Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 584643, 11 pages

doi:10.1155/2012/584643

Research Article

Some Identities on Bernoulli and Hermite Polynomials Associated with Jacobi Polynomials

Taekyun Kim,¹ Dae San Kim,² and Dmitry V. Dolgy³

¹Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

²Department of Mathematics, Sogang University, Seoul, Republic of Korea

³Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 17 July 2012; Accepted 9 August 2012

Academic Editor: Josef Diblík

Copyright © 2012 Taekyun 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.

Abstract.

We investigate some identities on the Bernoulli and the Hermite polynomials arising from the orthogonality of Jacobi polynomials in the inner product space P n.

1. Introduction

For with and , the Jacobi polynomials are defined as

(see [1–4]), where .

From (1.1), we note that

By (1.2), we see that is polynomial of degree with real coefficients. It is not difficult to show that the leading coefficient of is . From (1.2), we have .

By (1.1), we get

where is a positive integer (see [1–4]).

The Rodrigues' formula for is given by

It is easy to show that is a solution of the following differential equation:

As is well known, the generating function of is given by

where , (see [1–4]).

From (1.3), (1.4), and (1.6), we can derive the following identity:

where is the Kronecker symbol.

Let . Then is an inner product space with respect to the inner product , where . From (1.7), we note that is an orthogonal basis for .

The so-called Euler polynomials may be defined by means of

(see [5–22]), with the usual convention about replacing by . In the special case, , are called the Euler numbers.

The Bernoulli polynomials are also defined by the generating function to be

(see [11–21]), with the usual convention about replacing by .

From (1.8) and (1.9), we note that

For , we have

(see [23–29]) By the definition of Bernoulli and Euler polynomials, we get

In this paper we give some interesting identities on the Bernoulli and the Hermite polynomials arising from the orthogonality of Jacobi polynomials in the inner product space .

2. Bernoulli, Euler and Jacobi Polynomials

From (1.4), we have

By (2.1), we have

where we assume and circle around is taken so small that lie neither on it nor in its interior. It is not so difficult to show that .

For , let

From (1.7), we note that

Thus, by (2.4), we get

Therefore, by (1.7), (2.3), and (2.5), we obtain the following proposition.

Proposition 2.1. For , one has

where

Let us take . First, we consider the following integral:

From (2.5) and (16), we have

By Proposition 2.1, we get

From (1.9), we have

By (2.11), we get

Therefore, by (2.10) and (2.12), we obtain the following theorem.

Theorem 2.2. For , one has

Let us take . Then we evaluate the following integral:

Finding (2.5) and (21), we have

Theorem 2.3. For , one has

Let . From Proposition 2.1, we firstly evaluate the following integral:

By (2.1) and (2.17), we get

It is easy to show that

From (2.5), (2.18), and (2.19), we can derive the following equation:

Therefore, by Proposition 2.1, we obtain the following theorem.

Theorem 2.4. For , one has

Let be the Hermite polynomial with

where

Integrating by parts, one has

By (2.23) and (29), we get

Therefore, by (2.22) and (2.25), we obtain the following theorem.

Theorem 2.5. For , one has

where is the th Hermite number.

Remark 2.6. By the same method as Theorem 2.3, we get

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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