Arithmetic Identities Involving Bernoulli and Euler Numbers

Abstract

The purpose of this paper is to give some arithmatic identities for the Bernoulli and Euler numbers. These identities are derived from the several 𝑝-adic integral equations on ℤ_𝑝.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper, ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 will denote the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of ℚ_𝑝, respectively. The 𝑝-adic norm is normalized so that |𝑝|_𝑝=1/𝑝. Let ℕ be the set of natural numbers and ℤ₊=ℕ∪{0}.

Let UD(ℤ_𝑝) be the space of uniformly differentiable functions on ℤ_𝑝. For 𝑓∈UD(ℤ_𝑝), the bosonic 𝑝-adic integral on ℤ_𝑝 is defined by 𝐼𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇(𝑥)=lim^𝑝_𝑁→∞^𝑁−1_𝑥=0𝑓(𝑥)𝜇𝑥+𝑝^𝑁ℤ_𝑝=lim_𝑁→∞1𝑝^{𝑁𝑝𝑁−1}_𝑥=0𝑓(𝑥), (1.1) and the fermionic 𝑝-adic integral on ℤ_𝑝 is defined by Kim as follows (see [1–8]): 𝐼₋₁𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)=lim^𝑝_𝑁→∞^𝑁−1_𝑥=0𝑓(𝑥)(−1)^𝑥. (1.2)

The Euler polynomials, 𝐸_𝑛(𝑥), are defined by the generating function as follows (see [1–16]): 𝐹^𝐸()=2𝑡,𝑥𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛.𝑛! (1.3) In the special case, 𝑥=0, 𝐸_𝑛(0)=𝐸_𝑛 is called the 𝑛th Euler number.

By (1.3) and the definition of Euler numbers, we easily see that 𝐸_𝑛(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐸_𝑙𝑥^𝑛−𝑙=(𝐸+𝑥)^𝑛, (1.4) with the usual convention about replacing 𝐸^𝑙 by 𝐸_𝑙 (see [10]). Thus, by (1.3) and (1.4), we have 𝐸₀=1,(𝐸+1)^𝑛+𝐸_𝑛=2𝛿_0,𝑛, (1.5) where 𝛿_𝑘,𝑛 is the Kronecker symbol (see [9, 10, 17–19]).

From (1.2), we can also derive the following integral equation for the fermionic 𝑝-adic integral on ℤ_𝑝 as follows: 𝐼₋₁𝑓₁=−𝐼₋₁𝑓+2𝑓(0), (1.6) see [1, 2]. By (1.3) and (1.6), we get _ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑑𝜇₋₁𝑦=2𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛.𝑛! (1.7) Thus, by (1.7), we have _ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇₋₁𝑦=𝐸_𝑛(𝑥), (1.8) see [1–8, 13–16].

The Bernoulli polynomials, 𝐵_𝑛(𝑥), are defined by the generating function as follows: 𝐹^𝐵()=𝑡𝑡,𝑥𝑒^𝑡𝑒−1^𝑥𝑡=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛,𝑛! (1.9) see [18]. In the special case, 𝑥=0, 𝐵_𝑛(0)=𝐵_𝑛 is called the 𝑛th Bernoulli number. From (1.9) and the definition of Bernoulli numbers, we note that 𝐵_𝑛(𝑥)=^𝑛_𝑙=0𝑛𝑙𝑥^𝑛−𝑙𝐵_𝑙=(𝐵+𝑥)^𝑛, (1.10) see [1–19], with the usual convention about replacing 𝐵^𝑙 by 𝐵_𝑙. By (1.9) and (1.10), we easily see that 𝐵₀=1,(𝐵+1)^𝑛−𝐵_𝑛=𝛿_1,𝑛, (1.11) see [13].

From (1.1), we can derive the following integral equation on ℤ_𝑝: 𝐼𝑓₁=𝐼𝑓+𝑓^′(0), (1.12) where 𝑓₁(𝑥)=𝑓(𝑥+1) and 𝑓^′(0)=(𝑑𝑓(𝑥)/𝑑𝑥)|_𝑥=0.

