AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 865721 10.1155/2012/865721 865721 Research Article Applications of Umbral Calculus Associated with p-Adic Invariant Integrals on Zp Kim Dae San 1 Kim Taekyun 2 N'Guerekata Gaston 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 2012 13 12 2012 2012 08 11 2012 22 11 2012 2012 Copyright © 2012 Dae San Kim and Taekyun Kim. 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.

Recently, Dere and Simsek (2012) have studied the applications of umbral algebra to some special functions. In this paper, we investigate some properties of umbral calculus associated with p-adic invariant integrals on Zp. From our properties, we can also derive some interesting identities of Bernoulli polynomials.

1. Introduction

Let p be a fixed prime number. Throughout this paper, Zp,Qp, and Cp denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively.

Let N{0}. Let UD(Zp) be space of uniformly differentiable functions on Zp. For fUD(Zp), the p-adic invariant integral on Zp is defined by (1.1)Zpf(x)dμ(x)=limN1pNx=0pN-1f(x), see [1, 2].

From (1.1), we have (1.2)Zpf(x+n)dμ(x)-Zpf(x)dμ(x)=l=0nf(l),nN, where f'(l)=(df(x)/dx)x=l (see ). Let F be the set of all formal power series in the variable t over Cp with (1.3)F={f(t)=k=0akk!tkakCp}. Let =Cp[x] and let * denote the vector space of all linear functional on .

The formal power series, (1.4)f(t)=k=0akk!tkF, defines a linear functional on by setting (1.5)f(t)xn=an,n0, see [7, 8].

In particular, by (1.4) and (1.5), we get (1.6)tkxn=n!δn,k, where δn,k is the Kronecker symbol (see ). Here, F denotes both the algebra of formal power series in t and the vector space of all linear functional on , so an element f(t) of F will be thought of as both a formal power series and a linear functional. We shall call F the umbral algebra. The umbral calculus is the study of umbral algebra.

The order o(f(t)) of power series f(t)(0) is the smallest integer k for which ak does not vanish. We define o(f(t))= if f(t)=0. From the definition of order, we note that o(f(t)g(t))=o(f(t))+o(g(t)) and o(f(t)+g(t))min{o(f(t)),o(g(t))}.

The series f(t) has a multiplicative inverse, denoted by f(t)-1 or 1/f(t), if and only if o(f(t))=0.

Such a series is called invertible series. A series f(t) for which o(f(t))=1 is called a delta series (see [7, 8]). Let f(t),g(t)F. Then, we have (1.7)f(t)g(t)p(x)=f(t)g(t)p(x)=g(t)f(t)p(x). By (1.5) and (1.6), we get (1.8)eytxn=yn,eytp(x)=p(y), see .

Notice that for all f(t) in F, (1.9)f(t)=k=0f(t)xkk!tk, and for all polynomials p(x), (1.10)p(x)=k0tkp(x)k!xk, see [7, 8].

Let f1(t),f2(t),,fm(t)F. Then, we have (1.11)f1(t)f2(t)fm(t)xn=(ni1,,im)f1(t)xi1fm(t)xim, where the sum is over all nonnegative integers i1,i2,,im such that i1++im=n (see ).

By (1.10), we get (1.12)p(k)(x)=dkp(x)dxk=l=kntlp(x)l!l(l-1)(l-k+1)xl-k. Thus, from (1.12), we have (1.13)p(k)(0)=tkp(x)=1p(k)(x), see .

By (1.13), we get (1.14)tkp(x)=p(k)(x)=dk(p(x))dxk. Thus, by (1.14), we see that (1.15)eytp(x)=p(x+y). Let us assume that sn(x) is a polynomial of degree n. Suppose that f(t),g(t)F with o(f(t))=1 and o(g(t))=0. Then, there exists a unique sequence sn(x) of polynomials satisfying g(t)f(t)ksn(x)=n!δn,k for all n,k0.

The sequence sn(x) is called the Sheffer sequence for (g(t),f(t)), which is denoted by sn(x)~(g(t),f(t)).

The Sheffer sequence for (g(t),t) is called the Appell sequence for g(t), or sn(x) is Appell for g(t), which is indicated by sn(x)~(g(t),t).

For p(x), it is known that (1.16)f(t)xp(x)=tf(t)p(x)=f(t)p(x),eyt-1p(x)=p(y)-p(0),   see [7, 8].

