A Note on Some Properties of the Weighted 𝑞-Genocchi Numbers and Polynomials

Abstract

We consider the weighted 𝑞-Genocchi numbers and polynomials. From the construction of the weighted 𝑞-Genocchi numbers and polynomials, we investigate many interesting identities and relations satisfied by these new numbers and polynomials.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper, ℤ_𝑝, ℚ_𝑝, ℂ, and ℂ_𝑝, will, respectively, denote the ring of 𝑝-adic integers, the field, of 𝑝-adic rational numbers, the complex number field and the completion of algebraic closure of ℚ_𝑝. Let 𝜈_𝑝 be the normalized exponential valuation of ℂ_𝑝 such that |𝑝|_𝑝=𝑝^{−𝜈𝑝(𝑝)}=1/𝑝(see [1–16]).

As well-known definition, the Euler numbers and Genocchi numbers are defined by2𝑒^𝑡+1=𝑒^𝐸𝑡=^∞_𝑛=0𝐸_𝑛𝑡^𝑛,𝑛! (1.1) with the usual convention of replacing 𝐸^𝑛 by 𝐸_𝑛 and2𝑡𝑒^𝑡+1=𝑒^𝐺𝑡=^∞_𝑛=0𝐺_𝑛𝑡^𝑛,𝑛! (1.2) with the usual convention of replacing 𝐺^𝑛 by 𝐺_𝑛. We assume that 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1 and that the 𝑞-number of 𝑥 is defined by[𝑥]_𝑞=1−𝑞^𝑥,[𝑥]1−𝑞^−𝑞=1−−𝑞^𝑥1+𝑞 (1.3) (see [1–19]).

In [9], Kim introduced ordinary fermionic 𝑝-adic integral on ℤ_𝑝, and he studied some interesting relations and identities related to 𝑞-extension of Euler numbers and polynomials. In [8], he also introduced the 𝑞-extension of the ordinary fermionic 𝑝-adic integral on ℤ_𝑝 and he investigated many physical properties related to 𝑞-Euler numbers and polynomials. Recently, Kim firstly introduced the meaning of the weighted 𝑞-Euler numbers and polynomials associated with the weighted 𝑞-Bernstein polynomials by using the fermionic invariant 𝑝-adic integral on ℤ_𝑝(see [14, 15]). In [16], Ryoo tried to study the weighted 𝑞-Euler number and polynomials by the same method of Kim et al. in [14] and the 𝑞-extension of the fermionic 𝑝-adic invariant integrals on ℤ_𝑝. As well-known properties, the Genocchi numbers are integers. The first few Genocchi numbers for 𝑛=2,4,… are −1,1,−3,17,−155,2073,…. The first few prime Genocchi numbers are −3 and 17, which occur for 𝑛=6 and 8. There are no others with 𝑛<10⁵. These properties are very important to study in the area of fermionic distribution and 𝑝-adic numbers theory. By this reason, many mathematicians and physicians have studied Genocchi and Euler numbers which are in the different areas. By the same motivation, we consider weighted 𝑞-Genocchi polynomials and numbers by using the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 which are constructed by Kim and Ryoo (cf. [8, 16]).

In this paper, we consider the 𝑞-Genocchi numbers and polynomials with weighted 𝛼(𝛼∈ℚ). From the construction of the weighted 𝑞-Genocchi numbers and polynomials, we investigate many interesting identities and relations satisfied by these new numbers and polynomials.

