# Arithmetic Identities Involving Bernoulli and Euler Numbers

Authors: H.-M. Kim; D. S. Kim
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

---

## Abstract

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


---

## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this paper, ℤp, ℚp, and ℂp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of ℚp, 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}.LetUD(ℤp) be the space of uniformly differentiable functions on ℤp. For f∈UD(ℤp), the bosonic p-adic integral on ℤp is defined by
(1.1)I(f)=∫ℤp‍f(x)dμ(x)=limN→∞∑x=0pN-1‍f(x)μ(x+pNℤp)=limN→∞1pN∑x=0pN-1‍f(x),
and the fermionic p-adic integral on ℤp is defined by Kim as follows (see [1–8]):
(1.2)I-1(f)=∫ℤp‍f(x)dμ-1(x)=limN→∞∑x=0pN-1‍f(x)(-1)x.The Euler polynomials,En(x), are defined by the generating function as follows (see [1–16]):
(1.3)FE(t,x)=2et+1ext=∑n=0∞‍En(x)tnn!.
In the special case, x=0, En(0)=En is called the nth Euler number.By (1.3) and the definition of Euler numbers, we easily see that
(1.4)En(x)=∑l=0n‍(nl)Elxn-l=(E+x)n,
with the usual convention about replacing El by El (see [10]). Thus, by (1.3) and (1.4), we have
(1.5)E0=1,(E+1)n+En=2δ0,n,
where δk,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 ℤp as follows:
(1.6)I-1(f1)=-I-1(f)+2f(0),
see [1, 2]. By (1.3) and (1.6), we get
(1.7)∫ℤp‍e(x+y)tdμ-1(y)=2et+1ext=∑n=0∞‍En(x)tnn!.
Thus, by (1.7), we have
(1.8)∫ℤp‍(x+y)ndμ-1(y)=En(x),
see [1–8, 13–16].The Bernoulli polynomials,Bn(x), are defined by the generating function as follows:
(1.9)FB(t,x)=tet-1ext=∑n=0∞‍Bn(x)tnn!,
see [18]. In the special case, x=0, Bn(0)=Bn is called the nth Bernoulli number. From (1.9) and the definition of Bernoulli numbers, we note that
(1.10)Bn(x)=∑l=0n‍(nl)xn-lBl=(B+x)n,
see [1–19], with the usual convention about replacing Bl by Bl. By (1.9) and (1.10), we easily see that
(1.11)B0=1,(B+1)n-Bn=δ1,n,
see [13].From (1.1), we can derive the following integral equation on ℤp:
(1.12)I(f1)=I(f)+f′(0),
where f1(x)=f(x+1) and f'(0)=(df(x)/dx)|x=0.By (1.12), we have
(1.13)∫ℤp‍e(x+y)tdμ(y)=tet-1ext=∑n=0∞‍Bn(x)tnn!.
Thus, by (1.13), we can derive the following Witt’s formula for the Bernoulli polynomials:
(1.14)∫ℤp‍(x+y)ndμ(y)=Bn(x),forn∈ℤ+.In [19], it is known that for k,m∈ℤ+,
(1.15)∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]Bk+m+1-j(x)k+m+1-j=xk(x-1)m+(-1)m+1(k+m+1)(k+mk),
where (kj)=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 ofp-adic integral equations on ℤp.


