Abstract and Applied Analysis

Volume 2012 (2012), Article ID 957350, 15 pages

doi:10.1155/2012/957350

Research Article

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

Taekyun Kim¹ and Dae San Kim²

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

²Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

Received 14 June 2012; Accepted 9 August 2012

Academic Editor: Pekka Koskela

Copyright © 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 [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.

References

L. Carlitz, “Bilinear generating functions for Laguerre and Lauricella polynomials in several variables,” Rendiconti del Seminario Matematico della Università di Padova, vol. 43, pp. 269–276, 1970.

L. Carlitz, “Some generating functions for Laguerre polynomials,” Duke Mathematical Journal, vol. 35, pp. 825–827, 1968.

L. Carlitz, “The product of several Hermite or Laguerre polynomials,” Monatshefte für Mathematik, vol. 66, pp. 393–396, 1962.

L. Carlitz, “A characterization of the Laguerre polynomials,” Monatshefte für Mathematik, vol. 66, pp. 389–392, 1962.

L. Carlitz, “On the product of two Laguerre polynomials,” Journal of the London Mathematical Society, vol. 36, pp. 399–402, 1961.

L. Carlitz, “A note on the Laguerre polynomials,” The Michigan Mathematical Journal, vol. 7, pp. 219–223, 1960.

D. Zill and M. R. Cullen, Advanced Engineering Mathematics, Jonesand Bartlert, 2005.

S. Khan, M. W. Al-Saad, and R. Khan, “Laguerre-based Appell polynomials: properties and applications,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 247–259, 2010.

S. Khan, G. Yasmin, R. Khan, and N. A. Makboul Hassan, “Hermite-based Appell polynomials: properties and applications,” Journal of Mathematical Analysis and Applications, vol. 351, no. 2, pp. 756–764, 2009.

H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on -Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.

H. Ozden, I. N. Cangul, and Y. Simsek, “Multivariate interpolation functions of higher-order -Euler numbers and their applications,” Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages, 2008.

M. A. Özarslan and C. Kaanoğlu, “Multilateral generating functions for classes of polynomials involving multivariable Laguerre polynomials,” Journal of Computational Analysis and Applications, vol. 13, no. 4, pp. 683–691, 2011.

C. S. Ryoo, “Some identities of the twisted -Euler numbers and polynomials associated with -Bernstein polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 14, no. 2, pp. 239–248, 2011.

C. S. Ryoo, “Some relations between twisted -Euler numbers and Bernstein polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 217–223, 2011.

P. Rusev, “Laguerre's polynomials and the nonreal zeros of Riemann's -function,” Comptes Rendus de l'Académie Bulgare des Sciences, vol. 63, no. 11, pp. 1547–1550, 2010.

Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–278, 2008.

Y. Simsek and M. Acikgoz, “A new generating function of (-) Bernstein-type polynomials and their interpolation function,” Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010.

S. Araci, J. J. Seo, and D. Erdal, “New construction weighted -Genocchi numbers and polynomials related to zeta type functions,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 487490, 7 pages, 2011.

S. Araci, D. Erdal, and J. J. Seo, “A study on the fermionic -adic -integral representation on associated with weighted -Bernstein and -Genocchi polynomials,” Abstract and Applied Analysis, vol. 2011, Article ID 649248, 10 pages, 2011.

A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010.

A. Bayad and T. Kim, “Identities involving values of Bernstein, -Bernoulli, and -Euler polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 2, pp. 133–143, 2011.

I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-order --Genocchi numbers,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 39–57, 2009.

D. S. Kim, T. Kim, S. H. Rim, and S. H. Lee, “Hermite polynomials and their applications associated with Bernoulli and Euler numbers,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 974632, 13 pages, 2012.

T. Kim, J. Choi, Y. H. Kim, and C. S. Ryoo, “On -Bernstein and -Hermite polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 14, no. 2, pp. 215–221, 2011.

D. S. Kim, D. V. Dolgy, T. Kim, and S. H. Rim, “Some formulae for the product of two Bernoulli and Euler polynomials,” Abstract and Applied Analysis, vol. 2012, Article ID 784307, 14 pages, 2012.

L. Carlitz, “Congruence properties of Hermite and Laguerre polynomials,” Archiv für Mathematische Logik und Grundlagenforschung, vol. 10, pp. 460–465, 1959.

M. Cenkci, “The -adic generalized twisted -Euler--function and its applications,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 1, pp. 37–47, 2007.

M. W. Coffey, “On finite sums of Laguerre polynomials,” The Rocky Mountain Journal of Mathematics, vol. 41, no. 1, pp. 79–93, 2011.

N. S. Jung, H. Y. Lee, and C. S. Ryoo, “Some relations between twisted -Euler numbers with weight and -Bernstein polynomials with weight ,” Discrete Dynamics in Nature and Society. An International Multidisciplinary Research and Review Journal, vol. 2011, Article ID 176296, 11 pages, 2011.

T. Kim, “On the weighted -Bernoulli numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 207–215, 2011.

T. Kim, “Some identities on the -Euler polynomials of higher order and -Stirling numbers by the fermionic -adic integral on ,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009.

T. Kim, “Symmetry of power sum polynomials and multivariate fermionic -adic invariant integral on ,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009.