Arithmetic Identities Involving Genocchi and Stirling Numbers

Abstract

An explicit formula, the generalized Genocchi numbers, was established and some identities and congruences involving the Genocchi numbers, the Bernoulli numbers, and the Stirling numbers were obtained.

1. Introduction

The Genocchi numbers 𝐺_𝑛 and the Bernoulli numbers 𝐵_𝑛 (𝑛∈ℕ₀={0,1,2,…}) are defined by the following generating functions (see [1]):

2𝑡𝑒^𝑡=+1^∞_𝑛=0𝐺_𝑛𝑡^𝑛𝑡𝑛!(|𝑡|<𝜋),𝑒^𝑡=−1^∞_𝑛=0𝐵_𝑛𝑡^𝑛𝑛!(|𝑡|<2𝜋), (1.1)(1.2) respectively. By (1.1) and (1.2), we have

𝐺_2𝑛+1=𝐵_2𝑛+1=0,(𝑛∈ℕ)𝐺_𝑛=2(1−2^𝑛)𝐵_𝑛, (1.3) with ℕ being the set of positive integers.

The Genocchi numbers 𝐺_𝑛 satisfy the recurrence relation

𝐺₀=0,𝐺₁=1,𝐺_𝑛1=−2^𝑛−1_𝑘=1𝑛𝑘𝐺_𝑘(𝑛≥2), (1.4) so we find 𝐺₂=−1,𝐺₄=1,𝐺₆=−3,𝐺₈=17,𝐺₁₀=−155,𝐺₁₂=2073,𝐺₁₄=−38227,….

The Stirling numbers of the first kind 𝑠(𝑛,𝑘) can be defined by means of (see [2])

(𝑥)_𝑛=𝑥(𝑥−1)…(𝑥−𝑛+1)=^𝑛_𝑘=0𝑠(𝑛,𝑘)𝑥^𝑘, (1.5) or by the generating function

(log(1+𝑥))^𝑘=𝑘!^∞_𝑛=𝑘𝑥𝑠(𝑛,𝑘)^𝑛.𝑛! (1.6)

It follows from (1.5) or (1.6) that

𝑠(𝑛,𝑘)=𝑠(𝑛−1,𝑘−1)−(𝑛−1)𝑠(𝑛−1,𝑘), (1.7) with 𝑠(𝑛,0)=0(𝑛>0), 𝑠(𝑛,𝑛)=1(𝑛≥0), 𝑠(𝑛,1)=(−1)^𝑛−1(𝑛−1)!(𝑛>0), 𝑠(𝑛,𝑘)=0(𝑘>𝑛 or 𝑘<0).

Stirling numbers of the second kind 𝑆(𝑛,𝑘) can be defined by (see [2])

𝑥^𝑛=^𝑛_𝑘=0𝑆(𝑛,𝑘)(𝑥)_𝑘 (1.8) or by the generating function

(𝑒^𝑥−1)^𝑘=𝑘!^∞_𝑛=𝑘𝑥𝑆(𝑛,𝑘)^𝑛.𝑛! (1.9) It follows from (1.8) or (1.9) that

𝑆(𝑛,𝑘)=𝑆(𝑛−1,𝑘−1)+𝑘𝑆(𝑛−1,𝑘), (1.10) with 𝑆(𝑛,0)=0(𝑛>0), 𝑆(𝑛,𝑛)=1(𝑛≥0), 𝑆(𝑛,1)=1(𝑛>0), 𝑆(𝑛,𝑘)=0(𝑘>𝑛 or 𝑘<0).

The study of Genocchi numbers and polynomials has received much attention; numerous interesting (and useful) properties for Genocchi numbers can be found in many books (see [1, 3–16]). The main purpose of this paper is to prove an explicit formula for the generalized Genocchi numbers (cf. Section 2). We also obtain some identities congruences involving the Genocchi numbers, the Bernoulli numbers, and the Stirling numbers. That is, we will prove the following main conclusion.