## 2. Arithmetic Identities for Bernoulli and Euler Numbers

Let us take the bosonicp-adic integral on ℤp in (1.15) as follows:
(2.1)I1=∫ℤp‍xk(x-1)mdμ(x)+(-1)m+1(k+m+1)(k+mk)=∑l=0m‍(ml)(-1)l∫ℤp‍xk+m-ldμ(x)+(-1)m+1(k+m+1)(k+mk)=∑l=0m‍(ml)(-1)lBk+m-l+(-1)m+1(k+m+1)(k+mk).
On the other hand, we get
(2.2)I1=∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]1k+m+1-j∫ℤp‍Bk+m+1-j(x)dμ(x)=∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]1k+m+1-j×∑l=0k+m+1-j‍(k+m+1-jl)Bk+m+1-j-lBl.
By (2.1) and (2.2), we get
(2.3)∑j=1max{k,m}‍∑l=0k+m+1-j‍1k+m+1-j[(kj)+(-1)j+1(mj)]×(k+m+1-jl)Bk+m+1-j-lBl=∑l=0m‍(-1)l(ml)Bk+m-l+(-1)m+1(k+m+1)(k+mk).Therefore, by (2.3), we obtain the following theorem.Theorem 2.1.
Fork,m∈ℤ+, one has
(2.4)∑j=1max{k,m}‍∑l=0k+m+1-j‍1k+m+1-j[(kj)+(-1)j+1(mj)]×(k+m+1-jl)Bk+m+1-j-lBl-(-1)m+1(k+m+1)(k+mk)=∑l=0m‍(-1)l(ml)Bk+m-l.Now we consider the fermionicp-adic integral on ℤp in (1.15) as follows:
(2.5)I2=∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]1k+m+1-j∑l=0k+m+1-j‍(k+m+1-jl)×Bk+m+1-j-l∫ℤp‍xldμ-1(x)=∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]1k+m+1-j∑l=0k+m+1-j‍(k+m+1-jl)×Bk+m+1-j-lEl.
On the other hand, we get
(2.6)I2=∑l=0m‍(-1)l(ml)∫ℤp‍xm-l+kdμ-1(x)+(-1)m+1(k+m+1)(k+mk)=∑l=0m‍(-1)l(ml)Ek+m-l+(-1)m+1(k+m+1)(k+mk).
By (2.5) and (2.6), we get
(2.7)∑j=1max{k,m}‍∑l=0k+m+1-j‍1k+m+1-j[(kj)+(-1)j+1(mj)](k+m+1-jl)×Bk+m+1-j-lEl=∑l=0m‍(-1)l(ml)Ek+m-l+(-1)m+1(k+m+1)(k+mk).
Therefore, by (2.7), we obtain the following theorem.Theorem 2.2.
Fork,m∈ℤ+, one has
(2.8)∑j=1max{k,m}‍∑l=0k+m+1-j‍1k+m+1-j[(kj)+(-1)j+1(mj)](k+m+1-jl)×Bk+m+1-j-lEl-(-1)m+1(k+m+1)(k+mk)=∑l=0m‍(-1)l(ml)Ek+m-l.Replacingx by (1-x) in (1.15), we have the identity:
(2.9)∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]Bk+m+1-j(1-x)k+m+1-j=(-1)k+mxm(1-x)k+(-1)m+1(k+m+1)(k+mk).
Let us take the bosonic p-adic integral on ℤp in (2.9) as follows:
(2.10)I3=∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]1k+m+1-j×∑l=0k+m+1-j‍(k+m+1-jl)Bk+m+1-j-l∫ℤp‍(1-x)ldμ(x)=∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]1k+m+1-j×∑l=0k+m+1-j‍(k+m+1-jl)Bk+m+1-j-lBl+∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]1k+m+1-j×∑l=0k+m+1-j‍(k+m+1-jl)Bk+m+1-j-ll+∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]1k+m+1-j×∑l=0k+m+1-j‍(k+m+1-jl)Bk+m+1-j-lδ1,l=∑j=1max{k,m}‍∑l=0k+m+1-j‍1k+m+1-j[(kj)+(-1)j+1(mj)]×(k+m+1-jl)Bk+m+1-j-lBl+∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)](2Bk+m-j+δ1,(k+m-j))=∑j=1max{k,m}‍∑l=0k+m+1-j‍1k+m+1-j[(kj)+(-1)j+1(mj)]×(k+m+1-jl)Bk+m+1-j-lBl+2∑j=1max(k,m)‍[(kj)+(-1)j+1(mj)]×Bk+m-j+(kk+m-1)+(-1)k+m(mk+m-1).On the other hand, we see that(2.11)I3=(-1)k+m∑l=0k‍(-1)l(kl)Bk+m-l+(-1)m+1(k+m+1)(k+mk).
By (2.10) and (2.11), we get
(2.12)∑j=1max{k,m}‍∑l=0k+m+1-j‍1k+m+1-j[(kj)+(-1)j+1(mj)]×(k+m+1-jl)Bk+m+1-j-lBl+2∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]×Bk+m-j+(kk+m-1)+(-1)k+m(mk+m-1)=(-1)k+m∑l=0k‍(-1)l(kl)Bk+m-l+(-1)m+1(k+m+1)(k+mk).
Therefore, by (2.12), we obtain the following theorem.Theorem 2.3.
Fork,m∈ℤ+, one has
(2.13)∑j=1max{k,m}‍∑l=0k+m+1-j‍1k+m+1-j[(kj)+(-1)j+1(mj)]×(k+m+1-jl)Bk+m+1-j-lBl+2∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]×Bk+m-j+(kk+m-1)+(-1)k+m(mk+m-1)-(-1)m+1(k+m+1)(k+mk)=(-1)k+m∑l=0k‍(-1)l(kl)Bk+m-l.We consider the fermionicp-adic integral on ℤp in (2.9) as follows:
(2.14)I4=∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]1k+m+1-j×∑l=0k+m+1-j‍(k+m+1-jl)Bk+m+1-j-l∫ℤp‍(1-x)ldμ-1(x)=∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]1k+m+1-j×∑l=0k+m+1-j‍(k+m+1-jl)Bk+m+1-j-lEl+2∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]1k+m+1-j×∑l=0k+m+1-j‍(k+m+1-jl)Bk+m+1-j-l-2∑j=1max{k,m}‍[(kj)+(-1)j+1(mj)]1k+m+1-j×∑l=0k+m+1-j‍(k+m+1-jl)Bk+m+1-j-lδ0,l=∑j=1max{k,m}‍∑l=0k+m+1-j‍1k+m+1-j[(kj)+(-1)j+1(mj)]×(k+m+1-jl)Bk+m+1-j-lEl+2∑j=1max{k,m}‍1k+m+1-j[(kj)+(-1)j+1(mj)]δ1,(k+m+1-j)=∑j=1max{k,m}‍∑l=0k+m+1-j‍1k+m+1-j[(kj)+(-1)j+1(mj)]×(k+m+1-jl)Bk+m+1-j-lEl+2[(kk+m)+(-1)k+m+1(mk+m)].
On the other hand, we get
(2.15)I4=(-1)k+m∑l=0k‍(-1)l(kl)Ek+m-l+(-1)m+1(k+m+1)(k+mk).
By (2.14) and (2.15), we obtain the following theorem.Theorem 2.4.
Fork,m∈ℤ+, one has
(2.16)∑j=1max{k,m}‍∑l=0k+m+1-j‍1k+m+1-j[(kj)+(-1)j+1(mj)](k+m+1-jl)×Bk+m+1-j-lEl+2[(kk+m)+(-1)k+m+1(mk+m)]-(-1)m+1(k+m+1)(k+mk)=(-1)k+m∑l=0k‍(-1)l(kl)Ek+m-l.



---

*Source: 689797-2012-11-21.xml*