DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation62106810.1155/2009/621068621068Research ArticleArithmetic Identities Involving Genocchi and Stirling NumbersLiuGuodongBerezanskyLeonidDepartment of MathematicsHuizhou UniversityHuizhouGuangdong 516015Chinahzu.edu.cn200927092009200918062009120820092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Gn and the Bernoulli numbers Bn (n0={0,1,2,}) are defined by the following generating functions (see ):

2tet+1=n=0Gntnn!(|t|<π),tet-1=n=0Bntnn!        (|t|<2π), respectively. By (1.1) and (1.2), we have

G2n+1=B2n+1=0,(n)  Gn=2(1-2n)Bn, with being the set of positive integers.

The Genocchi numbers Gn satisfy the recurrence relation

G0=0,G1=1,Gn=-12k=1n-1(nk)Gk(n2), 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 )

(x)n=x(x-1)(x-n+1)=k=0ns(n,k)xk, or by the generating function

(log(1+x))k=k!n=ks(n,k)xnn!.

It follows from (1.5) or (1.6) that

s(n,k)=s(n-1,k-1)-(n-1)s(n-1,k), with s(n,0)=0(n>0), s(n,n)=1(n0), 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 )

xn=k=0nS(n,k)(x)k or by the generating function

(ex-1)k=k!n=kS(n,k)xnn!. It follows from (1.8) or (1.9) that

S(n,k)=S(n-1,k-1)+kS(n-1,k), with S(n,0)=0(n>0), S(n,n)=1(n0), 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, 316]). 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 nk  (n,k), then v1,,vkv1++vk=nGv1Gvk(v1vk)v1!vk!=(-1)n-k2kk!n!j=kn12jS(n,j)s(j,k).

Remark 1.2.

Setting k=1 in (1.11), and noting that s(j,1)=(-1)j-1(j-1)!, we obtain Gn=2nj=1n(-1)n-j(j-1)!2jS(n,j)(n).

Remark 1.3.

By (1.11) and (1.3), we have v1,,vkv1++vk=n(2v1-1)Bv1(2vk-1)Bvk(v1vk)v1!vk!=(-1)nk!n!j=kn12jS(n,j)s(j,k).

Theorem 1.4.

Let n,k, then j=0n(-1)j(k+j-1)!2jS(n,j)=2k-1j=0k-1(-1)js(k,k-j)Gn+k-jn+k-j.

Remark 1.5.

Setting k=1,2,3,4 in (1.14), we get j=0n(-1)jj!2jS(n,j)=1n+1Gn+1,j=0n(-1)j(j+1)!2jS(n,j)=2n+1Gn+1+2n+2Gn+2,j=0n(-1)j(j+2)!2jS(n,j)=8n+1Gn+1+12n+2Gn+2+4n+3Gn+3,j=0n(-1)j(j+3)!2jS(n,j)=48n+1Gn+1+88n+2Gn+2+48n+3Gn+3+8n+4Gn+4.

Theorem 1.6.

Let n, m0, then 2nn+1Gn+12n-mj=0m(mj)jn(mod    m+1).

Remark 1.7.

Setting m=p-1 in (1.16), we have 1n+1Gn+1j=0p-1(-1)jjn(mod  p), where p is any odd prime.

For a real or complex parameter x, we have the generalized Genocchi numbers Gn(x), which are defined by (2e2t+1)x=n=0Gn(x)tnn!(|t|<π2;1x:=1). By (1.1) and (2.1), we have nGn-1(1)=2n-1Gn.

Remark 2.2.

For an integer x, the higher-order Euler numbers E2n(x) are defined by the following generating functions (see ): (2et+e-t)x=n=0E2n(x)t2n(2n)!(|t|<π2). Then we have Gn(x)=(-1)nk=0[n/2](n2k)E2k(x)xn-2k, where [n/2] denotes the greatest integer not exceeding n/2.

Lemma 2.3.

Let nk(n,k), then Gn(x)=k=1nω(n,k)xk, where ω(n,k)=(-1)kj=kn2n-jS(n,j)s(j,k).

Proof.

By (2.1), (1.5), and (1.9) we have n=0Gn(x)tnn!=(2e2t+1)x=(11+(1/2)(e2t-1))x=j=0(-1)j2j(x+j-1j)(e2t-1)j=j=0(-1)j2j(x+j-1j)j!n=j2nS(n,j)tnn!=n=0j=0n(-1)jj!2n-j(x+j-1j)S(n,j)tnn!, which readily yields Gn(x)=j=0n(-1)jj!2n-j(x+j-1j)S(n,j)=j=0n(-1)j2n-jS(n,j)(x+j-1)(x+j-2)(x+1)x=j=0n(-1)j2n-jS(n,j)k=1j(-1)j-ks(j,k)xk=k=1n(-1)kj=kn2n-jS(n,j)s(j,k)xk=k=1nω(n,k)xk. This completes the proof of Lemma 2.3.

Remark 2.4.

From (1.7), (1.10), and Lemma 2.3 we know that Gn(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.

Remark 2.5.

Let n,m, then by (2.5), we have k=1nω(n,k)=2nn+1Gn+1. Therefore, if q is odd, then by (2.10) we get G2kq0(mod  q), where k.

3. Proof of the TheoremsProof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>.

By applying Lemma 2.3, we have k!ω(n,k)=dkdxkGn(x)|x=0. On the other hand, it follows from (2.1) that n=kdkdxkGn(x)|x=0tnn!=(log2e2t+1)k, where log(2/(e2t+1))   is the principal branch of logarithm of 2/(e2t+1).

Thus, by (3.1) and (3.2), we have k!n=kω(n,k)tnn!=(log2e2t+1)k.

Now note that ddtlog2e2t+1=2e2t+1-2=n=0Gn(1)tnn!-2=n=02nGn+1n+1tnn!-2, whence by integrating from 0 to t, we deduce that log2e2t+1=n=12n-1Gnntnn!-2t=n=1(-1)n2n-1Gnntnn!. Since G2n+1=0   (n). Substituting (3.5) in (3.3) we get ω(n,k)=(-1)nn!2n-kk!v1,,vkv1++vk=nGv1Gvk(v1vk)v1!vk!.

By (3.6) and (2.6), we may immediately obtain Theorem 1.1. This completes the proof of Theorem 1.1.

Proof of Theorem <xref ref-type="statement" rid="thm1.2">1.4</xref>.

By (2.1) and note the identity 2(2e2t+1)x+1xddt(2e2t+1)x=(2e2t+1)x+1, we have Gn(x+1)=2Gn(x)+1xGn+1(x).

By (3.8), (1.7), and note that Gn(1)=2n/(n+1)Gn+1, we obtain Gn(k)=1(k-1)!j=0k-1(-1)j2js(k,k-j)Gn+k-1-j(1)=2n+k-1(k-1)!j=0k-1(-1)js(k,k-j)Gn+k-jn+k-j. Comparing (3.9) and (2.8), we immediately obtain Theorem 1.4. This completes the proof of Theorem 1.4.

Proof of Theorem <xref ref-type="statement" rid="thm1.3">1.6</xref>.

By Lemma 2.3, we have Gn(m+x)=j=1nω(n,j)(m+x)jj=1nω(n,j)xj=Gn(x)(mod    m). Therefore Gn(k)=Gn(m+k-m)Gn(-m)=2n-mj=0m(mj)jn(mod    m+k). Taking k=1 in (3.11) and note that Gn(1)=2n/(n+1)Gn+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).

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