Theorem 1.1. Let 𝑛≥𝑘(𝑛,𝑘∈ℕ), then _{𝑣1,…,𝑣𝑘𝑣∈ℕ1+⋯+𝑣𝑘=𝑛}𝐺_𝑣1⋯𝐺_𝑣𝑘(𝑣₁⋯𝑣_𝑘)𝑣₁!⋯𝑣_𝑘!=(−1)^𝑛−𝑘2^𝑘𝑘!𝑛!^𝑛_𝑗=𝑘12^𝑗𝑆(𝑛,𝑗)𝑠(𝑗,𝑘). (1.11)

Remark 1.2. Setting 𝑘=1 in (1.11), and noting that 𝑠(𝑗,1)=(−1)^𝑗−1(𝑗−1)!, we obtain 𝐺_𝑛=2𝑛^𝑛_𝑗=1(−1)^𝑛−𝑗(𝑗−1)!2^𝑗𝑆(𝑛,𝑗)(𝑛∈ℕ). (1.12)

Remark 1.3. By (1.11) and (1.3), we have _{𝑣1,…,𝑣𝑘𝑣∈ℕ1+⋯+𝑣𝑘=𝑛}(2^𝑣1−1)𝐵_𝑣1⋯(2^𝑣𝑘−1)𝐵_𝑣𝑘(𝑣₁⋯𝑣_𝑘)𝑣₁!⋯𝑣_𝑘!=(−1)^𝑛𝑘!𝑛!^𝑛_𝑗=𝑘12^𝑗𝑆(𝑛,𝑗)𝑠(𝑗,𝑘). (1.13)

Theorem 1.4. Let 𝑛,𝑘∈ℕ, then ^𝑛_𝑗=0(−1)^𝑗(𝑘+𝑗−1)!2^𝑗𝑆(𝑛,𝑗)=2^{𝑘−1𝑘−1}_𝑗=0(−1)^𝑗𝐺𝑠(𝑘,𝑘−𝑗)^{𝑛+𝑘−𝑗}.𝑛+𝑘−𝑗 (1.14)

Remark 1.5. Setting 𝑘=1,2,3,4 in (1.14), we get ^𝑛_𝑗=0(−1)^𝑗𝑗!2^𝑗1𝑆(𝑛,𝑗)=𝐺𝑛+1^𝑛+1_𝑛𝑗=0,(−1)^𝑗(𝑗+1)!2^𝑗2𝑆(𝑛,𝑗)=𝐺𝑛+1^𝑛+1+2𝐺𝑛+2^𝑛+2_𝑛𝑗=0,(−1)^𝑗(𝑗+2)!2^𝑗8𝑆(𝑛,𝑗)=𝐺𝑛+1^𝑛+1+12𝐺𝑛+2^𝑛+2+4𝐺𝑛+3^𝑛+3_𝑛𝑗=0,(−1)^𝑗(𝑗+3)!2^𝑗𝑆(𝑛,𝑗)=48𝐺𝑛+1^𝑛+1+88𝐺𝑛+2^𝑛+2+48𝐺𝑛+3^𝑛+3+8𝐺𝑛+4^𝑛+4. (1.15)

Theorem 1.6. Let 𝑛∈ℕ, 𝑚∈ℕ₀, then 2^𝑛𝐺𝑛+1^𝑛+1≡2^{𝑚𝑛−𝑚}_𝑗=0𝑚𝑗𝑗^𝑛(mod𝑚+1). (1.16)

Remark 1.7. Setting 𝑚=𝑝−1 in (1.16), we have 1𝐺𝑛+1^𝑛+1≡^𝑝−1_𝑗=0(−1)^𝑗𝑗^𝑛(mod𝑝), (1.17) where 𝑝 is any odd prime.

2. Definition and Lemma

Definition 2.1. For a real or complex parameter 𝑥, we have the generalized Genocchi numbers 𝐺_𝑛^(𝑥), which are defined by (2𝑒^2𝑡)+1^𝑥=^∞_𝑛=0𝐺_𝑛^(𝑥)𝑡^𝑛𝜋𝑛!(|𝑡|<2;1^𝑥∶=1). (2.1) By (1.1) and (2.1), we have 𝑛𝐺⁽¹⁾_𝑛−1=2^𝑛−1𝐺_𝑛. (2.2)

Remark 2.2. For an integer 𝑥, the higher-order Euler numbers 𝐸^(𝑥)_2𝑛 are defined by the following generating functions (see [17]): (2𝑒^𝑡+𝑒^−𝑡)^𝑥=^∞_𝑛=0𝐸^(𝑥)_2𝑛𝑡^2𝑛𝜋(2𝑛)!(|𝑡|<2). (2.3) Then we have 𝐺_𝑛^(𝑥)=(−1)^𝑛[𝑛/2]_𝑘=0𝑛𝐸2𝑘^(𝑥)2𝑘𝑥^𝑛−2𝑘, (2.4) where [𝑛/2] denotes the greatest integer not exceeding 𝑛/2.

