𝑞-Genocchi Numbers and Polynomials Associated with Fermionic 𝑝-Adic Invariant Integrals on ℤ_𝑝

Abstract

The main purpose of this paper is to present a systemic study of some families of multiple Genocchi numbers and polynomials. In particular, by using the fermionic 𝑝-adic invariant integral on ℤ_𝑝, we construct 𝑝-adic Genocchi numbers and polynomials of higher order. Finally, we derive the following interesting formula: 𝐺^(𝑘)_{𝑛+𝑘,𝑞}(𝑥)=2^𝑘𝑘!(_𝑘^𝑛+𝑘)∑^∞_𝑙=0∑_{𝑑0+𝑑1+⋯+𝑑𝑘=𝑘−1,𝑑𝑖∈ℕ}(−1)^𝑙(𝑙+𝑥)^𝑛, where 𝐺^(𝑘)_{𝑛+𝑘,𝑞}(𝑥) are the 𝑞-Genocchi polynomials of order 𝑘.

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 rational integers, the field of 𝑝-adic rational numbers, the complex number field and the 𝑝-adic completion of the algebraic closure of ℚ_𝑝. Let 𝜈_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{𝜈𝑝(𝑝)}=1/𝑝. When one talks of 𝑞-extension, 𝑞 is variously considered as an indeterminate, a complex number 𝑞∈ℂ, or a 𝑝-adic number 𝑞∈ℂ_𝑝. If 𝑞∈ℂ, one normally assumes |𝑞|<1. If 𝑞∈ℂ_𝑝, then we assume that |1−𝑞|_𝑝<1, see [1–6].

In ℂ, the ordinary Euler polynomials are defined as 2𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_𝑛𝑡(𝑥)^𝑛𝑛!,(|𝑡|<𝜋). (1.1) In the case 𝑥=0,𝐸_𝑛(0)=𝐸_𝑛 are called Euler numbers, see [1–13]. Let 𝛿_0,𝑛 be the Kronecker symbol. From (1.1) we derive the following relation: 𝐸₀=1,(𝐸+1)^𝑛+𝐸_𝑛=2𝛿_0,𝑛,𝑛∈ℕ, (1.2) (cf. [7–13]). Here, we use the technique method notation by replacing 𝐸^𝑛 by 𝐸_𝑛(𝑛≥0), symbolically. The first few are 1,−1/2,0,1/4,…, and 𝐸_2𝑘=0 for 𝑘=1,2,…. A sequence consisting of the Genocchi numbers 𝐺_𝑛 satisfies the following relations: 2𝑡𝑒^𝑡=+1^∞_𝑛=0𝐺_𝑛𝑡^𝑛𝑛!,(|𝑡|<𝜋), (1.3) see [11, 12]. It satisfies 𝐺₁=1,𝐺₃=𝐺₅=𝐺₇=⋯=𝐺_2𝑘+1=0,𝑘=1,2,3,…, and even coefficients are given by 𝐺_2𝑛=2(1−2^2𝑛)𝐵_2𝑛=2𝑛𝐸_2𝑛−1(0), (1.4) where 𝐵_𝑛 is Bernoulli numbers. The first few Genocchi numbers for even integers are −1,1,−3,17,−155,2073,…. The first few prime Genocchi numbers are −3 and 17, which occur at 𝑛=6 and 8. There are no others with 𝑛<10⁵. We now define the Genocchi polynomials as follows: 2𝑡𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐺_𝑛𝑡(𝑥)^𝑛𝑛!,(|𝑡|<𝜋). (1.5) Thus, we note that 𝐺_𝑛(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐺_𝑙𝑥^𝑛−𝑙. (1.6)