Let sn(x)~(g(t),f(t)). Then, we have (1.17)h(t)=k=0h(t)sk(x)k!g(t)f(t)k,h(t)F,(1.18)p(x)=k=0g(t)f(t)kp(x)k!sk(x),p(x),(1.19)1g(f-(t))eyf-(t)=k=0sk(y)k!tk,for  any  yCp, where f-(t) is the compositional inverse of f(t), and (1.20)f(t)sn(x)=nsn-1(x), see [7, 8].

We recall that the Bernoulli polynomials are defined by the generating function to be (1.21)tet-1ext=eB(x)t=n=0Bn(x)tnn!, with the usual convention about replacing Bn(x) by Bn(x) (see ).

In the special case, x=0,Bn(0)=Bn are called the nth Bernoulli numbers. By (1.21), we easily get (1.22)Bn(x)=(B+x)n=l=0n(nl)Blxn-l=l=0n(nl)Bn-lxl. Thus, by (1.22), we see that Bn(x) is a monic polynomial of degree n. It is easy to show that (1.23)B0=1,Bn(1)-Bn=δ1,n, see .

From (1.2), we can derive the following equation: (1.24)Zpf(x+1)dμ(x)-Zpf(x)dμ(x)=f(0). Let us take f(x)=etxUD(Zp). Then, from (1.21), (1.22), (1.23), and (1.24), we have (1.25)Zpxndμ(x)=Bn,Zp(x+y)ndμ(y)=Bn(x), where n0 (see [1, 2]). Recently, Dere and simsek have studied applications of umbral algebra to some special functions (see ). In this paper, we investigate some properties of umbral calculus associated with p-adic invariant integrals on Zp. From our properties, we can derive some interesting identities of Bernoulli polynomials.

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Let sn(x) be an Appell sequence for g(t). By (1.19), we get (2.1)1g(t)xn=sn(x),iff    xn=g(t)sn(x). Let us take g(t)=((et-1)/t)F. Then, g(t) is clearly invertible series. From (1.21) and (2.1), we have (2.2)k=0Bk(x)k!tk=1g(t)ext. Thus, by (2.2), we get (2.3)1g(t)xn=Bn(x),tBn(x)=Bn(x)=nBn-1(x),(n0). From (1.21), (2.1), and (2.3), we note that Bn(x) is an Appell sequence for g(t)=(et-1)/t.

Let us take the derivative with respect to t on both sides of (2.2). Then, we have (2.4)k=1Bk(x)k!ktk-1=xg(t)ext-extg(t)g(t)2=k=0{xxkg(t)-xkg(t)g(t)g(t)}tkk!. Thus, by (2.4), we get (2.5)Bk+1(x)=xxkg(t)-xkg(t)g(t)g(t)=(x-g(t)g(t)  )Bk(x), where k0. (2.6)Zpe(x+y+1)tdμ(y)-Zpe(x+y)tdμ(y)=text. Thus, by (2.6), we get (2.7)Zp(x+y+1)ndμ(y)-Zp(x+y)nμ(y)=nxn-1,(n0). From (1.25) and (2.7), we have (2.8)Bn(x+1)-Bn(x)=nxn-1,(n0). By (2.5), we see that (2.9)g(t)Bk+1(x)=g(t)xBk(x)-g(t)Bk(x), Thus, by (2.9), we have (2.10)(et-1)Bk+1(x)=(et-1)xBk(x)-(et-g(t))Bk(x),(k0), and we can derive the following equation.

From (2.3) and (2.10), (2.11)Bk+1(x+1)-Bk+1(x)=(x+1)Bk(x+1)-xBk(x)-Bk(x+1)+xk,(k0). By (2.8) and (2.11), we see that (2.12)Bk+1(x+1)=Bk+1(x)+(k+1)xk. Therefore, by (2.5), we obtain the following theorem.

Theorem 2.1.

For kZ+, one has (2.13)Bk+1(x)=(x-g(t)g(t))Bk, where g(t)=dg(t)/dt.

Corollary 2.2.

For 0, one has (2.14)Bk+1(x+1)=Bk+1(x)+(k+1)xk.

Let us consider the linear functional f(t) that satisfies (2.15)f(t)p(x)=Zpp(u)dμ(u), for all polynomials p(x). It can be determined from (1.9) that (2.16)f(t)=k=0f(t)xkk!tk=k=0Zpukdμ(u)tkk!=Zpeutdμ(u). By (1.24) and (2.16), we get (2.17)f(t)=Zpeutdμ(u)=tet-1. Therefore, by (2.17), we obtain the following theorem.

Theorem 2.3.

For p(x)P, one has (2.18)Zpeutdμ(u)p(x)=Zpp(u)dμ(u). That is (2.19)tet-1p(x)=Zpp(u)dμ(u). In particular, one has (2.20)Bn=Zpeutdμ(u)xn.

