Research Article | Open Access

# Applications of Umbral Calculus Associated with -Adic Invariant Integrals on

**Academic Editor:**Gaston N'Guerekata

#### Abstract

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 -adic invariant integrals on . From our properties, we can also derive some interesting identities of Bernoulli polynomials.

#### 1. Introduction

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

Let . Let be space of uniformly differentiable functions on . For , the -adic invariant integral on is defined by see [1, 2].

From (1.1), we have where (see [1–6]). Let be the set of all formal power series in the variable over with Let and let denote the vector space of all linear functional on .

The formal power series, defines a linear functional on by setting see [7, 8].

In particular, by (1.4) and (1.5), we get where is the Kronecker symbol (see [7]). Here, denotes both the algebra of formal power series in and the vector space of all linear functional on , so an element of will be thought of as both a formal power series and a linear functional. We shall call the umbral algebra. The umbral calculus is the study of umbral algebra.

The order of power series is the smallest integer for which does not vanish. We define if . From the definition of order, we note that and .

The series has a multiplicative inverse, denoted by or , if and only if .

Such a series is called invertible series. A series for which is called a delta series (see [7, 8]). Let . Then, we have By (1.5) and (1.6), we get see [7].

Notice that for all in , and for all polynomials , see [7, 8].

Let . Then, we have where the sum is over all nonnegative integers such that (see [8]).

By (1.10), we get Thus, from (1.12), we have see [7].

By (1.13), we get Thus, by (1.14), we see that Let us assume that is a polynomial of degree . Suppose that with and . Then, there exists a unique sequence of polynomials satisfying for all .

The sequence is called the Sheffer sequence for , which is denoted by .

The Sheffer sequence for is called the Appell sequence for , or is Appell for , which is indicated by .

For , it is known that see [7, 8].

Let . Then, we have where is the compositional inverse of , and see [7, 8].

We recall that the Bernoulli polynomials are defined by the generating function to be with the usual convention about replacing by (see [1–16]).

In the special case, are called the th Bernoulli numbers. By (1.21), we easily get Thus, by (1.22), we see that is a monic polynomial of degree . It is easy to show that see [13–15].

From (1.2), we can derive the following equation: Let us take . Then, from (1.21), (1.22), (1.23), and (1.24), we have where (see [1, 2]). Recently, Dere and simsek have studied applications of umbral algebra to some special functions (see [7]). In this paper, we investigate some properties of umbral calculus associated with -adic invariant integrals on . From our properties, we can derive some interesting identities of Bernoulli polynomials.

#### 2. Applications of Umbral Calculus Associated with -Adic Invariant Integrals on

Let be an Appell sequence for . By (1.19), we get Let us take . Then, is clearly invertible series. From (1.21) and (2.1), we have Thus, by (2.2), we get From (1.21), (2.1), and (2.3), we note that is an Appell sequence for .

Let us take the derivative with respect to on both sides of (2.2). Then, we have Thus, by (2.4), we get where . Thus, by (2.6), we get From (1.25) and (2.7), we have By (2.5), we see that Thus, by (2.9), we have and we can derive the following equation.

From (2.3) and (2.10), By (2.8) and (2.11), we see that Therefore, by (2.5), we obtain the following theorem.

Theorem 2.1. * For , one has
**
where . *

Corollary 2.2. *For , one has
*

Let us consider the linear functional that satisfies for all polynomials . It can be determined from (1.9) that By (1.24) and (2.16), we get Therefore, by (2.17), we obtain the following theorem.

Theorem 2.3. *For , one has
**
That is
**
In particular, one has
*

From (1.24), one has By (1.25) and (2.21), we get where .

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

Theorem 2.4. *For , we have
**
In particular, one obtains
*

The higher order Bernoulli polynomials are defined by In the special case, , are called the th Bernoulli numbers of order (). From (2.25), we note that By (2.25) and (2.26), we get From (2.26) and (2.27), we note that is a monic polynomial of degree with coefficients in . For , let us assume that By (2.28), we easily see that is an invertible series. From (2.25) and (2.28), we have From (2.29), we note that is an Appell sequence for . Therefore, by (2.29), we obtain the following theorem.

Theorem 2.5. *For and , one has
**
In particular, the Bernoulli polynomials of order are given by
**
That is
*

Let us consider the linear functional that satisfies for all polynomials . It can be determined from (1.9) that Therefore, by (2.34), we obtain the following theorem.

Theorem 2.6. * For , one has
**
That is
**
In particular, one gets
*

*Remark 2.7. * From (1.11), we note that
By Theorems 2.3 and 2.6 and (2.38), we get
Let be the Sheffer sequence for .

Then the Sheffer identity is given by
see [7, 8], where . From Theorem 2.5 and (2.40), we have

#### 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.

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#### Copyright

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.