Abstract and Applied Analysis

VolumeΒ 2012, Article IDΒ 957350, 15 pages

http://dx.doi.org/10.1155/2012/957350

## Extended Laguerre Polynomials Associated with Hermite, Bernoulli, and Euler Numbers and Polynomials

^{1}Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea^{2}Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

Received 14 June 2012; Accepted 9 August 2012

Academic Editor: PekkaΒ Koskela

Copyright Β© 2012 Taekyun Kim and Dae San 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

Let be an inner product space with the inner product , where and with . In this paper we study the properties of the extended Laguerre polynomials which are an orthogonal basis for . From those properties, we derive some interesting relations and identities of the extended Laguerre polynomials associated with Hermite, Bernoulli, and Euler numbers and polynomials.

#### 1. Introduction/Preliminaries

For with , the extended Laguerre polynomials are defined by the generating function as follows: see [1β6].

From (1.1), we can derive the following: see [1β9].

As is well known, Rodrigues' formula for is given by see [1β6, 8, 9].

From (1.3), we note that where is the Kronecker symbol.

From (1.1), (1.2), and (1.3), we can derive the following identities: and is a solution of .

The derivatives of general Laguerre polynomials are given by The th Bernoulli polynomials, , are defined by the generating function to be see [10β17], with the usual convention about replacing by . In the special case, , are called the th Bernoulli numbers.

It is well known that the th Euler polynomials are also defined by the generating function to be see [18β22], with the usual convention about replacing by .

The Hermite polynomials are given by see [23, 24], with the usual convention about replacing by . In the special case, , are called the th Hermite numbers.

From (1.11), we note that see [23, 24], and is a solution of Hermite differential equation which is given by (see [1β6, 23β32]).

Throughout this paper we assume that with . Let . Then is an inner product space with the inner product , where . By (1.4) the set of the extended Laguerre polynomials is an orthogonal basis for . In this paper we study the properties of the extended Laguerre polynomials which are an orthogonal basis for . From those properties, we derive some new and interesting relations and identities of the extended Laguerre polynomials associated with Hermite, Bernoulli and Euler numbers and polynomials.

#### 2. On the Extended Laguerre Polynomials Associated with Hermite, Bernoulli, and Euler Polynomials

For , is given by From (1.3), (1.4), and (2.1), we note that Thus, by (2.2), we get Therefore, by (2.1) and (2.3), we obtain the following proposition.

Proposition 2.1. *For , let
**
Then one has the following:
*

To derive inverse formula of (1.2), let take one . Then, by Proposition 2.1, one gets Therefore, by (2.6), we obtain the following corollary.

Corollary 2.2 (Inverse formula of ). *For , one has
*

Let one takes Bernoulli polynomials of degree with . Then can be written as From Proposition 2.1, one has By the fundamental property of gamma function, one gets Therefore, by (2.8), (2.9), and (2.10), we obtain the following theorem.

Theorem 2.3. *For , with , one has
*

As is known, relationships between Hermite and Laguerre polynomials are given by see [1β6]. In the special case , by (2.12) and (2.13), we obtain the following corollary.

Corollary 2.4. *For , one has
*

By the same method as Theorem 2.3, one gets where are the th Euler polynomials. In the special case, , are called the th Euler numbers.

Let one considers the th Hermite polynomials with . Then can be written as From Proposition 2.1, one notes that It is not difficult to show that Therefore, by (2.16), (2.17), and (2.18), we obtain the following theorem.

Theorem 2.5. *For , with , one has
*

In the special case, , we obtain the following corollary.

Corollary 2.6. *For , one has
*

For with , let one takes Then is also written as From Proposition 2.1, one can determine the coefficients of (2.22) as follows: By the fundamental property of gamma function, one gets Therefore, by (2.22), (2.23), and (2.24), we obtain the following theorem.

Theorem 2.7. *For with , and , one has
*

In the special case, , one has Thus, by (2.26), we obtain the following corollary.

Corollary 2.8. *For , with , one has
*

Let one assumes that Then can be rewritten as a linear combination of as follows: By Proposition 2.1, one can determine the coefficients of (2.29) as follows: It is known that see [25].

From (2.30) and (2.31), one notes that For , one has Therefore, by (2.29) and (2.32), we obtain the following theorem.

Theorem 2.9. *For , with , one has
*

Let one takes the polynomial in as follows: From the orthogonality of , one notes that where It is known in [25] that From (2.35), (2.37), and (2.38), one notes that For , by (2.37) and (2.38), one gets Therefore, by (2.36), (2.39), and (2.40), we obtain the following theorem.

Theorem 2.10. *For , , and with , one has
*

For with , let one assumes that .

By Proposition 2.1, one sees that can be written as From the orthogonality of , one has By (1.2), (1.3), and (1.8), one gets Thus, from (2.44), one has By (2.44) and (2.45), one gets Therefore, by (2.42) and (2.46), we obtain the following theorem.

Theorem 2.11. *For with , with , one has
*

#### Acknowledgments

The authors would like to express their sincere gratitude to referee for his/her valuable comments and information. 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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