On Genocchi Numbers and Polynomials

Abstract

The main purpose of this paper is to study the distribution of Genocchi polynomials. Finally, we construct the Genocchi zeta function which interpolates Genocchi polynomials at negative integers.

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 completion of the algebraic closure of ℚ_𝑝. Let 𝑣_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{−𝑣𝑝(𝑝)}=1/𝑝. When one talks about 𝑞-extension, 𝑞 is variously considered as an indeterminate, a complex, 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 |𝑞−1|_𝑝<1. The ordinary Genocchi polynomials are defined as the generating function:𝐹(𝑡,𝑥)=2𝑡𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐺_𝑛𝑡(𝑥)^𝑛𝑛!,|𝑡|<𝜋. (1.1) For a fixed positive integer 𝑑 with (𝑝,𝑑)=1, set𝑋=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)} (1.2)(cf. [1–30]), where 𝑎∈ℤ satisfies the condition 0≤𝑎<𝑑𝑝^𝑁. We say that 𝑓 is uniformly differential function at 𝑎∈ℤ_𝑝 and write 𝑓∈𝑈𝐷(ℤ_𝑝) if the difference quotients, 𝐹_𝑓(𝑥,𝑦)=(𝑓(𝑥)−𝑓(𝑦))/(𝑥−𝑦), have a limit 𝑓^′(𝑎) as (𝑥,𝑦)→(𝑎,𝑎). Throughout this paper, we use the following notation:[𝑥]_−𝑞=1−(−𝑞)^𝑥1+𝑞,[𝑥]_𝑞=1−𝑞^𝑥1−𝑞. (1.3)For 𝑓∈𝑈𝐷(ℤ_𝑝), the fermionic 𝑝-adic invariant 𝑞-integral on ℤ_𝑝 is defined as𝐼_−𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1[𝑝^𝑁]^𝑝_−𝑞^𝑁−1_𝑥=0𝑓(𝑥)(−𝑞)^𝑥; (1.4)see [1–27]. Note that𝐼₋₁(𝑓)=lim_𝑞→1𝐼_−𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥). (1.5)In this paper, we investigate some interesting integral equations related to 𝐼₋₁(𝑓). From these integral equations related to 𝐼₋₁(𝑓), we can derive many interesting properties of Genocchi numbers and polynomials. The main purpose of this paper is to derive the distribution relations of the Genocchi polynomials, and to construct the Genocchi zeta function which interpolates the Genocchi polynomials at negative integers.