2. The Weighted 𝑞-Genocchi Numbers and Polynomials

Let UD(ℤ_𝑝) be the space of uniformly differentiable functions and, for 𝑓∈UD(ℤ_𝑝), the fermionic 𝑝-adic invariant integral of 𝑓 on ℤ_𝑝 is defined by Kim as follows:𝐼₋₁𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)=lim_𝑁→∞1𝑝^𝑁𝑝₋₁^𝑁−1_𝑥=0𝑓(𝑥)(−1)^𝑥 (2.1) (see [1–16]). If we take 𝑓(𝑥)=𝑡𝑒^𝑥𝑡, then we get𝑡_ℤ𝑝𝑒^𝑥𝑡𝑑𝜇₋₁(𝑥)=2𝑡𝑒^𝑡.+1 (2.2) By (1.2) and (2.2), we get^∞_𝑛=0𝐺_𝑛𝑡^𝑛=𝑛!^∞_{𝑛=0ℤ𝑝}𝑥^𝑛𝑑𝜇₋₁(𝑥)𝑡^𝑛+1=𝑛!_∞𝑛=0(𝑛+1)_ℤ𝑝𝑥^𝑛𝑑𝜇₋₁(𝑥)𝑡^𝑛+1()!=𝑛+1_∞𝑛=1𝑛_ℤ𝑝𝑥^𝑛−1𝑑𝜇₋₁(𝑥)𝑡^𝑛.𝑛! (2.3) From (2.3),𝐺₀𝐺=0,^𝑛𝑛=_ℤ𝑝𝑥^𝑛−1𝑑𝜇₋₁(𝑥),𝑛∈ℕ. (2.4) For 𝑓∈UD(ℤ_𝑝), the fermionic 𝑝-adic 𝑞-integral of 𝑓 on ℤ_𝑝 is defined by Kim as follows:𝐼_−𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁𝑝_−𝑞^𝑁−1_𝑥=0𝑓(𝑥)−𝑞^𝑥 (2.5) (see [1–16]). From (2.5), we note that𝑞^𝑛𝐼_−𝑞𝑓_𝑛=(−1)^𝑛𝐼_−𝑞𝑓+[2]_𝑞^𝑛−1_𝑙=0(−1)^{𝑛−1−𝑙}𝑞^𝑙𝑓(𝑙), (2.6) where 𝑛∈ℕ and 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛).

For 𝛼∈ℚ, we consider the following fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝:𝑡_ℤ𝑝𝑒^{[𝑥]𝑞𝛼𝑡}𝑑𝜇_−𝑞(𝑥)=^∞_𝑛=0𝐺^(𝛼)_𝑛,𝑞𝑡^𝑛,𝑛! (2.7) where 𝐺^(𝛼)_𝑛,𝑞 are called the 𝑛th 𝑞-Genocchi numbers with weight 𝛼. From (2.7), we get𝑡_ℤ𝑝𝑒^{[𝑥]𝑞𝛼𝑡}𝑑𝜇_−𝑞(𝑥)=𝑡^∞_{𝑛=0ℤ𝑝}[𝑥]^𝑛_𝑞𝛼𝑑𝜇_−𝑞(𝑥)𝑡^𝑛=𝑛!_∞𝑛=0(𝑛+1)_ℤ𝑝[𝑥]^𝑛_𝑞𝛼𝑑𝜇_−𝑞(𝑥)𝑡^𝑛+1()!=𝑛+1_∞𝑛=1𝑛_ℤ𝑝[𝑥]_𝑞^𝑛−1_𝛼𝑑𝜇_−𝑞(𝑥)𝑡^𝑛.𝑛! (2.8) By comparing the coefficients on the both sides of (2.7) and (2.8), we get𝑛_ℤ𝑝[𝑥]_𝑞^𝑛−1_𝛼𝑑𝜇_−𝑞(𝑥)=𝐺^(𝛼)_𝑛,𝑞,𝑛∈ℕ,𝐺^(𝛼)_0,𝑞=0. (2.9) From (2.9), we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℕ and 𝛼∈ℚ, one has _ℤ𝑝[𝑥]_𝑞^𝑛−1_𝛼𝑑𝜇_−𝑞(𝑥)=𝐺^(𝛼)_𝑛,𝑞𝑛,𝐺^(𝛼)_0,𝑞=0. (2.10)