By (1.12), we have _ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑦=𝑡𝑑𝜇𝑒^𝑡𝑒−1^𝑥𝑡=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛.𝑛! (1.13) Thus, by (1.13), we can derive the following Witt’s formula for the Bernoulli polynomials: _ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇𝑦=𝐵_𝑛(𝑥),for𝑛∈ℤ₊. (1.14)

In [19], it is known that for 𝑘,𝑚∈ℤ₊, ^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗𝐵_{𝑘+𝑚+1−𝑗}(𝑥)𝑘+𝑚+1−𝑗=𝑥^𝑘(𝑥−1)^𝑚+(−1)^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚, (1.15) where ^𝑘_𝑗=0 if 𝑗<0 or 𝑗>𝑘.

The purpose of this paper is to give some arithmetic identities involving Bernoulli and Euler numbers. To derive our identities, we use the properties of 𝑝-adic integral equations on ℤ_𝑝.

2. Arithmetic Identities for Bernoulli and Euler Numbers

Let us take the bosonic 𝑝-adic integral on ℤ_𝑝 in (1.15) as follows: 𝐼₁=_ℤ𝑝𝑥^𝑘(𝑥−1)^𝑚(𝑥)+()𝑑𝜇−1^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚=^𝑚_𝑙=0𝑚𝑙(−1)^𝑙_ℤ𝑝𝑥^{𝑘+𝑚−𝑙}(𝑥)+()𝑑𝜇−1^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚=^𝑚_𝑙=0𝑚𝑙(−1)^𝑙𝐵_{𝑘+𝑚−𝑙}+(−1)^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚. (2.1) On the other hand, we get 𝐼₁=^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗1𝑘+𝑚+1−𝑗_ℤ𝑝𝐵_{𝑘+𝑚+1−𝑗}(𝑥)(𝑥)=𝑑𝜇_{max{𝑘,𝑚}𝑗=1}𝑘𝑗+(−1)^𝑗+1𝑚𝑗1×𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗𝑙=0}𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝐵_𝑙. (2.2) By (2.1) and (2.2), we get ^{max{𝑘,𝑚}}_𝑗=1^{𝑘+𝑚+1−𝑗}_𝑙=01𝑘𝑗+()𝑘+𝑚+1−𝑗−1^𝑗+1𝑚𝑗×𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝐵_𝑙=^𝑚_𝑙=0(−1)^𝑙𝑚𝑙𝐵_{𝑘+𝑚−𝑙}+(−1)^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚. (2.3)

Therefore, by (2.3), we obtain the following theorem.

Theorem 2.1. For 𝑘,𝑚∈ℤ₊, one has ^{max{𝑘,𝑚}}_𝑗=1^{𝑘+𝑚+1−𝑗}_𝑙=01𝑘𝑗+()𝑘+𝑚+1−𝑗−1^𝑗+1𝑚𝑗×𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝐵_𝑙−(−1)^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚=^𝑚_𝑙=0(−1)^𝑙𝑚𝑙𝐵_{𝑘+𝑚−𝑙}. (2.4)