2. Genocchi Numbers and Polynomials

The Genocchi numbers are defined as𝐺₀=0,(𝐺+1)^𝑛+𝐺_𝑛=2if𝑛=1,0if𝑛>1, (2.1)where 𝐺^𝑛 is replaced by 𝐺_𝑛, symbolically. The Genocchi polynomials are also defined as𝐺_𝑛(𝑥)=^𝑛_𝑘=0(𝑛𝑘)𝑥^𝑛−𝑘𝐺_𝑘. (2.2)From (2.1), we note that 𝐺₁=1,𝐺₂=−1,𝐺₃=0,𝐺₄=1,…,𝐺_2𝑘+1=0, and 𝐺_2𝑘∈ℤ(𝑘=1,2,…). The fermionic 𝑝-adic invariant integral on ℤ_𝑝 is defined as𝐼₋₁(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)=lim_𝑁→∞1[𝑝^𝑁]^𝑝₋₁^𝑁−1_𝑥=0𝑓(𝑥)(−1)^𝑥,see[1]. (2.3)Let 𝑓₁(𝑥) be translation with 𝑓₁(𝑥)=𝑓(𝑥+1). Then we have the following integral equation. Note that 𝐼₋₁(𝑓₁)+𝐼₋₁(𝑓)=2𝑓(0). From (2.3), we can derive𝑡_ℤ𝑝𝑒^𝑥𝑡𝑑𝜇₋₁(𝑥)=^∞_𝑛=0𝐺_𝑛𝑡^𝑛𝑛!=𝑡^∞_𝑛=0𝐺_𝑛+1𝑡𝑛+1^𝑛,𝑡𝑛!_ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑑𝜇₋₁(𝑦)=^∞_𝑛=0𝐺_𝑛𝑡(𝑥)^𝑛𝑛!=𝑡^∞_𝑛=0𝐺_𝑛+1(𝑥)𝑡𝑛+1^𝑛.𝑛! (2.4)(2.5)Thus, we obtain_ℤ𝑝𝑥^𝑛𝑑𝜇₋₁𝐺(𝑥)=^𝑛+1,𝑛+1_ℤ𝑝(𝑥+𝑦)^𝑛𝑑𝜇₋₁𝐺(𝑦)=^𝑛+1(𝑥)𝑛+1. (2.6)For 𝑛∈ℕ, we have_ℤ𝑝𝑓(𝑥+𝑛)𝑑𝜇₋₁(𝑥)=(−1)^𝑛_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)+2^𝑛−1_ℓ=0(−1)^𝑛−1+ℓ𝑓(ℓ),see[1-27]. (2.7)By (2.6) and (2.7), if we take 𝑓(𝑥)=𝑥^𝑘(𝑘∈ℤ⁺), we easily see that_ℤ𝑝(𝑥+𝑛)^𝑘𝑑𝜇₋₁(𝑥)−_ℤ𝑝𝑥^𝑘𝑑𝜇₋₁(𝑥)=2^𝑛−1_ℓ=0(−1)^ℓ−1ℓ^𝑘if𝑛≡0(mod2). (2.8)Thus, we have𝐺_𝑘+1(𝑛)−𝐺𝑘+1𝑘+1^𝑘+1=2^𝑛−1_ℓ=0(−1)^ℓ−1ℓ^𝑘if𝑛≡0(mod2). (2.9)If 𝑛≡1(mod2), then we know that_ℤ𝑝(𝑥+𝑛)^𝑘𝑑𝜇₋₁(𝑥)+_ℤ𝑝𝑥^𝑘𝑑𝜇₋₁(𝑥)=2^𝑛−1_ℓ=0(−1)^ℓℓ^𝑘if𝑛≡1(mod2). (2.10)Thus, we get𝐺_𝑘+1(𝑛)+𝐺𝑘+1𝑘+1^𝑘+1=2^𝑛−1_ℓ=0(−1)^ℓℓ^𝑘,see[1-30]. (2.11)Let 𝜒 be the Dirichlet character with conductor 𝑑∈ℕ, with 𝑑≡1(mod2). Then, we consider the generalized Genocchi numbers attached to 𝜒 as follows:𝐺_𝑛+1,𝜒=𝑛+1_𝑋𝜒(𝑥)𝑥^𝑛𝑑𝜇₋₁(𝑥),𝐺_0,𝜒=0, (2.12)where 𝑛∈ℤ₊. From (2.7) and (2.12), we note that𝑡_𝑋𝑒^𝑥𝑡𝜒(𝑥)𝑑𝜇₋₁2∑(𝑥)=^{𝑑−1ℓ=0}(−1)^ℓ𝜒(ℓ)𝑒^ℓ𝑡𝑒^𝑑𝑡+1𝑡=^∞_𝑛=0𝐺_𝑛,𝜒𝑡𝑛!^𝑛. (2.13)By (2.12) and (2.13), it is not difficult to show that𝐺_𝑛+1,𝜒=𝑛+1_𝑋𝜒(𝑥)𝑥^𝑛𝑑𝜇₋₁(𝑥)=𝑑^𝑛𝑑−1_𝑎=0𝜒(𝑎)(−1)^𝑎_ℤ𝑝(𝑎𝑑+𝑥)^𝑛𝑑𝜇₋₁(𝑥)=𝑑^𝑛𝑑−1_𝑎=0𝜒(𝑎)(−1)^𝑎𝐺_𝑛+1(𝑎/𝑑),𝑛+1_𝑋(𝑥+𝑦)^𝑛𝑑𝜇₋₁(𝑦)=𝑑^𝑛𝑑−1_𝑛=0(−1)^𝑎_ℤ𝑝(𝑥+𝑎𝑑+𝑦)^𝑛𝑑𝜇₋₁(𝑦). (2.14)(2.15)By (2.6) and (2.15), we obtain the following theorem.

Theorem 2.1. Let 𝑑∈ℕ with 𝑑≡1(mod2), and let 𝜒 be the Dirichlet character with conductor 𝑑. Then, one has 𝐺_𝑛+1(𝑥)𝑛+1=𝑑^𝑛𝑑−1_𝑎=0(−1)^𝑎𝐺_𝑛+1((𝑥+𝑎)/𝑑)𝐺𝑛+1(distributionrelationforGenocchipolynomials),_𝑛+1,𝜒𝑛+1=𝑑^𝑛𝑑−1_𝑎=0𝜒(𝑎)(−1)^𝑎𝐺_𝑛+1(𝑎/𝑑).𝑛+1 (2.16)(2.17)

3. Genocchi Zeta Function

Let 𝐹(𝑡,𝑥) be the generating function of 𝐺_𝑘(𝑥) in complex plane as follows:𝐹(𝑡,𝑥)=2𝑡𝑒^𝑥𝑡𝑒^𝑡=+1^∞_𝑛=0𝐺_𝑛𝑡(𝑥)^𝑛𝑛!=𝑡^∞_𝑛=0𝐺_𝑛+1(𝑥)𝑡𝑛+1𝑛^𝑛!,|𝑡|<𝜋. (3.1)Then, we show that𝐹(𝑡,𝑥)=2𝑡^∞_𝑛=0(−1)^𝑛𝑒^{(𝑛+𝑥)𝑡}. (3.2)By (3.1) and (3.2), we easily see that𝐺_𝑘𝑑(𝑥)=^𝑘𝐹(𝑡,𝑥)𝑑𝑡^𝑘|_𝑡=0=2𝑘^∞_𝑛=0(−1)^𝑛(𝑛+𝑥)^𝑘−1. (3.3)Therefore, we obtain the following proposition.

