# Identities on the Bernoulli and Genocchi Numbers and Polynomials

Authors: Seog-Hoon Rim; Joohee Jeong; Sun-Jung Lee
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We give some interesting identities on the Bernoulli numbers and polynomials, on the Genocchi numbers and polynomials by using symmetric properties of the Bernoulli and Genocchi polynomials.


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## 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 the algebraic closure of ℚp. Let ℕ be the set of natural numbers and ℤ+=ℕ∪{0}. The p-adic norm on Cp is normalized so that |p|p=p-1. Let C(ℤp) be the space of continuous functions on ℤp. For f∈C(ℤp), the fermionic p-adic integral on ℤp is defined by Kim as follows:
(1.1)I-1(f)=∫ℤpf(x)dμ-1(x)=limN→∞∑‍pN-1x=0f(x)(-1)x (see [1–16]).  From (1.1), we have
(1.2)I-1(f1)=-I-1(f)+2f(0) (see [1–16]),  where f1(x)=f(x+1).Let us takef(x)=ext. Then, by (1.2), we get
(1.3)t∫ℤpextdμ-1(x)=2tet+1=∑‍∞n=0Gntnn!,
where Gn are the nth ordinary Genocchi numbers (see [8, 15]).From the same method of (1.3), we can also derive the following equation:
(1.4)t∫ℤp‍e(x+y)tdμ-1(y)=2tet+1ext=∑‍∞n=0Gn(x)tnn!,
where Gn(x) are called the nth Genocchi polynomials (see [14, 15]).By (1.3), we easily see that
(1.5)Gn(x)=∑l=0n‍(nl)Glxn-l (see [15]).  By (1.3) and (1.4), we get Witt's formula for the nth Genocchi numbers and polynomials as follows:
(1.6)∫ℤpxndμ-1(x)=Gn+1n+1,∫ℤp(x+y)ndμ-1(y)=Gn+1(x)n+1,forn∈ℤ+.
From (1.2), we have
(1.7)∫ℤp‍(x+1)ndμ-1(x)+∫ℤp‍xndμ-1(x)=2δ0,n,
where the symbol δ0,n is the Kronecker symbol (see [4, 5]).Thus, by (1.5) and (1.7), we get
(1.8)(G+1)n+Gn=2δ1,n(see [15]). From (1.4), we can derive the following equation:
(1.9)∫ℤp‍(1-x+y)ndμ-1(y)=(-1)n∫ℤp‍(x+y)ndμ-1(y).
By (1.6) and (1.9), we see that
(1.10)Gn+1(1-x)n+1=(-1)nGn+1(x)n+1.
Thus, by (1.10), we get Gn+1(2)/(n+1)=(-1)n(Gn+1(-1)/(n+1)).From (1.5) and (1.8), we have
(1.11)Gn+1(2)n+1=2-Gn+1(1)n+1=2+Gn+1n+1-2δ1,n+1.
The Bernoulli polynomials Bn(x) are defined by
(1.12)tet-1ext=eB(x)t=∑‍∞n=0Bn(x)tnn! (see [6, 9, 12]) with the usual convention about replacing Bn(x) by Bn(x).In the special case,x=0, Bn(0)=Bn is called the n-th Bernoulli number. By (1.12), we easily see that
(1.13)Bn(x)=∑l=0n‍(nl)xn-lBl=(B+x)n (see [6]).  Thus, by (1.12) and (1.13), we get reflection symmetric formula for the Bernoulli polynomials as follows:
(1.14)Bn(1-x)=(-1)nBn(x),(1.15)B0=1,(B+1)n-Bn=δ1,n(see [6, 9, 12]). From (1.14) and (1.15), we can also derive the following identity:
(1.16)(-1)nBn(-1)=Bn(2)=n+Bn(1)=n+Bn+δ1,n.
In this paper, we investigate some properties of the fermionic p-adic integrals on ℤp. By using these properties, we give some new identities on the Bernoulli and the Euler numbers which are useful in studying combinatorics.