From (1.24), one has (2.21)n=0Zp(x+y)ndμ(y)tnn!=Zpe(x+y)tdμ(y)=n=0Zpeytdμ(y)xntnn!. By (1.25) and (2.21), we get (2.22)Bn(x)=Zp(x+y)ndμ(y)=Zpeytdμ(y)xn, where n0.

Therefore, by (2.22), we obtain the following theorem.

Theorem 2.4.

For p(x), we have (2.23)Zpp(x+y)dμ(y)=Zpeytdμ(y)p(x)=tet-1p(x). In particular, one obtains (2.24)Bn(x)=Zp(x+y)ndμ(y)=Zpeytdμ(y)xn=tet-1xn.

The higher order Bernoulli polynomials Bn(r)(x) are defined by (2.25)ZpZpe(x1+x2++xr+x)tdμ(x1)dμ(xr)  =(tet-1)rext=n=0Bn(r)(x)tnn!. In the special case, x=0, Bn(r)(0)=Bn(r) are called the nth Bernoulli numbers of order r (N). From (2.25), we note that (2.26)ZpZp(x1++xr)ndμ(x1)dμ(xr)=i1++ir=n(ni1,,ir)Zpx1i1dμ(x1)Zpx2i2dμ(x2)Zpxrirdμ(xr)=i1++ir=n(ni1,,ir)Bi1Bir=Bn(r). By (2.25) and (2.26), we get (2.27)Bn(r)(x)=l=0n(nl)Bn-l(r)xl. From (2.26) and (2.27), we note that Bn(r)(x) is a monic polynomial of degree n with coefficients in Q. For rN, let us assume that (2.28)g(r)(t)=(ZpZpe(x1++xr)tdμ(x1)dμ(xr))-1=(et-1t)r. By (2.28), we easily see that g(r)(t) is an invertible series. From (2.25) and (2.28), we have (2.29)extg(r)(t)=ZpZpe(x1++xr+x)tdμ(x1)dμ(xr)=n=0Bn(r)(x)tnn!,tBn(r)(x)=nBn-1(r)(x). From (2.29), we note that Bn(r) is an Appell sequence for g(r)(t). Therefore, by (2.29), we obtain the following theorem.

Theorem 2.5.

For p(x) and rN, one has (2.30)ZpZpp(x1++xr+x)dμ(x1)dμ(xr)=(tet-1)rp(x). In particular, the Bernoulli polynomials of order r are given by (2.31)Bn(r)(x)=(tet-1)rxn=ZpZpe(x1++xr)tdμ(x1)dμ(xr)xn. That is (2.32)Bn(r)(x)~((et-1t)r,t).

Let us consider the linear functional f(r)(t) that satisfies (2.33)f(r)(t)p(x)=ZpZpp(x1++xr)dμ(x1)dμ(xr), for all polynomials p(x). It can be determined from (1.9) that (2.34)f(r)(t)=k=0f(r)(t)xkk!tk=k=0ZpZp(x1++xr)kdμ(x1)dμ(xr)tkk!=ZpZpe(x1++xr)tdμ(x1)dμ(xr)=(tet-1)r. Therefore, by (2.34), we obtain the following theorem.

Theorem 2.6.

For p(x), one has (2.35)ZpZpe(x1++xr)tdμ(x1)dμ(xr)p(x)=ZpZpp(x1++xr)dμ(x1)dμ(xr). That is (2.36)(tet-1)rp(x)=ZpZpp(x1++xr)dμ(x1)dμ(xr). In particular, one gets (2.37)Bn(r)=ZpZpe(x1++xr)tdμ(x1)dμ(xr)xn.

Remark 2.7.

From (1.11), we note that (2.38)ZpZpe(x1++xr)tdμ(x1)dμ(xr)xn=n=i1++ir(ni1,,ir)Zpex1tdμ(x1)xi1Zpexrtdμ(xr)xir. By Theorems 2.3 and 2.6 and (2.38), we get (2.39)Bn(r)=n=i1++ir(ni1,,ir)Bi1Bir. Let sn(x) be the Sheffer sequence for (g(t),f(t)).

Then the Sheffer identity is given by (2.40)sn(x+y)=k=0n(nk)pk(y)sn-k(x),   see [7, 8], where pk(y)=g(t)sk(y). From Theorem 2.5 and (2.40), we have (2.41)Bn(r)(x+y)=k=0n(nk)Bn-k(r)(x)xk.

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.