AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 269640 10.1155/2012/269640 269640 Research Article A Note on Eulerian Polynomials Kim D. S. 1 Kim T. 2 Kim W. J. 3 Dolgy D. V. 4 Diblík Josef 1 Department of Mathematics Sogang University Seoul 121-742 Republic of Korea sogang.ac.kr 2 Department of Mathematics Kwangwoon University Seoul 139-701 Republic of Korea kw.ac.kr 3 Division of General Education-Mathematics Kwangwoon University Seoul 139-701 Republic of Korea kw.ac.kr 4 Hanrimwon, Kwangwoon University Seoul 139-701 Republic of Korea kw.ac.kr 2012 17 7 2012 2012 29 05 2012 25 06 2012 2012 Copyright © 2012 D. S. Kim et al. This 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.

We study Genocchi, Euler, and tangent numbers. From those numbers we derive some identities on Eulerian polynomials in connection with Genocchi and tangent numbers.

1. Introduction

As is well known, the Eulerian polynomials, An(t), are defined by generating function as follows: (1.1)1-texp(x(t-1))-t=eA(t)x=n=0An(t)xnn!, with the usual convention about replacing An(t) by An(t) (see ). From (1.1), we note that (1.2)(A(t)+(t-1))n-tAn(t)=(1-t)δ0,n, where δn,k is the Kronecker symbol (see ).

Thus, by (1.2), we get (1.3)A0(t)=1,An(t)=1t-1l=0n-1(nl)Al(t)(t-1)n-l,(n1).

By (1.1), (1.2), and (1.3), we see that (1.4)i=1minti=l=1n(-1)n+l(nl)tm+1An-l(t)(t-1)n-l+1ml+(-1)nt(tm-1)(t-1)n+1An(t), where m1 and n0 (see ).

The Genocchi polynomials are defined by (1.5)2tet+1ext=eG(x)t=n=0Gn(x)tnn!,(see ). In the special case, x=0, Gn(0)=Gn are called the nth Genocchi numbers (see [14, 17, 18]).

It is well known that the Euler polynomials are also defined by (1.6)2et+1ext=eE(x)t=n=0En(x)tnn!,(see [15, 1924]). Here x=0, then En(0)=En is called the nth Euler number. From (1.6), we have (1.7)E0=1,(E+1)n+En=2δ0,n,(see [35, 1923]).

As is well known, the Bernoulli numbers are defined by (1.8)B0=1,(B+1)n-Bn=δ0,n,(see [5, 18, 19]), with the usual convention about replacing Bn by Bn.

From (1.8), we note that the Bernoulli polynomials are also defined as (1.9)Bn(x)=l=0n(nl)Blxn-l=(B+x)n,(see [5, 18, 19]).

The tangent numbers T2n-1    (n1) are defined as the coefficients of the Taylor expansion of tanx: (1.10)tanx=n=1T2n-1(2n-1)!x2n-1=x1!+x33!2+x55!16+,(see [13, 5]).

In this paper, we give some identities on the Eulerian polynomials at t=-1 associated with Genocchi, Euler, and tangent numbers.

2. Witt's Formula for Eulerian Polynomials

In this section, we assume that p, p, and p will, respectively, denote the ring of p-adic integers, the field of p-adic numbers, and the completion of algebraic closure of p. The p-adic norm is normalized so that |p|p=1/p.

Let q be an indeterminate with |1-q|p<1. Then the q-number is defined by (2.1)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q,(see ).

Let C(p) be the space of continuous functions on p. For fC(p), the fermionic p-adic q-integral on p is defined by (2.2)I-q(f)=Zpf(x)dμ-q(x)=limN1[pN]-qx=0pN-1f(x)(-q)x, (see [7, 1013]). From (2.2), we can derive the following: (2.3)q-1I-q-1(f1)+I-q-1(f)=q-1f(0), where f1(x)=f(x+1).