## 2. Identities on the Bernoulli and Genocchi Numbers and Polynomials

Let us consider the following fermionicp-adic integral on ℤp as follows:
(2.1)I1=∫ℤp‍Bn(x)dμ-1(x)=∑l=0n‍(nl)Bn-l∫ℤp‍xldμ-1(x)=∑l=0n‍(nl)Bn-lGl+1l+1,forn∈ℤ+=ℕ∪{0}.
On the other hand, by (1.14) and (1.15), we get
(2.2)I1=(-1)n∫ℤp‍Bn(1-x)dμ-1(x)=(-1)n∑l=0n‍(nl)Bn-l∫ℤp‍(1-x)ldμ-1(x)=(-1)n∑l=0n‍(nl)Bn-l(-1)lGl+1(-1)l+1=(-1)n∑l=0n‍(nl)Bn-l(2+Gl+1l+1-2δ1,l+1)=2(-1)n(Bn+δ1,n)+(-1)n∑l=0n‍(nl)Bn-lGl+1l+1+2(-1)n+1Bn.
Equating (2.1) and (2.2), we obtain the following theorem.Theorem 2.1.
Forn∈ℤ+, one has(2.3)(1+(-1)n+1)∑l=0n‍(nl)Bn-lGl+1l+1=2(-1)nδ1,n.By using the reflection symmetric property for the Euler polynomials, we can also obtain some interesting identities on the Euler numbers.Now, we consider the fermionicp-adic integral on ℤp for the polynomials as follows:
(2.4)I2=∫ℤp‍Gn(x)dμ-1(x)=∑l=0n‍(nl)Gn-l∫ℤp‍xldμ-1(x)=∑l=0n‍(nl)Gn-lGl+1l+1,forn∈ℤ+.
On the other hand, by (1.8), (1.10), and (1.11), we get
(2.5)I2=(-1)n-1∫ℤp‍Gn(1-x)dμ-1(x)=(-1)n-1∑l=0n‍(nl)Gn-l∫ℤp‍(1-x)ldμ-1(x)=(-1)n-1∑l=0n‍(nl)Gn-l(-1)lGl+1(-1)l+1=(-1)n-1∑l=0n‍(nl)Gn-l(2+Gl+1l+1-2δ1,l+1)=2(-1)n-1(2δ1,n-Gn)+2(-1)nGn+(-1)n-1∑l=0n‍(nl)Gn-lGl+1l+1.
Equating (2.4) and (2.5), we obtain the following theorem.Theorem 2.2.
Forn∈ℤ+, one has(2.6)(1+(-1)n)∑l=0n‍(nl)Gn-lGl+1l+1=4(-1)nGn+4(-1)n+1δ1,n.Let us consider the fermionicp-adic integral on ℤp for the product of Bn(x) and Gn(x) as follows:
(2.7)I3=∫ℤp‍Bm(x)Gn(x)dμ-1(x)=∑k=0m‍∑l=0n‍(mk)(nl)Bm-kGn-l∫ℤpxk+ldμ-1(x)=∑k=0m‍∑l=0n‍(mk)(nl)Bm-kGn-lGk+l+1k+l+1.
On the other hand, by (1.10) and (1.14), we get
(2.8)I3=∫ℤpBm(x)Gn(x)dμ-1(x)=(-1)n+m-1∫ℤp‍Bm(1-x)Gn(1-x)dμ-1(x)=(-1)n+m-1∑k=0m‍∑l=0n‍(mk)(nl)Bm-kGn-l∫ℤp(1-x)k+ldμ-1(x)=2(-1)n+m-1Bm(1)Gn(1)+2(-1)m+nBmGn+(-1)n+m-1∑k=0m‍∑l=0n‍(mk)(nl)Bm-kGn-lGk+l+1k+l+1.
By (2.7) and (2.8), we easily see that
(2.9)(1+(-1)n+m+1)∑k=0m‍∑l=0n‍(mk)(nl)Bm-kGn-lGk+l+1k+l+1=2(-1)m+n-1(δ1,m+Bm)(2δ1,n-Gn)+2(-1)m+nBmGn=4(-1)m+n-1Bmδ1,n+2(-1)m+nBmGn+4(-1)m+n-1δ1,mδ1,n+2(-1)m+nδ1,mGn+2(-1)m+nBmGn.
Therefore, by (2.9), we obtain the following theorem.Theorem 2.3.
Forn,m∈ℤ+, one has(2.10)(1+(-1)n+m+1)∑k=0m‍∑l=0n‍(mk)(nl)Bm-kGn-l+1n-l+1Gk+l+1k+l+1=4(-1)m+nBmGn+4(-1)m+n-1Bmδ1,n+4(-1)m+n-1δ1,mδ1,n+2(-1)m+nδ1,mGn.Corollary 2.4.