Now we consider the fermionic 𝑝-adic integral on ℤ_𝑝 in (1.15) as follows: 𝐼₂=^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗1𝑘+𝑚+1−𝑗^{𝑘+𝑚+1−𝑗}_𝑙=0𝑙𝑘+𝑚+1−𝑗×𝐵_{𝑘+𝑚+1−𝑗−𝑙ℤ𝑝}𝑥^𝑙𝑑𝜇₋₁(𝑥)=^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗1𝑘+𝑚+1−𝑗^{𝑘+𝑚+1−𝑗}_𝑙=0𝑙𝑘+𝑚+1−𝑗×𝐵_{𝑘+𝑚+1−𝑗−𝑙}𝐸_𝑙. (2.5) On the other hand, we get 𝐼₂=^𝑚_𝑙=0(−1)^𝑙𝑚𝑙^ℤ𝑝𝑥^{𝑚−𝑙+𝑘}𝑑𝜇₋₁(𝑥)+(−1)^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚=^𝑚_𝑙=0(−1)^𝑙𝑚𝑙𝐸_{𝑘+𝑚−𝑙}+(−1)^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚. (2.6) By (2.5) and (2.6), we get ^{max{𝑘,𝑚}}_𝑗=1^{𝑘+𝑚+1−𝑗}_𝑙=01𝑘𝑗+()𝑘+𝑚+1−𝑗−1^𝑗+1𝑚𝑗𝑙𝑘+𝑚+1−𝑗×𝐵_{𝑘+𝑚+1−𝑗−𝑙}𝐸_𝑙=^𝑚_𝑙=0(−1)^𝑙𝑚𝑙𝐸_{𝑘+𝑚−𝑙}+(−1)^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚. (2.7) Therefore, by (2.7), we obtain the following theorem.

Theorem 2.2. For 𝑘,𝑚∈ℤ₊, one has ^{max{𝑘,𝑚}}_𝑗=1^{𝑘+𝑚+1−𝑗}_𝑙=01𝑘𝑗+()𝑘+𝑚+1−𝑗−1^𝑗+1𝑚𝑗𝑙𝑘+𝑚+1−𝑗×𝐵_{𝑘+𝑚+1−𝑗−𝑙}𝐸_𝑙−(−1)^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚=^𝑚_𝑙=0(−1)^𝑙𝑚𝑙𝐸_{𝑘+𝑚−𝑙}. (2.8)

Replacing 𝑥 by (1−𝑥) in (1.15), we have the identity: ^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗𝐵_{𝑘+𝑚+1−𝑗}(1−𝑥)=()𝑘+𝑚+1−𝑗−1^𝑘+𝑚𝑥^𝑚(1−𝑥)^𝑘+(−1)^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚. (2.9) Let us take the bosonic 𝑝-adic integral on ℤ_𝑝 in (2.9) as follows: 𝐼₃=^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗1×𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗𝑙=0}𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙ℤ𝑝}(1−𝑥)^𝑙(𝑥)=𝑑𝜇_{max{𝑘,𝑚}𝑗=1}𝑘𝑗+(−1)^𝑗+1𝑚𝑗1×𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗𝑙=0}𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝐵_𝑙+^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗1×𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗𝑙=0}𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝑙+^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗1×𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗𝑙=0}𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝛿_1,𝑙=^{max{𝑘,𝑚}}_𝑗=1^{𝑘+𝑚+1−𝑗}_𝑙=01𝑘𝑗+()𝑘+𝑚+1−𝑗−1^𝑗+1𝑚𝑗×𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝐵_𝑙+^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗2𝐵_{𝑘+𝑚−𝑗}+𝛿_{1,(𝑘+𝑚−𝑗)}=^{max{𝑘,𝑚}}_𝑗=1^{𝑘+𝑚+1−𝑗}_𝑙=01𝑘𝑗+()𝑘+𝑚+1−𝑗−1^𝑗+1𝑚𝑗×𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝐵_𝑙+2^{max(𝑘,𝑚)}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗×𝐵_{𝑘+𝑚−𝑗}+𝑘+()𝑘+𝑚−1−1^𝑘+𝑚𝑚.𝑘+𝑚−1 (2.10)

On the other hand, we see that 𝐼₃=(−1)^{𝑘𝑘+𝑚}_𝑙=0(−1)^𝑙𝑘𝑙𝐵_{𝑘+𝑚−𝑙}+(−1)^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚. (2.11) By (2.10) and (2.11), we get ^{max{𝑘,𝑚}}_𝑗=1^{𝑘+𝑚+1−𝑗}_𝑙=01𝑘𝑗+()𝑘+𝑚+1−𝑗−1^𝑗+1𝑚𝑗×𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝐵_𝑙+2^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗×𝐵_{𝑘+𝑚−𝑗}+𝑘+()𝑘+𝑚−1−1^𝑘+𝑚𝑚=()𝑘+𝑚−1−1^{𝑘𝑘+𝑚}_𝑙=0(−1)^𝑙𝑘𝑙𝐵_{𝑘+𝑚−𝑙}+(−1)^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚. (2.12) Therefore, by (2.12), we obtain the following theorem.

