Abstract and Applied Analysis

Volume 2012 (2012), Article ID 865721, 12 pages

doi:10.1155/2012/865721

Research Article

Applications of Umbral Calculus Associated with -Adic Invariant Integrals on

Dae San Kim¹ and Taekyun Kim²

¹Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

²Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 8 November 2012; Accepted 22 November 2012

Academic Editor: Gaston N'Guerekata

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.

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.

References

T. Kim, “Symmetry -adic invariant integral on for Bernoulli and Euler polynomials,” Journal of Difference Equations and Applications, vol. 14, no. 12, pp. 1267–1277, 2008.

T. Kim, “-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.

T. Kim, “An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic -adic invariant -integrals on ,” The Rocky Mountain Journal of Mathematics, vol. 41, no. 1, pp. 239–247, 2011.

S.-H. Rim and J. Jeong, “On the modified -Euler numbers of higher order with weight,” Advanced Studies in Contemporary Mathematics, vol. 22, no. 1, pp. 93–98, 2012.

S.-H. Rim and S.-J. Lee, “Some identities on the twisted -Genocchi numbers and polynomials associated with -Bernstein polynomials,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 482840, 8 pages, 2011.

H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on -Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.

R. Dere and Y. Simsek, “Applications of umbral algebra to some special polynomials,” Advanced Studies in Contemporary Mathematics, vol. 22, no. 3, pp. 433–438, 2012.

S. Roman, The Umbral Calculus, Academic Press, New York, NY, USA, 2005.

J. Choi, D. S. Kim, T. Kim, and Y. H. Kim, “Some arithmetic identities on Bernoulli and Euler numbers arising from the -adic integrals on ,” Advanced Studies in Contemporary Mathematics, vol. 22, no. 2, pp. 239–247, 2012.

D. Ding and J. Yang, “Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 1, pp. 7–21, 2010.

G. Kim, B. Kim, and J. Choi, “The DC algorithm for computing sums of powers of consecutive integers and Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 137–145, 2008.

C. S. Ryoo, “On the generalized Barnes type multiple -Euler polynomials twisted by ramified roots of unity,” Proceedings of the Jangjeon Mathematical Society, vol. 13, no. 2, pp. 255–263, 2010.

Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–278, 2008.

Y. Simsek, “Special functions related to Dedekind-type DC-sums and their applications,” Russian Journal of Mathematical Physics, vol. 17, no. 4, pp. 495–508, 2010.

K. Shiratani and S. Yokoyama, “An application of -adic convolutions,” Memoirs of the Faculty of Science, Kyushu University Series A, vol. 36, no. 1, pp. 73–83, 1982.

Z. Zhang and H. Yang, “Some closed formulas for generalized Bernoulli-Euler numbers and polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 11, no. 2, pp. 191–198, 2008.