Forn,m∈ℕ, one has(2.11)∑k=02m‍∑l=02n‍(2mk)(2nl)B2m-kG2n-lGk+l+1k+l+1=2B2mG2n.Let us consider the fermionicp-adic integral on ℤp for the product of the Bernoulli polynomials and the Bernstein polynomials. For n,k∈ℤ+, with 0≤k≤n, Bk,n(x)=(nk)xk(1-x)n-k are called the Bernstein polynomials of degree n, see [11]. It is easy to show that Bk,n(x)=Bn-k,n(1-x),
(2.12)I4=∫ℤpBm(x)Bk,n(x)dμ-1(x)=(nk)∑l=0m‍(ml)Bm-l∫ℤpxk+l(1-x)n-kdμ-1(x)=(nk)∑l=0m‍∑j=0n-k‍(ml)(n-kj)(-1)jBm-l∫ℤpxk+l+jdμ-1(x)=(nk)∑l=0m‍∑j=0n-k‍(ml)(n-kj)(-1)jBm-lGk+l+j+1k+l+j+1.
On the other hand, by (1.14) and (2.12), we get
(2.13)I4=(-1)m∫ℤpBm(1-x)Bn-k,n(1-x)dμ-1(x)=(-1)m(nk)∑l=0m‍∑j=0k‍(ml)(kj)(-1)jBm-l∫ℤp(1-x)n-k+l+jdμ-1(x)=(-1)m(nk)∑l=0m‍∑j=0k‍(ml)(kj)(-1)jBm-l×(2-2δ1,n-k+l+j+1+Gn-k+l+j+1n-k+l+j+1)=2(-1)m(nk)Bm(1)δ0,k+2(-1)m+1(nk)Bmδk,n+(-1)m(nk)∑l=0m‍∑j=0k‍(ml)(kj)(-1)jBm-lGn-k+l+j+1n-k+l+j+1.
Equating (2.12) and (2.13), we see that
(2.14)∑l=0m‍∑j=0n-k‍(ml)(n-kj)(-1)jBm-lGk+l+j+1k+l+j+1=2(-1)mBm(1)δ0,k+2(-1)m+1Bmδk,n+(-1)m∑l=0m‍∑j=0k‍(ml)(kj)(-1)jBm-lGn-k+l+j+1n-k+l+j+1.
Thus, from (2.14), we obtain the following theorem.Theorem 2.5.
Forn,m∈ℕ, one has(2.15)∑l=02m‍∑j=0n‍(2ml)(nj)(-1)jB2m-lGl+j+1l+j+1=2B2m(1)+∑l=02m‍(2ml)B2m-lGn+l+1n+l+1.Finally, we consider the fermionicp-adic integral on ℤp for the product of the Euler polynomials and the Bernstein polynomials as follows:
(2.16)I5=∫ℤp‍Gm(x)Bk,n(x)dμ-1(x)=(nk)∑l=0m‍(ml)Gm-l∫ℤp‍xk+l(1-x)n-kdμ-1(x)=(nk)∑l=0m‍∑j=0n-k‍(ml)(n-kj)(-1)jGm-l∫ℤp‍xk+l+jdμ-1(x)=(nk)∑l=0m‍∑j=0n-k‍(ml)(n-kj)(-1)jGm-lGk+l+j+1k+l+j+1.
On the other hand, by (1.10) and (2.12), we get
(2.17)I5=(-1)m-1∫ℤp‍Gm(1-x)Bn-k,n(1-x)dμ-1(x)=(-1)m-1(nk)∑l=0m‍(ml)Gm-l∑j=0k‍(kj)(-1)j∫ℤp‍(1-x)n-k+l+jdμ-1(x)=(-1)m-1(nk)∑l=0m‍∑j=0k‍(ml)(kj)(-1)jGm-l×(2+Gn-k+l+j+1n-k+l+j+1-2δ1,n-k+l+j+1)=2(-1)m-1(nk)Gm(1)δ0,k+2(-1)m(nk)Gmδk,n+(-1)m-1(nk)∑l=0m‍∑j=0k‍(ml)(kj)(-1)jGm-lGn-k+l+j+1n-k+l+j+1.
Equating (2.16) and (2.17), we obtain
(2.18)∑l=0m‍∑j=0n-k‍(ml)(n-kj)(-1)jGm-lGk+l+j+1k+l+j+1=2(-1)m-1Gm(1)δ0,k+2(-1)mGmδk,n+(-1)m-1∑l=0m‍∑j=0k‍(ml)(kj)(-1)jGm-lGn-k+l+j+1n-k+l+j+1.
Therefore, by (2.18), we obtain the following theorem.Theorem 2.6.
Forn,m∈ℕ, one has(2.19)∑l=02m‍∑j=0n‍(2ml)(nj)(-1)jG2m-lGl+j+1l+j+1=-2G2m(1)-∑l=02m‍(2ml)G2m-lGn+l+1n+l+1.



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