Theorem 2.3. For 𝑘,𝑚∈ℤ₊, one has ^{max{𝑘,𝑚}}_𝑗=1^{𝑘+𝑚+1−𝑗}_𝑙=01𝑘𝑗+()𝑘+𝑚+1−𝑗−1^𝑗+1𝑚𝑗×𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝐵_𝑙+2^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗×𝐵_{𝑘+𝑚−𝑗}+𝑘+()𝑘+𝑚−1−1^𝑘+𝑚𝑚−()𝑘+𝑚−1−1^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚=(−1)^{𝑘𝑘+𝑚}_𝑙=0(−1)^𝑙𝑘𝑙𝐵_{𝑘+𝑚−𝑙}. (2.13)

We consider the fermionic 𝑝-adic integral on ℤ_𝑝 in (2.9) as follows: 𝐼₄=^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗1×𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗𝑙=0}𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙ℤ𝑝}(1−𝑥)^𝑙𝑑𝜇₋₁(𝑥)=^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗1×𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗𝑙=0}𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝐸_𝑙+2^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗1×𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗𝑙=0}𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}−2^{max{𝑘,𝑚}}_𝑗=1𝑘𝑗+(−1)^𝑗+1𝑚𝑗1×𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗𝑙=0}𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝛿_0,𝑙=^{max{𝑘,𝑚}}_𝑗=1^{𝑘+𝑚+1−𝑗}_𝑙=01𝑘𝑗+()𝑘+𝑚+1−𝑗−1^𝑗+1𝑚𝑗×𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝐸_𝑙+2^{max{𝑘,𝑚}}_𝑗=11𝑘𝑗+()𝑘+𝑚+1−𝑗−1^𝑗+1𝑚𝑗𝛿_{1,(𝑘+𝑚+1−𝑗)}=^{max{𝑘,𝑚}}_𝑗=1^{𝑘+𝑚+1−𝑗}_𝑙=01𝑘𝑗+()𝑘+𝑚+1−𝑗−1^𝑗+1𝑚𝑗×𝑙𝐵𝑘+𝑚+1−𝑗_{𝑘+𝑚+1−𝑗−𝑙}𝐸_𝑙𝑘+2+()𝑘+𝑚−1^𝑘+𝑚+1𝑚.𝑘+𝑚 (2.14) On the other hand, we get 𝐼₄=(−1)^{𝑘𝑘+𝑚}_𝑙=0(−1)^𝑙𝑘𝑙𝐸_{𝑘+𝑚−𝑙}+(−1)^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚. (2.15) By (2.14) and (2.15), we obtain the following theorem.

Theorem 2.4. For 𝑘,𝑚∈ℤ₊, one has ^{max{𝑘,𝑚}}_𝑗=1^{𝑘+𝑚+1−𝑗}_𝑙=01𝑘𝑗+()𝑘+𝑚+1−𝑗−1^𝑗+1𝑚𝑗𝑙𝑘+𝑚+1−𝑗×𝐵_{𝑘+𝑚+1−𝑗−𝑙}𝐸_𝑙𝑘+2+()𝑘+𝑚−1^𝑘+𝑚+1𝑚−()𝑘+𝑚−1^𝑚+1(𝑘+𝑚+1)_𝑘^𝑘+𝑚=(−1)^{𝑘𝑘+𝑚}_𝑙=0(−1)^𝑙𝑘𝑙𝐸_{𝑘+𝑚−𝑙}. (2.16)

Acknowledgment

This Research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2012R1A1A2003786).