By the definition of fermionic 𝑝-adic 𝑞-integrals, we get𝑡_ℤ𝑝[𝑥]_𝑞^𝑛−1_𝛼𝑑𝜇_−𝑞(𝑥)=11−𝑞^{𝛼𝑛−1𝑛−1}_𝑙=0𝑙()𝑛−1−1^𝑙_ℤ𝑝𝑞^𝛼𝑙𝑥𝑑𝜇_−𝑞(𝑥)=[2]_𝑞1−𝑞^{𝛼𝑛−1𝑛−1}_𝑙=0𝑙()𝑛−1−1^𝑙11+𝑞^𝛼𝑙+1. (2.11) Therefore, we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℕ and 𝛼∈ℚ, we have 𝐺^(𝛼)_𝑛,𝑞𝑛=[2]_𝑞1−𝑞^𝑛−1[𝛼]_𝑞^{𝑛−1𝑛−1}_𝑙=0𝑙()𝑛−1−1^𝑙11+𝑞^𝛼𝑙+1. (2.12)

By Theorem 2.2, we have the generating function of 𝐺^(𝛼)_𝑛,𝑞 as follows:^∞_𝑛=0𝐺^(𝛼)_𝑛,𝑞𝑡^𝑛=[2]𝑛!^𝑞∞_𝑛=0𝑛1−𝑞^{𝛼𝑛−1𝑛−1}_𝑙=0𝑙()𝑛−1−1^𝑙∞_𝑚=0(−1)^𝑚𝑞^{𝛼𝑙𝑚+𝑚}𝑡^𝑛=[2]𝑛!_{𝑞∞𝑚=0}(−1)^𝑚𝑞^𝑚∞_𝑛=0𝑛1−𝑞^{𝛼𝑛−1𝑛−1}_𝑙=0𝑙()𝑛−1−1^𝑙𝑞^𝛼𝑙𝑚𝑡^𝑛=[2]𝑛!_{𝑞∞𝑚=0}(−1)^𝑚𝑞^𝑚∞_𝑛=0𝑛1−𝑞^𝛼𝑛−11−𝑞^{𝛼𝑚𝑛−1}𝑡^𝑛=[2]𝑛!_{𝑞∞𝑚=0}(−1)^𝑚𝑞^𝑚∞_𝑛=0𝑛[𝑚]_𝑞^𝑛−1_𝛼𝑡^𝑛=[2]𝑛!_{𝑞∞𝑚=0}(−1)^𝑚𝑞^𝑚∞_𝑛=1[𝑚]_𝑞^𝑛−1_𝛼𝑡^𝑛()!=[2]𝑛−1_{𝑞∞𝑚=0}(−1)^𝑚𝑞^𝑚∞_𝑛=0[𝑚]^𝑛_𝑞𝛼𝑡^𝑛+1=[2]𝑛!_𝑞𝑡^∞_𝑚=0(−1)^𝑚𝑞^𝑚𝑒^{[𝑚]𝑞𝛼𝑡}. (2.13) Let 𝐹_𝑞^(𝛼)(𝑡) be the generating function of 𝐺^(𝛼)_𝑛,𝑞. Then, by (2.9) and (2.13), we get𝐹_𝑞^(𝛼)(𝑡)=[2]_𝑞𝑡^∞_𝑚=0(−1)^𝑚𝑞^𝑚𝑒^{[𝑚]𝑞𝛼𝑡}=^∞_𝑛=0𝐺^(𝛼)_𝑛,𝑞𝑡^𝑛.𝑛! (2.14)

The 𝑞-Genocchi polynomials with weight 𝛼 are defined by𝐹_𝑞^(𝛼)(𝑡,𝑥)=𝑡_ℤ𝑝𝑒^{[𝑥+𝑦]𝑞𝛼𝑡}𝑑𝜇_−𝑞𝑦=^∞_𝑛=0𝐺^(𝛼)_𝑛,𝑞(𝑥)𝑡^𝑛.𝑛! (2.15) From (2.15), we get𝑡_ℤ𝑝𝑒^{[𝑥+𝑦]𝑞𝛼𝑡}𝑑𝜇_−𝑞𝑦=^∞_{𝑛=0ℤ𝑝}𝑥+𝑦^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝑦𝑡^𝑛+1=𝑛!_∞𝑛=0(𝑛+1)_ℤ𝑝𝑥+𝑦^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝑦𝑡^𝑛+1()!=𝑛+1_∞𝑛=1𝑛_ℤ𝑝𝑥+𝑦_𝑞^𝑛−1_𝛼𝑑𝜇_−𝑞𝑦𝑡^𝑛.𝑛! (2.16) By (2.15) and (2.16), we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℕ and 𝛼∈ℚ, one has 𝑛_ℤ𝑝𝑥+𝑦_𝑞^𝑛−1_𝛼𝑑𝜇_−𝑞𝑦=𝐺^(𝛼)_𝑛,𝑞(𝑥),𝐺^(𝛼)_0,𝑞(𝑥)=0. (2.17)

