Extended Laguerre Polynomials Associated with Hermite, Bernoulli, and Euler Numbers and Polynomials
Taekyun Kim1and Dae San Kim2
Academic Editor: Pekka Koskela
Received14 Jun 2012
Accepted09 Aug 2012
Published09 Sept 2012
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 [1–6].
From (1.1), we can derive the following:
see [1–9].
As is well known, Rodrigues' formula for is given by
see [1–6, 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 [10–17], 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 [18–22], 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 [1–6, 23–32]).
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 [1–6]. 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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