In this paper, we use the following notations: [𝑥]_𝑞=(1−𝑞^𝑥)/(1−𝑞) and [𝑥]_−𝑞=(1+(−𝑞)^𝑥)/(1+𝑞). Let 𝑈𝐷(ℤ_𝑝) be the space of uniformly differentiable functions on ℤ_𝑝. For 𝑓∈𝑈𝐷(ℤ_𝑝), the 𝑝-adic 𝑞-deformed fermionic integral on ℤ_𝑝 is defined as 𝐼_−𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1[𝑝^𝑁]_𝑞^𝑝𝑁−1_𝑥=0𝑓(𝑥)(−𝑞)^𝑥, (1.7) see [1–4]. The fermionic 𝑝-adic invariant integral on ℤ_𝑝 can be obtained as 𝑞→1. That is, 𝐼₋₁(𝑓)=lim_𝑞→1𝐼_−𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥). (1.8) From (1.8), we easily derive the following integral equation related to fermionic invariant 𝑝-adic integral on ℤ_𝑝: 𝐼₋₁(𝑓₁)+𝐼₋₁(𝑓)=2𝑓(0), (1.9) where 𝑓₁(𝑥)=𝑓(𝑥+1), see [5].

The purpose of this paper is to present a systemic study of some families of multiple Genocchi numbers and polynomials by using the fermionic multivariate 𝑝-adic invariant integral on ℤ_𝑝. In addition, we will investigate some interesting identities related to Genocchi numbers and polynomials.

2. Genocchi Numbers Associated with Fermionic 𝑝-Adic Invariant Integral on ℤ_𝑝

From (1.9) we can derive 𝑡_ℤ𝑝𝑒^𝑥𝑡𝑑𝜇₋₁(𝑥)=2𝑡𝑒^𝑡=+1^∞_𝑛=0𝐺_𝑛𝑡^𝑛,𝑛!𝑡_ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑑𝜇₋₁(𝑦)=2𝑡𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐺_𝑛𝑡(𝑥)^𝑛,𝑛! (2.1) where 𝐺_𝑛(𝑥) are Genocchi polynomials. It is easy to check that 𝑡_ℤ𝑝𝑒^{(2𝑥+1)𝑡}𝑑𝜇₋₁(𝑥)=2𝑡𝑒^𝑡+𝑒^−𝑡=𝑡sech1𝑡=2^∞_𝑛=0(^𝑛_𝑙=0𝑛𝑙2^𝑙𝐺_𝑙)𝑡^𝑛.𝑛! (2.2) By comparing the coefficient on both sides in (2.1), we easily see that _ℤ𝑝(𝑥+𝑦)^𝑛𝑑𝜇₋₁𝐺(𝑥)=^𝑛+1(𝑥).𝑛+1 (2.3) Therefore, we obtain the following proposition.

Proposition 2.1. For 𝑘∈ℤ₊,
(i)∫_ℤ𝑝(𝑥+𝑦)^𝑛𝑑𝜇₋₁(𝑥)=𝐺_𝑛+1(𝑥)/(𝑛+1) (Witt's formula for Genocchi polynomials);(ii)∫_ℤ𝑝𝑒^{(2𝑥+1)𝑡}𝑑𝜇₋₁(𝑥)=(1/(𝑛+1))((1/2)∑^𝑛+1_𝑙=0(_𝑙^𝑛+1)2^𝑙𝐺_𝑙), where (^𝑛_𝑙)=(𝑛(𝑛−1)⋯(𝑛−𝑙+1))/𝑙!.