We note that_ℤ𝑝𝑥+𝑦_𝑞^𝑛−1_𝛼𝑑𝜇_−𝑞𝑦=^𝑛−1_𝑙=0𝑙[𝑥]𝑛−1_{𝑞𝑛−1−𝑙𝛼}𝑞^𝛼𝑙𝑥_ℤ𝑝𝑦^𝑙_𝑞𝛼𝑑𝜇_−𝑞𝑦=^𝑛−1_𝑙=0𝑙[𝑥]𝑛−1_{𝑞𝑛−1−𝑙𝛼}𝑞^𝛼𝑙𝑥𝐺^(𝛼)_𝑙+1,𝑞.𝑙+1 (2.18) From (2.17) and (2.18), we obtain the following theorem.

Theorem 2.4. For 𝑛∈ℕ and 𝛼∈ℚ, one has 𝐺^(𝛼)_𝑛,𝑞(𝑥)𝑛=^𝑛−1_𝑙=0𝑙[𝑥]𝑛−1_{𝑞𝑛−1−𝑙𝛼}𝑞^𝛼𝑙𝑥𝐺^(𝛼)_𝑙+1,𝑞.𝑙+1 (2.19)

From (2.15), we note that𝐹_𝑞^(𝛼)(𝑡,𝑥)=^∞_𝑛=0𝐺^(𝛼)_𝑛,𝑞(𝑥)𝑡^𝑛=𝑛!_∞𝑛=0𝑛_ℤ𝑝𝑥+𝑦_𝑞^𝑛−1_𝛼𝑑𝜇_−𝑞𝑦𝑡^𝑛=𝑛!_∞𝑛=0𝑛11−𝑞^{𝛼𝑛−1𝑛−1}_𝑙=0𝑙𝑞𝑛−1_𝛼𝑙𝑥(−1)^𝑙_ℤ𝑝𝑞^𝛼𝑙𝑦𝑑𝜇_−𝑞𝑦𝑡^𝑛=[2]𝑛!_{𝑞∞𝑛=0}𝑛11−𝑞^{𝛼𝑛−1𝑛−1}_𝑙=0𝑙𝑞𝑛−1_𝛼𝑙𝑥(−1)^𝑙1+𝑞^𝛼𝑙+1𝑡^𝑛=[2]𝑛!_{𝑞∞𝑛=0}𝑛11−𝑞^{𝛼𝑛−1𝑛−1}_𝑙=0𝑙𝑞𝑛−1_𝛼𝑙𝑥(−1)^𝑙∞_𝑚=0(−1)^𝑚𝑞^{𝛼𝑙𝑚+𝑚}𝑡^𝑛=[2]𝑛!_{𝑞∞𝑚=0}(−1)^𝑚𝑞^𝑚∞_𝑛=0𝑛11−𝑞^{𝛼𝑛−1𝑛−1}_𝑙=0𝑙()𝑛−1−1^𝑙𝑞^{𝛼(𝑥+𝑚)𝑙}𝑡^𝑛=[2]𝑛!_{𝑞∞𝑚=0}(−1)^𝑚𝑞^𝑚∞_𝑛=0𝑛11−𝑞^𝛼𝑛−11−𝑞^{𝛼(𝑥+𝑚)𝑛−1}𝑡^𝑛=[2]𝑛!_{𝑞∞𝑚=0}(−1)^𝑚𝑞^𝑚∞_𝑛=0𝑛[𝑥+𝑚]_𝑞^𝑛−1_𝛼𝑡^𝑛=[2]𝑛!_{𝑞∞𝑚=0}(−1)^𝑚𝑞^𝑚∞_𝑛=1[𝑥+𝑚]_𝑞^𝑛−1_𝛼𝑡^𝑛()!=[2]𝑛−1_{𝑞∞𝑚=0}(−1)^𝑚𝑞^𝑚∞_𝑛=0[𝑥+𝑚]^𝑛_𝑞𝛼𝑡^𝑛+1=[2]𝑛!_𝑞𝑡^∞_𝑚=0(−1)^𝑚𝑞^𝑚𝑒^{[𝑥+𝑚]𝑞𝛼𝑡}. (2.20) Therefore, we obtain the following theorem.

