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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 865721, 12 pages
Applications of Umbral Calculus Associated with -Adic Invariant Integrals on
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.
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.
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.
In particular, by (1.4) and (1.5), we get where is the Kronecker symbol (see ). 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 .
Let . Then, we have where the sum is over all nonnegative integers such that (see ).
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 .
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 ). 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.
Theorem 2.1. For , one has where .
Corollary 2.2. For , one has
Theorem 2.3. For , one has That is In particular, one has
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
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
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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