Lemma 2.3. Let 𝑛≥𝑘(𝑛,𝑘∈ℕ), then 𝐺_𝑛^(𝑥)=^𝑛_𝑘=1𝜔(𝑛,𝑘)𝑥^𝑘, (2.5) where 𝜔(𝑛,𝑘)=(−1)^𝑘𝑛_𝑗=𝑘2^𝑛−𝑗𝑆(𝑛,𝑗)𝑠(𝑗,𝑘). (2.6)

Proof. By (2.1), (1.5), and (1.9) we have ^∞_𝑛=0𝐺_𝑛^(𝑥)𝑡^𝑛2𝑛!=(𝑒^2𝑡)+1^𝑥1=(1+(1/2)(𝑒^2𝑡)−1)^𝑥=^∞_𝑗=0(−1)^𝑗2^𝑗𝑗𝑥+𝑗−1(𝑒^2𝑡−1)^𝑗=^∞_𝑗=0(−1)^𝑗2^𝑗𝑗𝑥+𝑗−1𝑗!^∞_𝑛=𝑗2^𝑛𝑡𝑆(𝑛,𝑗)^𝑛=𝑛!_{∞𝑛𝑛=0𝑗=0}(−1)^𝑗𝑗!2^𝑛−𝑗𝑗𝑥+𝑗−1𝑡𝑆(𝑛,𝑗)^𝑛,𝑛! (2.7) which readily yields 𝐺_𝑛^(𝑥)=^𝑛_𝑗=0(−1)^𝑗𝑗!2^𝑛−𝑗𝑗=𝑥+𝑗−1𝑆(𝑛,𝑗)_𝑛𝑗=0(−1)^𝑗2^𝑛−𝑗=𝑆(𝑛,𝑗)(𝑥+𝑗−1)(𝑥+𝑗−2)⋯(𝑥+1)𝑥_𝑛𝑗=0(−1)^𝑗2^𝑛−𝑗𝑆(𝑛,𝑗)^𝑗_𝑘=1(−1)^𝑗−𝑘𝑠(𝑗,𝑘)𝑥^𝑘=^𝑛_𝑘=1(−1)^𝑘𝑛_𝑗=𝑘2^𝑛−𝑗𝑆(𝑛,𝑗)𝑠(𝑗,𝑘)𝑥^𝑘=^𝑛_𝑘=1𝜔(𝑛,𝑘)𝑥^𝑘. (2.8) This completes the proof of Lemma 2.3.

Remark 2.4. From (1.7), (1.10), and Lemma 2.3 we know that 𝐺_𝑛^(𝑥) is a polynomial of 𝑥 with integral coefficients. For example, setting 𝑛=1,2,3,4 in Lemma 2.3, we get 𝐺₁^(𝑥)=−𝑥,𝐺₂^(𝑥)=−𝑥+𝑥²,𝐺₃^(𝑥)=3𝑥²−𝑥³,𝐺₄^(𝑥)=2𝑥+3𝑥²−6𝑥³+𝑥⁴. (2.9)

Remark 2.5. Let 𝑛,𝑚∈ℕ, then by (2.5), we have ^𝑛_𝑘=12𝜔(𝑛,𝑘)=^𝑛𝐺𝑛+1^𝑛+1. (2.10) Therefore, if 𝑞∈ℕ is odd, then by (2.10) we get 𝐺₂^𝑘_𝑞≡0(mod𝑞), (2.11) where 𝑘∈ℕ.

