Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 619197, 10 pages

doi:10.1155/2012/619197

Research Article

Some Identities on Laguerre Polynomials in Connection with Bernoulli and Euler Numbers

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

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

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

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

Received 15 May 2012; Accepted 28 June 2012

Academic Editor: Lee Chae Jang

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 study some interesting identities and properties of Laguerre polynomials in connection with Bernoulli and Euler numbers. These identities are derived from the orthogonality of Laguerre polynomials with respect to inner product .

1. Introduction/Preliminaries

As is well known, Laguerre polynomials are defined by the generating function as

(see [1, 2]). By (1.1), we get

Thus, from (1.2), we have

By (1.3), we see that is a polynomial of degree with rational coefficients and the leading coefficient . It is well known that Rodrigues' formula is given by

(see [1–27]). From (1.1), we can derive the following of Laguerre polynomials:

By (1.7), we easily see that is a solution of the following differential equation of order 2:

The Bernoulli numbers, , are defined by the generating function as

(see [1–28, 28]), with the usual convention about replacing by .

It is well known that Bernoulli polynomials of degree are given by

(see [2, 26]). Thus, from (1.10), we have

(see [3–12]). From (1.9) and (1.10), we can derive the following recurrence relation:

where is Kronecker's symbol.

The Euler polynomials are also defined by the generating function as

(see [27, 28]), with the usual convention about replacing by .

In this special case, , are called the th Euler numbers. From (1.13), we note that the recurrence formula of is given by

(see [24]). Finally, we introduce Hermite polynomials, which are defined by

(see [29]). In the special case, , is called the -th Hermite number. By (1.15), we get

(see [29]). It is not difficult to show that

In the present paper, we investigate some interesting identities and properties of Laguerre polynomials in connection with Bernoulli, Euler, and Hermite polynomials. These identities and properties are derived from (1.17).

2. Some Formulae on Laguerre Polynomials in Connection with Bernoulli, Euler, and Hermite Polynomials

Let

Then is an inner product space with the inner product

By (1.17), (2.1), and (2.2), we see that are orthogonal basis for .

For , it is given by

where

Let us take . From (2.3) and (2.4), we note that

Therefore, by (2.3), (2.4), and (2.5), we obtain the following theorem.

Theorem 2.1. For , one has

Let us consider . Then, by (2.3) and (2.4), we get

Therefore, by (2.3), (2.4), and (2.7), we obtain the following theorem.

Theorem 2.2. For , one has

Let us take . By the same method, we easily see that

For , we have

where

Therefore, by (2.10) and (2.11), we obtain the following theorem.

Theorem 2.3. For , one has

Let . Then we have

where

In [15], it is known that

By (2.14) and (2.15), we get

From (2.16), we can derive the following equations ((2.17)-(2.18)):

For , we have

Therefore, by (2.13), (2.17), and (2.18), we obtain the following theorem.

Theorem 2.4. For , one has

Let us take . By (2.3) and (2.4), we get

where

It is known (see [15]) that

From (2.20), (2.21), and (2.22), we can derive the following equations ((2.23)-(2.24)):

For , we have

Therefore, by (2.20) and (2.24), we obtain the following theorem.

Theorem 2.5. For , one has

It is known that

(see [15]). From (2.20), (2.21), and (2.23), we have

Therefore, by (2.20) and (2.27), we obtain the following theorem.

Theorem 2.6. For , one has

Remark 2.7. Laguerre's differential equation

is known to possess polynomial solutions when is a nonnegative integer. These solutions are naturally called Laguerre polynomials and are denoted by . That is, are solutions of (2.29) which are given by

From (2.30), we note that Laplace transform of is given by

It is not difficult to show that

Thus, we conclude that

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