Theorem 2.5. For 𝛼∈ℚ, one has 𝐹_𝑞^(𝛼)(𝑡,𝑥)=[2]_𝑞𝑡^∞_𝑚=0(−1)^𝑚𝑞^𝑚𝑒^{[𝑥+𝑚]𝑞𝛼𝑡}. (2.21)

From (2.15) and (2.21), we obtain that𝐺^(𝛼)_𝑛,𝑞(𝑥)=𝑑^𝑛𝑑𝑡^𝑛𝐹_𝑞^(𝛼)(𝑡,𝑥)_𝑡=0=𝑛[2]_𝑞^∞_𝑚=0(−1)^𝑚𝑞^𝑚[𝑥+𝑚]_𝑞^𝑚−1_𝛼=𝑛[2]_𝑞11−𝑞^{𝛼𝑛−1𝑛−1}_𝑙=0^𝑙𝑛−1𝑞^𝛼𝑙𝑥(−1)^𝑙1+𝑞^𝛼𝑙+1=𝑛[2]_𝑞1−𝑞^𝑛−1[𝛼]_𝑞^{𝑛−1𝑛−1}_𝑙=0𝑙()𝑛−1−1^𝑙𝑞^𝛼𝑙𝑥11+𝑞^𝛼𝑙+1. (2.22) Therefore, we obtain the following theorem.

Theorem 2.6. For 𝑛∈ℕ and 𝛼∈ℚ, one has 𝐺^(𝛼)_𝑛,𝑞(𝑥)=𝑛[2]_𝑞1−𝑞^𝑛−1[𝛼]_𝑞^{𝑛−1𝑛−1}_𝑙=0𝑙𝑛−1(−1)^𝑙𝑞^𝛼𝑙𝑥1+𝑞^𝛼𝑙+1. (2.23)

From (2.6), if we take 𝑓(𝑥)=[𝑥]^𝑚_𝑞𝛼=((1−𝑞^𝛼𝑥)/(1−𝑞^𝛼))^𝑚, then we get𝑞^𝑛_ℤ𝑝[𝑥+𝑛]^𝑚_𝑞𝛼𝑑𝜇_−𝑞(𝑥)=(−1)^𝑛_ℤ𝑝[𝑥]^𝑚_𝑞𝛼𝑑𝜇_−𝑞(𝑥)+[2]_𝑞^𝑛−1_𝑙=0(−1)^𝑙𝑞^𝑙[𝑙]^𝑚_𝑞𝛼. (2.24) By (2.17) and (2.24), we obtain the following theorem.