3. Proof of the Theorems

Proof of Theorem 1.1. By applying Lemma 2.3, we have 𝑑𝑘!𝜔(𝑛,𝑘)=^𝑘𝑑𝑥^𝑘𝐺_𝑛^(𝑥)|_𝑥=0. (3.1) On the other hand, it follows from (2.1) that ^∞_𝑛=𝑘𝑑^𝑘𝑑𝑥^𝑘𝐺_𝑛^(𝑥)|_𝑥=0𝑡^𝑛2𝑛!=(log𝑒^2𝑡)+1^𝑘, (3.2) where log(2/(𝑒^2𝑡+1)) is the principal branch of logarithm of 2/(𝑒^2𝑡+1).
Thus, by (3.1) and (3.2), we have 𝑘!^∞_𝑛=𝑘𝑡𝜔(𝑛,𝑘)^𝑛2𝑛!=(log𝑒^2𝑡)+1^𝑘. (3.3)
Now note that 𝑑2𝑑𝑡log𝑒^2𝑡=2+1𝑒^2𝑡+1−2=^∞_𝑛=0𝐺_𝑛⁽¹⁾𝑡^𝑛𝑛!−2=^∞_𝑛=02^𝑛𝐺_𝑛+1𝑡𝑛+1𝑛^𝑛!−2, (3.4) whence by integrating from 0 to 𝑡, we deduce that 2log𝑒^2𝑡=+1^∞_𝑛=12^𝑛−1𝐺_𝑛𝑛𝑡^𝑛𝑛!−2𝑡=^∞_𝑛=1(−1)^𝑛2^𝑛−1𝐺_𝑛𝑛𝑡^𝑛.𝑛! (3.5) Since 𝐺_2𝑛+1=0   (𝑛∈ℕ). Substituting (3.5) in (3.3) we get 𝜔(𝑛,𝑘)=(−1)^𝑛𝑛!2^𝑛−𝑘𝑘!_{𝑣1,…,𝑣𝑘𝑣∈ℕ1+⋯+𝑣𝑘=𝑛}𝐺_𝑣1⋯𝐺_𝑣𝑘(𝑣₁⋯𝑣_𝑘)𝑣₁!⋯𝑣_𝑘!. (3.6)
By (3.6) and (2.6), we may immediately obtain Theorem 1.1. This completes the proof of Theorem 1.1.

Proof of Theorem 1.4. By (2.1) and note the identity 22(𝑒^2𝑡)+1^𝑥+1𝑥𝑑(2𝑑𝑡𝑒^2𝑡)+1^𝑥2=(𝑒^2𝑡)+1^𝑥+1, (3.7) we have 𝐺_𝑛^(𝑥+1)=2𝐺_𝑛^(𝑥)+1𝑥𝐺^(𝑥)_𝑛+1. (3.8)
By (3.8), (1.7), and note that 𝐺_𝑛⁽¹⁾=2^𝑛/(𝑛+1)𝐺_𝑛+1, we obtain 𝐺_𝑛^(𝑘)=1(𝑘−1)!^𝑘−1_𝑗=0(−1)^𝑗2^𝑗𝑠(𝑘,𝑘−𝑗)𝐺⁽¹⁾_{𝑛+𝑘−1−𝑗}=2^{𝑛+𝑘−1}(𝑘−1)!^𝑘−1_𝑗=0(−1)^𝑗𝐺𝑠(𝑘,𝑘−𝑗)^{𝑛+𝑘−𝑗}.𝑛+𝑘−𝑗 (3.9) Comparing (3.9) and (2.8), we immediately obtain Theorem 1.4. This completes the proof of Theorem 1.4.

Proof of Theorem 1.6. By Lemma 2.3, we have 𝐺_𝑛^(𝑚+𝑥)=^𝑛_𝑗=1𝜔(𝑛,𝑗)(𝑚+𝑥)^𝑗≡^𝑛_𝑗=1𝜔(𝑛,𝑗)𝑥^𝑗=𝐺_𝑛^(𝑥)(mod𝑚). (3.10) Therefore 𝐺_𝑛^(𝑘)=𝐺_𝑛^{(𝑚+𝑘−𝑚)}≡𝐺_𝑛^(−𝑚)=2^{𝑚𝑛−𝑚}_𝑗=0𝑚𝑗𝑗^𝑛(mod𝑚+𝑘). (3.11) Taking 𝑘=1 in (3.11) and note that 𝐺_𝑛⁽¹⁾=2^𝑛/(𝑛+1)𝐺_𝑛+1, we immediately obtain Theorem 1.6. This completes the proof of Theorem 1.6.

Acknowledgments

The author would like to thank the anonymous referee for valuable suggestions. This work was supported by the Guangdong Provincial Natural Science Foundation (no. 8151601501000002).