Let us take f(x)=e-x(1+q)t. Then, by (2.3), we get (2.4)(q+e-(1+q)tq)Zpe-x(1+q)tdμ-q-1(x)=q-1.

Thus, from (2.4), we have (2.5)Zpe-x(1+q)tdμ-q-1(x)=1+qe-(1+q)t+q=n=0An(-q)tnn!.

By Taylor expansion on the left-hand side of (2.5), we get (2.6)n=0(-1)nZpxndμ-q-1(x)(1+q)ntnn!=n=0An(-q)tnn!.

Comparing coefficients on the both sides of (2.6), we have (2.7)Zpxndμ-q-1(x)=(-1)n(1+q)nAn(-q). Therefore, by (2.7), we obtain the following theorem.

Theorem 2.1.

For n+, one has (2.8)Zpxndμ-q-1(x)=(-1)n(1+q)nAn(-q), where An(-q) is an Eulerian polynomials.

It seems interesting to study Theorem 2.1 at q=1. By (2.3), we get (2.9)I-1(f1)+I-1(f)=2f(0), where f1(x)=f(x+1). From (2.9), we can derive the following equation: (2.10)Zpf(x+n)dμ-1(x)+(-1)n-1Zpf(x)dμ-1(x)=2l=0n-1(-1)n-l+1f(l), where n+ (see ).

From (2.9), we can derive the following: (2.11)0=psina(x+1)dμ-1(x)+psinaxdμ-1(x)=(cosa+1)psinaxdμ-1(x)+sinapcosaxdμ-1(x),2=Zpcosa(x+1)dμ-1(x)+Zpcosaxdμ-1(x)=(cosa+1)Zpcosaxdμ-1(x)-sinaZpsinaxdμ-1(x). By (2.11), we get (2.12)Zpsinaxdμ-1(x)=-sinacosa+1=-tana2. From (1.10) and (2.12), we have (2.13)n=1T2n-1(2n-1)!(a2)2n-1=-Zpsinaxdμ-1(x)=n=1(-1)na2n-1(2n-1)!Zpx2n-1dμ-1(x). By comparing coefficients on the both sides of (2.13), we get (2.14)Zpx2n-1dμ-1(x)=(-1)nT2n-122n-1,fornN, where T2n-1 is the (2n-1)th tangent number.

Therefore, by (2.14), we obtain the following theorem.

Theorem 2.2.

For n, one has (2.15)Zpx2n-1dμ-1(x)=(-1)nT2n-122n-1, where T2n-1 is the (2n-1)th tangent numbers.

From Theorem 2.1, one has (2.16)Zpxndμ-1(x)=(-1)n2nAn(-1). Therefore, by Theorem 2.2 and (2.16), we obtain the following corollary.

Corollary 2.3.

For n, one has (2.17)A2n-1(-1)=(-1)n-1T2n-1.

From (1.6) and (2.9), we have (2.18)Zpextdμ-1(x)=2et+1=n=0Entnn!, (see ). Thus, by (2.16) and (2.18), we get (2.19)Zpx2n-1dμ-1(x)=E2n-1=(-1)nT2n-122n-1. Therefore, by Corollary 2.3 and (2.19), we obtain the following corollary.

Corollary 2.4.

For n, one has (2.20)E2n-1=(-1)nT2n-122n-1=-A2n-1(-1)22n-1.

By (1.5) and (2.9), we get (2.21)tZpextdμ-1(x)=2te2t-1et-2te2t-1=n=0Bn(12)2ntnn!-n=02nBnn!tn=n=0(Bn(12)-Bn)2ntnn!.

By (2.21), we get (2.22)Zpxndμ-1(x)=(Bn+1(1/2)-Bn+1)n+12n+1.

Thus, from (2.19), Theorem 2.2 and Corollary 2.3, we have (2.23)(B2n(1/2)-B2n)22n2n=(-1)nT2n-122n-1=-A2n-1(-1)22n-1.