Theorem 2.7. For 𝑛∈ℕ,𝑚∈ℤ₊=ℕ∪{0}, and 𝛼∈ℚ, one has 𝑞^𝑛𝐺^(𝛼)_𝑚+1,𝑞(𝑛)=()𝑚+1−1^𝑛𝐺^(𝛼)_𝑚+1,𝑞+[2]𝑚+1^𝑞𝑛−1_𝑙=0(−1)^𝑙𝑞^𝑙[𝑙]^𝑚_𝑞𝛼. (2.25)

We remark that if we take 𝑛=2𝑠(𝑠∈ℤ₊) in Theorem 2.7, then we have𝑞^2𝑠𝐺^(𝛼)_𝑚+1,𝑞(2𝑠)=𝐺𝑚+1^{(𝛼)𝑚+1,𝑞}+[2]𝑚+1^{𝑞2𝑠−1}_𝑙=0(−1)^𝑙𝑞^𝑙[𝑙]^𝑚_𝑞𝛼 (2.26) and if we take 𝑛=2𝑠+1(𝑠∈ℤ₊) in Theorem 2.7, then we have𝑞^2𝑠+1𝐺^(𝛼)_𝑚+1,𝑞(2𝑠+1)+𝐺𝑚+1^{(𝛼)𝑚+1,𝑞}=[2]𝑚+1^𝑞2𝑠_𝑙=0(−1)^𝑙𝑞^𝑙[𝑙]^𝑚_𝑞𝛼. (2.27) From (2.27) with 𝑠=0, we obtain the following corollary.

Corollary 2.8. For 𝛼∈ℚ and 𝑚∈ℤ₊, one has 𝑞𝐺^(𝛼)_𝑚+1,𝑞(1)+𝐺𝑚+1^{(𝛼)𝑚+1,𝑞}=[2]𝑚+1^𝑞if0𝑚=0,if𝑚>0. (2.28)

From (2.19), we note that𝐺^(𝛼)_𝑚+1,𝑞(𝑛)=𝑚+1^𝑚_𝑙=0𝑚𝑙[𝑥]_𝑞^𝑚−𝑙_𝛼𝐺^(𝛼)_𝑙+1,𝑞𝑞𝑙+1^𝛼𝑙𝑥=1𝑚+1^𝑚_𝑙=0[𝑥]𝑚+1𝑙+1^{𝑞𝑚−𝑙𝛼}𝐺^(𝛼)_𝑙+1,𝑞𝑞^𝛼𝑙𝑥=1𝑚+1^𝑚_𝑙=1𝑙[𝑥]𝑚+1_{𝑞𝑚+1−𝑙𝛼}𝐺^(𝛼)_𝑙,𝑞𝑞^{𝛼(𝑙−1)𝑥}=1𝑞^𝛼1𝑚+1^𝑚_𝑙=0𝑙[𝑥]𝑚+1_{𝑞𝑚+1−𝑙𝛼}𝐺^(𝛼)_𝑙,𝑞𝑞^𝛼𝑙𝑥. (2.29) From (2.29), we get𝑞^𝛼𝐺^(𝛼)_𝑚+1,𝑞(𝑥)=^𝑚+1_𝑙=0𝑙[𝑥]𝑚+1_{𝑞𝑚+1−𝑙𝛼}𝐺^(𝛼)_𝑙,𝑞𝑞^𝛼𝑙𝑥=[𝑥]_𝑞𝛼+𝑞^𝛼𝑥𝐺_𝑞^(𝛼)𝑚+1, (2.30) with the usual convention about replacing (𝐺_𝑞^(𝛼))^𝑛 by 𝐺^(𝛼)_𝑛,𝑞. By (2.28) and (2.30), we get𝑞^1−𝛼𝑞^𝛼𝐺^(𝛼)_𝑚+1,𝑞(1)𝑚+𝐺^(𝛼)_𝑚+1,𝑞=𝑞𝑚+1^1−𝛼1+𝑞^𝛼𝐺_𝑞^(𝛼)𝑚+1+𝐺𝑚+1^{(𝛼)𝑚+1,𝑞}.𝑚+1 (2.31) From (2.28) and (2.31), we obtain the following theorem.

Theorem 2.9. For 𝛼∈ℚ and 𝑚∈ℤ₊, one has 𝑞^1−𝛼1+𝑞^𝛼𝐺_𝑞^(𝛼)𝑚+1+𝐺^(𝛼)_𝑚+1,𝑞=[2]_𝑞if0𝑚=0,if𝑚>0. (2.32)

Acknowledgment

This paper was supported by the Konkuk University in 2011.