`Discrete Dynamics in Nature and SocietyVolume 2012 (2012), Article ID 584643, 11 pageshttp://dx.doi.org/10.1155/2012/584643`
Research Article

## Some Identities on Bernoulli and Hermite Polynomials Associated with Jacobi Polynomials

1Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2Department of Mathematics, Sogang University, Seoul, Republic of Korea
3Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 17 July 2012; Accepted 9 August 2012

Copyright © 2012 Taekyun 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 investigate some identities on the Bernoulli and the Hermite polynomials arising from the orthogonality of Jacobi polynomials in the inner product space Pn.

#### 1. Introduction

For with and , the Jacobi polynomials are defined as (see [14]), where .

From (1.1), we note that By (1.2), we see that is polynomial of degree with real coefficients. It is not difficult to show that the leading coefficient of is . From (1.2), we have .

By (1.1), we get where is a positive integer (see [14]).

The Rodrigues' formula for is given by It is easy to show that is a solution of the following differential equation: As is well known, the generating function of is given by where , (see [14]).

From (1.3), (1.4), and (1.6), we can derive the following identity: where is the Kronecker symbol.

Let . Then is an inner product space with respect to the inner product , where . From (1.7), we note that is an orthogonal basis for .

The so-called Euler polynomials may be defined by means of (see [522]), with the usual convention about replacing by . In the special case, , are called the Euler numbers.

The Bernoulli polynomials are also defined by the generating function to be (see [1121]), with the usual convention about replacing by .

From (1.8) and (1.9), we note that For , we have (see [2329]) By the definition of Bernoulli and Euler polynomials, we get

In this paper we give some interesting identities on the Bernoulli and the Hermite polynomials arising from the orthogonality of Jacobi polynomials in the inner product space .

#### 2. Bernoulli, Euler and Jacobi Polynomials

From (1.4), we have By (2.1), we have where we assume and circle around is taken so small that lie neither on it nor in its interior. It is not so difficult to show that .

For , let From (1.7), we note that Thus, by (2.4), we get Therefore, by (1.7), (2.3), and (2.5), we obtain the following proposition.

Proposition 2.1. For , one has where

Let us take . First, we consider the following integral: From (2.5) and (16), we have By Proposition 2.1, we get From (1.9), we have By (2.11), we get Therefore, by (2.10) and (2.12), we obtain the following theorem.

Theorem 2.2. For , one has

Let us take . Then we evaluate the following integral: Finding (2.5) and (21), we have

Theorem 2.3. For , one has

Let . From Proposition 2.1, we firstly evaluate the following integral:

By (2.1) and (2.17), we get It is easy to show that From (2.5), (2.18), and (2.19), we can derive the following equation: Therefore, by Proposition 2.1, we obtain the following theorem.

Theorem 2.4. For , one has

Let be the Hermite polynomial with where Integrating by parts, one has By (2.23) and (29), we get Therefore, by (2.22) and (2.25), we obtain the following theorem.

Theorem 2.5. For , one has where is the th Hermite number.

Remark 2.6. By the same method as Theorem 2.3, we get

#### Acknowledgments

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

1. L. Carlitz, “The generating function for the Jacobi polynomial,” Rendiconti del Seminario Matematico della Università di Padova, vol. 38, pp. 86–88, 1967.
2. L. Carlitz, “An integral formula for the Jacobi polynomial,” Bollettino dell'Unione Matematica Italiana, vol. 17, no. 3, pp. 273–275, 1962.
3. L. Carlitz, “Some generating functions for the Jacobi polynomials,” Bollettino dell'Unione Matematica Italiana, vol. 16, no. 2, pp. 150–155, 1961.
4. L. Carlitz, “On Laguerre and Jacobi polynomials,” Bollettino dell'Unione Matematica Italiana, vol. 12, no. 1, pp. 34–40, 1957.
5. T. Kim, “On the weighted $q$-Bernoulli numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 207–215, 2011.
6. T. Kim, “An identity of the symmetry for the Frobenius-Euler polynomials associated with the Fermionic $p$-adic invariant $q$-integrals on ${ℤ}_{p}$,” The Rocky Mountain Journal of Mathematics, vol. 41, no. 1, pp. 239–247, 2011.
7. T. Kim, “Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on ${ℤ}_{p}$,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009.
8. T. Kim, “Symmetry of power sum polynomials and multivariate fermionic $p$-adic invariant integral on ${ℤ}_{p}$,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009.
9. T. Kim, “On Euler-Barnes multiple zeta functions,” Russian Journal of Mathematical Physics, vol. 10, no. 3, pp. 261–267, 2003.
10. T. Kim, “$q$-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.
11. H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on $q$-Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.
12. H. Ozden, I. N. Cangul, and Y. Simsek, “Multivariate interpolation functions of higher-order $q$-Euler numbers and their applications,” Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages, 2008.
13. C. S. Ryoo, “A note on the Frobenius-Euler polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 14, no. 4, pp. 495–501, 2011.
14. C. S. Ryoo, “Some identities of the twisted $q$-Euler numbers and polynomials associated with $q$-Bernstein polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 14, no. 2, pp. 239–248, 2011.
15. C. S. Ryoo, “On the generalized Barnes type multiple $q$-Euler polynomials twisted by ramified roots of unity,” Proceedings of the Jangjeon Mathematical Society, vol. 13, no. 2, pp. 255–263, 2010.
16. S.-H. Rim and S.-J. Lee, “Some identities on the twisted $\left(h,q\right)$-Genocchi numbers and polynomials associated with $q$-Bernstein polynomials,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 482840, 8 pages, 2011.
17. S.-H. Rim, E.-J. Moon, J.-H. Jin, and S.-J. Lee, “On the symmetric properties for the generalized Genocchi polynomials,” Journal of Computational Analysis and Applications, vol. 13, no. 7, pp. 1240–1245, 2011.
18. S.-H. Rim, J.-H. Jin, E.-J. Moon, and S.-J. Lee, “Some identities on the $q$-Genocchi polynomials of higher-order and $q$-Stirling numbers by the fermionic $p$-adic integral on ${ℤ}_{p}$,” International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 860280, 14 pages, 2010.
19. 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.
20. Y. Simsek, “Special functions related to Dedekind-type DC-sums and their applications,” Russian Journal of Mathematical Physics, vol. 17, no. 4, pp. 495–508, 2010.
21. Y. Simsek and M. Acikgoz, “A new generating function of ($q$-) Bernstein-type polynomials and their interpolation function,” Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010.
22. Z. Zhang and H. Yang, “Some closed formulas for generalized Bernoulli-Euler numbers and polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 11, no. 2, pp. 191–198, 2008.
23. W. A. Al-Salam and L. Carlitz, “Finite summation formulas and congruences for Legendre and Jacobi polynomials,” vol. 62, pp. 108–118, 1958.
24. L. N. Karmazina, Tables of Jacobi Polynomials, Izdat. Akademii Nauk SSSR, Moscow, Russia, 1954.
25. A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010.
26. A. Bayad and T. Kim, “Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 2, pp. 247–253, 2010.
27. A. Bayad and T. Kim, “Identities involving values of Bernstein, $q$-Bernoulli, and $q$-Euler polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 2, pp. 133–143, 2011.
28. I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-order $w$-$q$-Genocchi numbers,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 39–57, 2009.
29. M. Can, M. Cenkci, V. Kurt, and Y. Simsek, “Twisted Dedekind type sums associated with Barnes' type multiple Frobenius-Euler $l$-functions,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 135–160, 2009.