Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 974632, 13 pages

doi:10.1155/2012/974632

Research Article

Hermite Polynomials and their Applications Associated with Bernoulli and Euler Numbers

Dae San Kim,¹ Taekyun Kim,² Seog-Hoon Rim,³ and Sang Hun Lee⁴

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

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

³Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea

⁴Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 7 May 2012; Accepted 15 May 2012

Academic Editor: Garyfalos Papaschinopoulos

Copyright © 2012 Dae San 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 derive some interesting identities and arithmetic properties of Bernoulli and Euler polynomials from the orthogonality of Hermite polynomials. Let be the -dimensional vector space over . Then we show that is a good basis for the space for our purpose of arithmetical and combinatorial applications.

1. Introduction

As is well known, the Euler polynomials, , are defined by the generating function as follows:

(see [1–8]), with the usual convention about replacing by .

In the special case, , is called the nth Euler number. From (1.1) and definition of Euler numbers, we note that

with the usual convention about replacing by .

The Bernoulli numbers are defined as

(see [9–14]), where is a Kronecker symbol.

As is well known, Bernoulli polynomials are also defined by

with the usual convention about replacing by (see [1, 15–18]).

The Hermite polynomials are defined by the generating function as follows:

(see [5, 19]), with the usual convention about replacing by .

From (1.5), we can derive the following identities:

Let us consider two operators as follows:

By (1.7), we get . In particular, if we take , then we have

We note that

From (1.8), we note that

Thus, by (1.10), we get

(see [5, 19–23]). In the special case, , are called the Hermite numbers.

From (1.5), we can derive the following identities:

(cf. [5, 19]), with the usual convention about replacing by . It is easy to show that

By comparing coefficients on the both sides of (1.13), we get

where . From (1.12), we have

Let be the -dimensional vector space over . Probably, is the most natural basis for this space. But is also a good basis for the space , for our purpose of arithmetical and combinatorial applications.

For ,

for some uniquely determined .

The purpose of this paper is to develop methods for computing from the information of . By using these methods, we define some interesting identities.

2. Properties of Hermite Polynomials

From (1.5) and (1.13), we note that

Thus, by (2.1), we obtain the following recurrence formula.

Proposition 2.1. For , one has

By, (1.5), we get

From (2.3), we can derive the following reflection symmetric identity of :

By (1.5), we easily see that

Thus, by (1.5) and (2.5), we get

Thus, by (2.6) and (2.7), we get

From (1.15) and (2.9), we note that

Differentiating on both sides, we have

Thus, we have

From (2.12), we note that is a solution of the following second-order linear differential equation:

From (1.5), we note that

Thus, by (2.14), we get

3. Main Results

By (1.6), we easily get

From (3.1), we note that

It is easy to show that

where . By (3.3), we get

From (3.2), we note that are orthogonal basis for the space with respect to the inner product

For , the polynomial is given by

where

Let us take . For , we compute in (3.6) as follows

Let . Then we have

Therefore, by (3.6), (3.8), and (3.9), we obtain the following proposition.

Proposition 3.1. One has

Let us take . From (3.4), can be rewritten by

where

By integrating by parts, we get

Thus, from (3.11) and (3.13), we have

Therefore, by (3.11) and (3.14), we obtain the following theorem.

Theorem 3.2. For , one has

Remark 3.3. Let us take . Then, by the same method, we obtain the following identity:

Now, we consider . From (3.6), we note that can be rewritten as

where

By integrating by parts, we get

From (3.17) and (3.19), we note that

Therefore, by (3.17) and (3.20), we obtain the following theorem.

Theorem 3.4. For , one has

From Theorem 3.4, we note that

Thus, we have, for ,

Let with . Then we easily see that

Let us consider the following polynomial of degree in :

From (3.6), we note that can be rewritten as

where

In [15], it is known that

From (3.23) and (3.29), we have the following:

By (3.24) and (3.30), we get

For , we have

Therefore, by (3.27) and (3.32), we obtain the following theorem.

Theorem 3.5. For , one has

Acknowledgment

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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