Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 974632 | 13 pages | https://doi.org/10.1155/2012/974632

Hermite Polynomials and their Applications Associated with Bernoulli and Euler Numbers

Academic Editor: Garyfalos Papaschinopoulos
Received07 May 2012
Accepted15 May 2012
Published14 Aug 2012

Abstract

We derive some interesting identities and arithmetic properties of Bernoulli and Euler polynomials from the orthogonality of Hermite polynomials. Let be the -dimensional vector space over . Then we show that is a good basis for the space for our purpose of arithmetical and combinatorial applications.

1. Introduction

As is well known, the Euler polynomials, , are defined by the generating function as follows: (see [18]), with the usual convention about replacing by .

In the special case, , is called the nth Euler number. From (1.1) and definition of Euler numbers, we note that with the usual convention about replacing by .

The Bernoulli numbers are defined as (see [914]), where is a Kronecker symbol.

As is well known, Bernoulli polynomials are also defined by with the usual convention about replacing by (see [1, 1518]).

The Hermite polynomials are defined by the generating function as follows: (see [5, 19]), with the usual convention about replacing by .

From (1.5), we can derive the following identities:

Let us consider two operators as follows:

By (1.7), we get . In particular, if we take , then we have

We note that

From (1.8), we note that

Thus, by (1.10), we get (see [5, 1923]). In the special case, , are called the Hermite numbers.

From (1.5), we can derive the following identities: (cf. [5, 19]), with the usual convention about replacing by . It is easy to show that

By comparing coefficients on the both sides of (1.13), we get where . From (1.12), we have

Let be the -dimensional vector space over . Probably, is the most natural basis for this space. But is also a good basis for the space , for our purpose of arithmetical and combinatorial applications.

For , for some uniquely determined .

The purpose of this paper is to develop methods for computing from the information of . By using these methods, we define some interesting identities.

2. Properties of Hermite Polynomials

From (1.5) and (1.13), we note that

Thus, by (2.1), we obtain the following recurrence formula.

Proposition 2.1. For , one has

By, (1.5), we get

From (2.3), we can derive the following reflection symmetric identity of :

By (1.5), we easily see that

Thus, by (1.5) and (2.5), we get

Thus, by (2.6) and (2.7), we get

From (1.15) and (2.9), we note that

Differentiating on both sides, we have

Thus, we have

From (2.12), we note that is a solution of the following second-order linear differential equation:

From (1.5), we note that

Thus, by (2.14), we get

3. Main Results

By (1.6), we easily get

From (3.1), we note that

It is easy to show that where . By (3.3), we get

From (3.2), we note that are orthogonal basis for the space with respect to the inner product

For , the polynomial is given by where

Let us take . For , we compute in (3.6) as follows

Let . Then we have

Therefore, by (3.6), (3.8), and (3.9), we obtain the following proposition.

Proposition 3.1. One has

Let us take . From (3.4), can be rewritten by where

By integrating by parts, we get

Thus, from (3.11) and (3.13), we have

Therefore, by (3.11) and (3.14), we obtain the following theorem.

Theorem 3.2. For , one has

Remark 3.3. Let us take . Then, by the same method, we obtain the following identity:

Now, we consider . From (3.6), we note that can be rewritten as where

By integrating by parts, we get

From (3.17) and (3.19), we note that

Therefore, by (3.17) and (3.20), we obtain the following theorem.

Theorem 3.4. For , one has

From Theorem 3.4, we note that

Thus, we have, for ,

Let with . Then we easily see that

Let us consider the following polynomial of degree in :

From (3.6), we note that can be rewritten as where

In [15], it is known that

From (3.23) and (3.29), we have the following:

By (3.24) and (3.30), we get

For , we have

Therefore, by (3.27) and (3.32), we obtain the following theorem.

