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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 619197, 10 pages
http://dx.doi.org/10.1155/2012/619197
Research Article

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

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 [127]). 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 [128, 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 [312]). 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

References

  1. 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. View at Google Scholar · View at Zentralblatt MATH
  2. A. K. Shukla and S. K. Meher, “Group-theoretic origin of some generating functions for Laguerre polynomials of two variables,” Applied Mathematical Sciences, vol. 5, no. 13–16, pp. 775–784, 2011. View at Google Scholar · View at Zentralblatt MATH
  3. S. Araci, “Novel identities for q-genocchi numbers and polynomials,” Journal of Functions Spaces and Applications. In press.
  4. S. Araci, D. Erdal, and J. J. Seo, “A study on the fermionic p-adic q-integral representation on Zp associated with weighted q-Bernstein and q-Genocchi polynomials,” Abstract and Applied Analysis, vol. 2011, Article ID 649248, 10 pages, 2011. View at Publisher · View at Google Scholar
  5. A. Bayad, “Arithmetical properties of elliptic Bernoulli and Euler numbers,” International Journal of Algebra, vol. 4, no. 5–8, pp. 353–372, 2010. View at Google Scholar
  6. A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010. View at Google Scholar
  7. A. Bayad, “Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials,” Mathematics of Computation, vol. 80, no. 276, pp. 2219–2221, 2011. View at Publisher · View at Google Scholar
  8. 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. View at Publisher · View at Google Scholar
  9. 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. View at Google Scholar
  10. M. Cenkci, “The p-adic generalized twisted (h,q)-Euler-l-function and its applications,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 1, pp. 37–47, 2007. View at Google Scholar
  11. M. W. Coffey, “On finite sums of Laguerre polynomials,” Rocky Mountain Journal of Mathematics, vol. 41, no. 1, pp. 79–93, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. M. Gülsu, B. Gürbüz, Y. Öztürk, and M. Sezer, “Laguerre polynomial approach for solving linear delay difference equations,” Applied Mathematics and Computation, vol. 217, no. 15, pp. 6765–6776, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. N. S. Jung, H. Y. Lee, and C. S. Ryoo, “Some relations between twisted (h,q)-Euler numbers with weight α and q-Bernstein polynomials with weight α,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 176296, 11 pages, 2011. View at Publisher · View at Google Scholar
  14. 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, 15 pages, 2012. View at Publisher · View at Google Scholar
  15. D. S. Kim, D. V. Dolgy, H. M. Kim, S. H. Lee, and T. Kim, “Integral formulae of bernoulli polynomials,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 269847, 15 pages, 2012. View at Publisher · View at Google Scholar
  16. M.-S. Kim, “A note on sums of products of Bernoulli numbers,” Applied Mathematics Letters, vol. 24, no. 1, pp. 55–61, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. T. Kim, “On the weighted q-Bernoulli numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 207–215, 2011. View at Google Scholar
  18. T. Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009. View at Publisher · View at Google Scholar
  19. T. Kim, “Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on Zp,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009. View at Publisher · View at Google Scholar
  20. H. Y. Lee, N. S. Jung, and C. S. Ryoo, “A note on the q-Euler numbers and polynomials with weak weight α,” Journal of Applied Mathematics, Article ID 497409, 14 pages, 2011. View at Publisher · View at Google Scholar
  21. H. Ozden, Y. Simsek, and I. N. Cangul, “Euler polynomials associated with p-adic q-Euler measure,” General Mathematics, vol. 15, no. 2, pp. 24–37, 2007. View at Google Scholar
  22. 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. 2088, Article ID 390857, 16 pages, 2008. View at Publisher · View at Google Scholar
  23. 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. View at Google Scholar
  24. C. S. Ryoo, “Some relations between twisted q-Euler numbers and Bernstein polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 217–223, 2011. View at Google Scholar
  25. P. Rusev, “Laguerre's polynomials and the nonreal zeros of Riemann's ζ-function,” Comptes Rendus de l'Académie Bulgare des Sciences, vol. 63, no. 11, pp. 1547–1550, 2010. View at Google Scholar
  26. D. Zill and M. R. Cullen, Advanced Engineering Mathematics, Jonesand Bartlert, 2005.
  27. Y. Simsek, “Construction a new generating function of Bernstein type polynomials,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 1072–1076, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. 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. View at Publisher · View at Google Scholar
  29. R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer, Berlin, Germany, 2010.