽𝑚[𝑥+𝑚]^𝑘𝑞_𝛼=𝐺^{(𝛼,𝛽)𝑘+1,𝑞}(𝑥),(𝑘∈ℕ),(3.4)𝑑𝑑𝑡^𝑘+1^∞_𝑛=0𝐺^{(𝛼,𝛽)𝑛,𝑞}(𝑥)𝑛𝑡^𝑛!|||||_𝑡=0=𝐺^{(𝛼,𝛽)𝑘+1,𝑞}(𝑥),for𝑘∈ℕ.(3.5)

By (3.2) and (3.5), we are now ready to define the Hurwitz 𝑞-Genocchi zeta functions.

Definition 3.3. Let 𝑠∈ℂ. We define 𝜁^{𝑞(𝛼,𝛽)}(𝑠,𝑥)=[2]_𝑞𝛽^∞_𝑛=0(−1)^𝑛𝑞^𝛽𝑛[𝑛+𝑥]^𝑠𝑞_𝛼.(3.6) Note that 𝜁^{𝑞(𝛼,𝛽)}(𝑠,𝑥) is a meromorphic function on ℂ.

Remark 3.4. It holds that lim_𝑞→1𝜁^{𝑞(𝛼,𝛽)}(𝑠,𝑥)=2^∞_𝑛=0(−1)^𝑛(𝑛+𝑥)^𝑠.(3.7)

Relation between 𝜁^𝑞(𝛼)(𝑠,𝑥) and 𝐺^{(𝛼)𝑘,𝑞}(𝑥) is given by the following theorem.

Theorem 3.5. For 𝑘∈ℕ, one has 𝜁^{𝑞(𝛼,𝛽)}(−𝑘,𝑥)=𝐺^{(𝛼,𝛽)𝑘+1,𝑞}(𝑥)𝑘+1.(3.8) Observe that 𝜁^{𝑞(𝛼,𝛽)}(−𝑘,𝑥) function interpolates 𝐺^{(𝛼,𝛽)𝑘,𝑞}(𝑥) numbers at nonnegative integers.