Proposition 3.1. For 𝑘∈ℕ, one has 𝐺_𝑘(𝑥)𝑘=2^∞_𝑛=0(−1)^𝑛(𝑛+𝑥)^𝑘−1. (3.4)

From Proposition 3.1, we can derive the Genocchi zeta function which interpolates Genocchi polynomials at negative integers.

For 𝑠∈ℂ, we define the Hurwitz-type Genocchi zeta function as follows.

Definition 3.2. For 𝑠∈ℂ,𝜁_𝐺(𝑠,𝑥)=2^∞_𝑛=0(−1)^𝑛(𝑛+𝑥)^𝑠,𝜁_𝐺(𝑠)=2^∞_𝑛=1(−1)^𝑛𝑛^𝑠. (3.5)

By Proposition 3.1 and Definition 3.2, we obtain the following theorem.

Theorem 3.3. For 𝑘∈ℕ, one has 𝜁_𝐺𝐺(1−𝑘,𝑥)=^𝑘(𝑥)𝑘,𝜁_𝐺𝐺(1−𝑘)=^𝑘𝑘. (3.6)

Let 𝜒 be the Dirichlet character with conductor 𝑑∈ℕ, with 𝑑≡1(mod2), and let 𝐹_𝜒(𝑡) be the generating function in ℂ of 𝐺_𝑛,𝜒. Then, we have𝐹_𝜒∑(𝑡)=2^{𝑑−1ℓ=0}(−1)^ℓ𝜒(ℓ)𝑒^ℓ𝑡𝑒^𝑑𝑡+1𝑡=^∞_𝑛=0𝐺_𝑛,𝜒𝑡^𝑛𝜋𝑛!,|𝑡|<𝑑. (3.7)From (3.7), we derive𝐹_𝜒(𝑡)=^∞_𝑛=0𝐺_𝑛,𝜒𝑡^𝑛=𝑛!^∞_𝑛=1𝐺_𝑛,𝜒𝑡^𝑛𝑛!=𝑡^∞_𝑛=0𝐺_𝑛+1,𝜒𝑡𝑛+1𝑛^𝑛!=𝑡^∞_𝑛=0(𝑑^𝑛𝑑−1_𝑎=0𝜒(𝑎)(−1)^𝑎𝐺_𝑛+1(𝑎/𝑑))𝑡𝑛+1^𝑛=𝑛!_{𝑑−1𝑎=0}𝜒(𝑎)(−1)^𝑎(𝑑𝑡^∞_𝑛=0𝐺_𝑛+1(𝑎/𝑑)𝑑𝑛+1^𝑛𝑡^𝑛𝑛!). (3.8)By (3.1), (3.2), and (3.8), we easily see that𝐹_𝜒(𝑡)=^𝑑−1_𝑎=0𝜒(𝑎)(−1)^𝑎(2𝑡^∞_𝑘=0(−1)^𝑘𝑒^{(𝑘+𝑎/𝑑)𝑑𝑡})=2𝑡^∞_𝑘=0^𝑑−1_𝑎=0𝜒(𝑎+𝑑𝑘)(−1)^{𝑎+𝑑𝑘}𝑒^{(𝑑𝑘+𝑎)𝑡}=2𝑡^∞_𝑛=0𝜒(𝑛)(−1)^𝑛𝑒^𝑛𝑡=2𝑡^∞_𝑛=1𝜒(𝑛)(−1)^𝑛𝑒^𝑛𝑡. (3.9)From (3.9), we can derive𝐺_𝑘,𝜒=𝑑^𝑘𝑑𝑡^𝑘𝐹_𝜒(𝑡)|_𝑡=0=𝑘(2^∞_𝑛=1𝜒(𝑛)(−1)^𝑛𝑛^𝑘−1). (3.10)Thus, we have𝐺_𝑘,𝜒𝑘=2^∞_𝑘=0𝜒(𝑛)(−1)^𝑛𝑛^𝑘−1. (3.11)Now, we consider the Dirichlet-type Genocchi ℓ-function in complex plane as follows. For 𝑠∈ℂ, defineℓ_𝐺,𝜒(𝑠)=2^∞_𝑛=1(−1)^𝑛𝜒(𝑛)𝑛^𝑠. (3.12)By (3.11) and (3.12), we obtain the following theorem.

Theorem 3.4. Let 𝜒 be the Dirichlet character with conductor 𝑑∈ℕ, with 𝑑≡1(mod2), and let 𝑘∈ℤ⁺. Then, one has ℓ_𝐺,𝜒𝐺(1−𝑘)=^𝑘,𝜒𝑘. (3.13)

Remark 3.5. In [1], we can observe the value of Genocchi zeta function at positive integers as follows:𝜁_𝐺(2𝑛)=(−1)^𝑛−1𝜋^2𝑛(2−4^𝑛)2(2𝑛)!(1−4^𝑛)𝐺_2𝑛,cf.[1]. (3.14)