Let 𝜗_ℂ𝑝={𝑥∈ℂ_𝑝∣|𝑥|_𝑝≤1} be the integer ring of ℂ_𝑝. We note that 𝑖=(−1)^1/2∈𝜗_ℂ𝑝. By using Taylor expansion, we see that 𝑒^𝑖𝑥=∑^∞_𝑛=0𝑖^𝑛𝑥^𝑛=𝑛!^∞_𝑛=0(−1)^𝑛𝑥^2𝑛(2𝑛)!+𝑖^∞_𝑛=0(−1)^𝑛𝑥^2𝑛+1.(2𝑛+1)! (2.4) In the 𝑝-adic number field, sin𝑥 and cos𝑥 are defined as 𝑥sin𝑥=−𝑥1!³+𝑥3!⁵−𝑥5!⁷𝑥7!+⋯,cos𝑥=1−²+𝑥2!⁴−𝑥4!6⁶!+⋯. (2.5) From (2.4) and (2.5), we derive 𝑒^𝑖𝑥=cos𝑥+𝑖sin𝑥. (2.6) This is equivalent to 𝑒cos𝑥=^𝑖𝑥+𝑒^−𝑖𝑥2𝑒,sin𝑥=^𝑖𝑥−𝑒^−𝑖𝑥.2𝑖 (2.7) By (2.7), we easily see that =2sec𝑡𝑒^𝑖𝑡+𝑒^−𝑖𝑡=∫_ℤ𝑝𝑒^{(2𝑥+1)𝑖𝑡}𝑑𝜇₋₁(𝑥)=^∞_{𝑛=0ℤ𝑝}(2𝑥+1)^𝑛𝑑𝜇₋₁𝑖(𝑥)^𝑛𝑡^𝑛.𝑛! (2.8) It is not difficult to show that ∫_ℤ𝑝(2𝑥+1)^2𝑛+1𝑑𝜇₋₁(𝑥)=0 for 𝑛∈ℤ₊=𝑁∪{0}. From (2.8), we note that sec𝑡=^∞_𝑛=0𝑖^𝑛_ℤ𝑝(2𝑥+1)^𝑛𝑑𝜇₋₁𝑡(𝑥)^𝑛=𝑛!^∞_𝑛=0(−1)^𝑛_ℤ𝑝(2𝑥+1)^2𝑛𝑑𝜇₋₁𝑡(𝑥)^2𝑛.(2𝑛)! (2.9) Thus, we have 1𝑡sec𝑡=2^∞_𝑛=0(−1)^𝑛(^2𝑛+1_𝑙=0𝑙22𝑛+1_𝑙𝐺_𝑙)𝑡^2𝑛+1.(2𝑛+1)! (2.10)

Now we consider the fermionic multivariate 𝑝-adic invariant integral on ℤ_𝑝 as follows: 𝑡^𝑘_ℤ𝑝⋯_{ℤ𝑝𝑘-times}𝑒^{(𝑥1+𝑥2+⋯+𝑥𝑘)𝑡}𝑑𝜇₋₁(𝑥₁)⋯𝑑𝜇₋₁(𝑥_𝑘)=2^𝑘𝑡^𝑘(𝑒^𝑡+1)⋯(𝑒^𝑡+1)_𝑘-times=^∞_𝑛=0𝐺_𝑛^(𝑘)𝑡^𝑛,𝑛! (2.11) where 𝐺_𝑛^(𝑘) are the 𝑛th Genocchi number of order 𝑘. By comparing the coefficient on both sides in (2.11), we see that 𝐺₀^(𝑘)=𝐺₁^(𝑘)=⋯=𝐺_𝑛^(𝑘)=0, and _{ℤ𝑝ℤ𝑝}⋯_𝑘-times(𝑥₁+𝑥₂+⋯+𝑥_𝑘)^𝑛𝑑𝜇₋₁(𝑥₁)⋯𝑑𝜇₋₁(𝑥_𝑘)(𝑛+𝑘)_𝑘=𝐺^(𝑘)_𝑛+𝑘, (2.12) where (𝑛+𝑘)_𝑘 is the Jordan factor which is defined by (𝑛+𝑘)_𝑘=(𝑛+𝑘)⋯(𝑛+1). Thus, we note that _{ℤ𝑝ℤ𝑝}⋯_𝑘-times(𝑥₁+𝑥₂+⋯+𝑥_𝑘)^𝑛𝑑𝜇₋₁(𝑥₁)⋯𝑑𝜇₋₁(𝑥_𝑘𝐺)=^{(𝑘)𝑛+𝑘}(_𝑛^𝑛+𝑘,)𝑛! (2.13) for 𝑘∈𝑁,𝑛∈ℤ₊.

Theorem 2.2. For 𝑛∈ℤ₊,𝑘∈ℕ, _{ℤ𝑝ℤ𝑝}⋯_𝑘-times(𝑥₁+𝑥₂+⋯+𝑥_𝑘)^𝑛𝑑𝜇₋₁(𝑥₁)⋯𝑑𝜇₋₁(𝑥_𝑘𝐺)=^{(𝑘)𝑛+𝑘}(_𝑛^𝑛+𝑘.)𝑛! (2.14)

The multinomial coefficient is well known as (𝑥₁+𝑥₂+⋯+𝑥_𝑘)^𝑛=_{𝑙1+⋯+𝑙𝑘=𝑛𝑙1𝑘,…,𝑙>0}𝑛𝑙₁,…,𝑙_𝑘𝑥^𝑙1₁𝑥^𝑙2₂⋯𝑥^𝑙𝑘_𝑘. (2.15) Therefore, we obtain the following corollary.