Therefore, by (2.23), we obtain the following theorem.

Theorem 2.5.

For n, one has (2.24)(B2n(1/2)-B2n)22nn=(-1)nT2n-122n-2=-A2n-1(-1)22n-2.

From (1.5), we note that (2.25)tZpextdμ-1(x)=2tet+1=n=0Gntnn! (see [13, 14]). Thus, by (2.25), we get (2.26)G0=0,(G+1)n+Gn=2δ1,n, (see [13, 14]), with the usual convention about replacing Gn by Gn.

From (1.5) and (2.9), one has (2.27)tZpextdμ-1(x)=2(tet-1-2te2t-1)=2n=0(Bn-2nBn)tnn!. Thus, by (2.27), we get (2.28)Zpxndμ-1(x)=2(Bn+1-2n+1Bn+1n+1).

From (2.28), we have (2.29)G2n2n=Zpx2n-1dμ-1(x)=B2n-22nB2nn,forN. Therefore, by (2.19), Corollary 2.3 and (2.29), we obtain the following theorem.

Theorem 2.6.

For n, we have (2.30)G2n=2(B2n-22nB2n).

In particular, (2.31)-122n-1(A2n-1(-1))=((-1)nT2n-1)122n-1=G2n2n.

3. Further Remark

In complex plane, we note that (3.1)tanx=1i(eix-e-ixeix+e-ix)=1i(1-2e-ixeix+e-ix)=1i(1-n=0Enn!2ninxn)=1i(-n=1Enn!2ninxn)=n=1(-1)n(2n-1)!E2n-122n-1x2n-1. By (1.10) and (3.1), we also get (3.2)T2n-1=(-1)nE2n-122n-1,fornN. From (1.5), we have (3.3)n=1t2n(2n)!G2n=n=1(it)2n(2n)!(-1)nG2n=2it1+eit-it=it(1-eit)1+eit=it(e-it/2-eit/2)eit/2+e-it/2=t((eit/2-e-it/2)/2i(eit/2+e-it/2)/2)=ttan(t2). Thus, by (1.10) and (3.3), we get (3.4)n=1t2n(2n)!G2n=t    tan(t2)=tn=1(t/2)2n-1(2n-1)!T2n-1=n=1t2n(2n-1)!22n-1T2n-1.

From (3.4), we have (3.5)nT2n-1=22n-2G2n=22n-1(1-22n)B2n. By (1.1), we see that (3.6)21+e-2it=n=0An(-1)intnn!. Thus, we note that (3.7)n=1in-1An(-1)tnn!=1i(21+e-2it-1)=1-e-2it(1+e-2it)i=((eit-e-it)/2)((eit+e-it)/2)i=tant=n=1T2n-1t2n-1(2n-1)!.

From (3.7), we have (3.8)A2n(-1)=0,A2n-1(-1)=(-1)n-1T2n-1,(n1). It is easy to show that (3.9)k=1mkn(-1)k=(-1)mk=0n(nk)Ak(-1)2k+1mn-k-{(-1)m-1}2n+1An(-1). For simple calculation, we can derive the following equation: (3.10)itanx=eix-e-ixeix+e-ix=1-2e2ix-1+4e4ix-1. By (3.10), we get (3.11)xtanx=-ix+2ixe2ix-1-4ixe4ix-1=n=1(-1)nB2n4n(1-4n)(2n)!x2n. Thus, from (3.11),we have (3.12)tanx=n=1(-1)nB2n4n(1-4n)(2n)!x2n-1.

By (1.10) and (3.12), we get (3.13)T2n-1=(-1)nB2n4n(1-4n)2n,fornN. From Corollary 2.3 and (3.13), we can derive the following identity: (3.14)A2n-1(-1)=-B2n22n-1(1-4n)n.

Acknowledgments

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. Also, the authors would like to thank the referees for their valuable comments and suggestions.