Corollary 2.3. For 𝑛∈ℤ₊, 𝑘∈ℕ, _{𝑙1+⋯+𝑙𝑘=𝑛𝑙1𝑘,…,𝑙>0}𝑛𝑙₁,…,𝑙_𝑘(𝐺_𝑙1+1𝑙₁𝐺+1)(^𝑙2+1𝑙₂𝐺+1)⋯(^𝑙𝑘+1𝑙_𝑘𝐺+1)=^{(𝑘)𝑛+𝑘}(_𝑛^𝑛+𝑘.)𝑛! (2.16)

For 𝑞∈ℂ_𝑝 with |1−𝑞|<1, it is not difficult to show that 𝑡_ℤ𝑝𝑞^𝑥𝑒^𝑥𝑡𝑑𝜇₋₁(𝑥)=2𝑡𝑞𝑒^𝑡.+1 (2.17) Now, we define the 𝑞-extension of the Genocchi numbers as follows: 2𝑡𝑞𝑒^𝑡=+1^∞_𝑛=0𝐺_𝑛,𝑞𝑡^𝑛.𝑛! (2.18) By (2.17) and (2.18), we easily see that 𝐺_𝑛+1,𝑞=𝑛+1_ℤ𝑝𝑞^𝑥𝑥^𝑛𝑑𝜇₋₁(𝑥). (2.19) With the same motivation to construct the Genocchi polynomials of higher order, we can consider the 𝑞-extension of higher-order Genocchi numbers as follows: 𝑡^𝑘_ℤ𝑝⋯_{ℤ𝑝𝑘-times}𝑞^{𝑥2+2𝑥3+⋯+(𝑘−1)𝑥𝑘}𝑒^{(𝑥+𝑥1+𝑥2+⋯+𝑥𝑘)𝑡}𝑑𝜇₋₁(𝑥₁)⋯𝑑𝜇₋₁(𝑥_𝑘)=𝑡^𝑘2^𝑘(𝑒^𝑡+1)(𝑞𝑒^𝑡+1)⋯(𝑞^𝑘−1𝑒^𝑡+1)_𝑘-times𝑒^𝑥𝑡=^∞_𝑛=0𝐺^(𝑘)_𝑛,𝑞𝑡^𝑛,𝑛! (2.20) where 𝐺^(𝑘)_𝑛,𝑞 are the 𝑞-Genocchi polynomials of order 𝑘. The basic 𝑞-natural numbers are defined as [𝑛]_𝑞=1−𝑞^𝑛1−𝑞=1+𝑞+𝑞²+⋯+𝑞^𝑛−1,𝑛∈𝑁. (2.21) The 𝑞-factorial of 𝑛 is defined as [𝑛]_𝑞!=[𝑛]_𝑞[𝑛−1]_𝑞⋯[2]_𝑞[1]_𝑞=(1+𝑞+⋯+𝑞^𝑛−1)⋯(1+𝑞)⋅1. (2.22)The 𝑞-binomial coefficient is also defined as 𝑛𝑘_𝑞=[𝑛]_𝑞![𝑛−𝑘]_𝑞![𝑘]_𝑞!=[𝑛]_𝑞[𝑛−1]_𝑞⋯[𝑛−𝑘+1]_𝑞[𝑘]_𝑞!. (2.23) Note that lim_𝑞→1(^𝑛_𝑘)_𝑞=(^𝑛_𝑘)=(𝑛(𝑛−1)⋯(𝑛−𝑘+1))/𝑘!. The 𝑞-binomial coefficient satisfies the following recurrsion formula: 𝑘𝑛+1_𝑞=𝑛𝑘−1_𝑞+𝑞^𝑘𝑛𝑘_𝑞=𝑞^𝑛−𝑘𝑛𝑘−1_𝑞+𝑛𝑘_𝑞. (2.24) From this recurrsion formula, we can derive 𝑛𝑘_𝑞=_{𝑑0+𝑑1+⋯+𝑑𝑘=𝑘−1𝑑𝑖∈𝑁}𝑞^{0⋅𝑑0+1⋅𝑑1+⋯+𝑘⋅𝑑𝑘}. (2.25) The 𝑞-binomial expansion is given by ^𝑛_𝑖=1(𝑎+𝑏𝑞^𝑖−1)=^𝑛_𝑘=0𝑛𝑘_𝑞𝑞^(𝑘2)𝑎^𝑛−𝑘𝑏^𝑘,_𝑛𝑖=1(1−𝑏𝑞^𝑖−1)⁻¹=^∞_𝑘=0𝑘𝑛+𝑘−1_𝑞𝑏^𝑘. (2.26) By (2.20) and (2.26), we see that 𝑡^𝑘_ℤ𝑝⋯_{ℤ𝑝𝑘-times}𝑞^{𝑥2+2𝑥3+⋯+(𝑘−1)𝑥𝑘}𝑒^{(𝑥+𝑥1+𝑥2+⋯+𝑥𝑘)𝑡}𝑑𝜇₋₁(𝑥₁)⋯𝑑𝜇₋₁(𝑥_𝑘)=𝑡^𝑘2^𝑘𝑘_𝑖=1(1−(−𝑞^𝑖−1)𝑒^𝑡)⁻¹𝑒^𝑥𝑡=𝑡^𝑘2^𝑘∞_𝑙=0𝑙𝑘+𝑙−1_𝑞(−1)^𝑙𝑒^{(𝑙+𝑥)𝑡}=𝑡^𝑘∞_𝑛=0(2^𝑘∞_𝑙=0𝑙𝑘+𝑙−1_𝑞(−1)^𝑙(𝑙+𝑥)^𝑛)𝑡^𝑛.𝑛! (2.27) Therefore, we obtain the following theorem.

Theorem 2.4. For 𝑛∈ℤ₊,𝑘∈𝑁, we have _{ℤ𝑝ℤ𝑝}⋯_𝑘-times𝑞^{𝑥2+2𝑥3+⋯+(𝑘−1)𝑥𝑘}(𝑥₁+𝑥₂+⋯+𝑥_𝑘)^𝑛𝑑𝜇₋₁(𝑥₁)⋯𝑑𝜇₋₁(𝑥_𝑘)=2^𝑘∞_𝑙=0𝑙𝑘+𝑙−1_𝑞(−1)^𝑙(𝑙+𝑥)^𝑛. (2.28)

By (2.20), it is not difficult to show that 𝐺^(𝑘)_{𝑛+𝑘,𝑞}𝑘(𝑥)=𝑘!𝑛+𝑘_{ℤ𝑝ℤ𝑝}⋯_𝑘-times𝑞^{𝑥2+2𝑥3+⋯+(𝑘−1)𝑥𝑘}×(𝑥+𝑥₁+𝑥₂+⋯+𝑥_𝑘)^𝑘×𝑑𝜇₋₁(𝑥₁)⋯𝑑𝜇₋₁(𝑥_𝑘)(𝑛+𝑘)_𝑘,𝐺^(𝑘)_0,𝑞=𝐺^(𝑘)_1,𝑞=⋯=𝐺^(𝑘)_𝑛,𝑞=0, (2.29) where 𝑛=0,1,2,…. Therefore, we obtain the following corollary.

Corollary 2.5. For 𝑛∈ℤ₊,𝑘∈ℕ, 𝐺^(𝑘)_{𝑛+𝑘,𝑞}(𝑥)𝑘!(_𝑘^𝑛+𝑘)=2^𝑘∞_𝑙=0𝑙𝑘+𝑙−1_𝑞(−1)^𝑙(𝑙+𝑥)^𝑛. (2.30)

Corollary 2.6. For 𝑛∈ℤ₊,𝑘∈ℕ, 𝐺^(𝑘)_{𝑛+𝑘,𝑞}(𝑥)=2^𝑘𝑘𝑘!𝑛+𝑘^∞_{𝑙=0𝑑0+𝑑1+⋯+𝑑𝑘=𝑘−1𝑑𝑖∈𝑁}𝑞^{0⋅𝑑0+1⋅𝑑1+⋯+𝑘⋅𝑑𝑘}(−1)^𝑙(𝑙+𝑥)^𝑛. (2.31)

Acknowledgment

The present research has been conducted by the research grant of Kwangwoon University in 2008.