Theorem 3.5. For , one has

Acknowledgment

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. H. Ozden, I. N. Cangul, and Y. Simsek, “On the behavior of two variable twisted p-adic Euler q-l-functions,” Nonlinear Analysis, vol. 71, no. 12, pp. e942–e951, 2009. View at: Publisher Site | p-adic%20Euler%20q-l-functions&author=H. Ozden&author=I. N. Cangul&author=&author=Y. Simsek&publication_year=2009" target="_blank">Google Scholar
  2. S.-H. Rim, A. Bayad, E.-J. Moon, J.-H. Jin, and S.-J. Lee, “A new construction on the q-Bernoulli polynomials,” Advances in Difference Equations, vol. 2011, article 34, 2011. View at: q-Bernoulli%20polynomials&author=S.-H. Rim&author=A. Bayad&author=E.-J. Moon&author=J.-H. Jin&author=&author=S.-J. Lee&publication_year=2011" target="_blank">Google Scholar
  3. 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: q-Euler%20numbers%20and%20Bernstein%20polynomials&author=C. S. Ryoo&publication_year=2011" target="_blank">Google Scholar
  4. 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 Site | Google Scholar | Zentralblatt MATH
  5. 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 Site | q-)%20Bernstein-type%20polynomials%20and%20their%20interpolation%20function&author=Y. Simsek &author=M. Acikgoz&publication_year=2010" target="_blank">Google Scholar
  6. 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. View at: Google Scholar | Zentralblatt MATH
  7. C. Vignat, “Old and new results about relativistic Hermite polynomials,” Journal of Mathematical Physics, vol. 52, no. 9, Article ID 093503, 16 pages, 2011. View at: Publisher Site | Google Scholar
  8. T. Kim, “A note on q-Bernstein polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 1, pp. 73–82, 2011. View at: Publisher Site | q-Bernstein%20polynomials&author=T. Kim&publication_year=2011" target="_blank">Google Scholar
  9. S. Araci, D. Erdal, and J. J. Seo, “A study on the fermionic p-adic q-integral representation on p associated with weighted q-Bernstein and q-Genocchi polynomials,” Abstract and Applied Analysis, Article ID 649248, 10 pages, 2011. View at: p-adic%20q-integral%20representation%20on%20p%20associated%20with%20weighted%20q-Bernstein%20and%20q-Genocchi%20polynomials&author=S. Araci&author=D. Erdal&author=&author=J. J. Seo&publication_year=2011" target="_blank">Google Scholar
  10. 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
  11. 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 Site | q-Bernoulli,%20and%20q-Euler%20polynomials&author=A. Bayad &author=T. Kim&publication_year=2011" target="_blank">Google Scholar
  12. L. Carlitz, “Note on the integral of the product of several Bernoulli polynomials,” Journal of the London Mathematical Society, vol. 34, pp. 361–363, 1959. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. L. Carlitz, “Multiplication formulas for products of Bernoulli and Euler polynomials,” Pacific Journal of Mathematics, vol. 9, pp. 661–666, 1959. View at: Google Scholar
  14. L. Carlitz, “Arithmetic properties of generalized Bernoulli numbers,” Journal für die Reine und Angewandte Mathematik, vol. 202, pp. 174–182, 1959. View at: Google Scholar | Zentralblatt MATH
  15. 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. In press. View at: Google Scholar
  16. T. Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on p,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009. View at: Publisher Site | q-Euler%20polynomials%20of%20higher%20order%20and%20q-Stirling%20numbers%20by%20the%20fermionic%20p-adic%20integral%20on%20p&author=T. Kim&publication_year=2009" target="_blank">Google Scholar
  17. 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, vol. 2011, Article ID 497409, 14 pages, 2011. View at: Publisher Site | q-Euler%20numbers%20and%20polynomials%20with%20weak%20weight%20α&author=H. Y. Lee&author=N. S. Jung&author=&author=C. S. Ryoo&publication_year=2011" target="_blank">Google Scholar
  18. H. Ozden, “p-adic distribution of the unification of the Bernoulli, Euler and Genocchi polynomials,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 970–973, 2011. View at: Publisher Site | p-adic%20distribution%20of%20the%20unification%20of%20the%20Bernoulli,%20Euler%20and%20Genocchi%20polynomials&author=H. Ozden&publication_year=2011" target="_blank">Google Scholar
  19. T. Kim, J. Choi, Y. H. Kim, and C. S. Ryoo, “On q-Bernstein and q-Hermite polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 14, no. 2, pp. 215–221, 2011. View at: q-Bernstein%20and%20q-Hermite%20polynomials&author=T. Kim&author=J. Choi&author=Y. H. Kim&author=&author=C. S. Ryoo&publication_year=2011" target="_blank">Google Scholar
  20. K. Coulembier, H. De Bie, and F. Sommen, “Orthogonality of Hermite polynomials in superspace and Mehler type formulae,” Proceedings of the London Mathematical Society. Third Series, vol. 103, no. 5, pp. 786–825, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  21. H. Chaggara and W. Koepf, “On linearization and connection coefficients for generalized Hermite polynomials,” Journal of Computational and Applied Mathematics, vol. 236, no. 1, pp. 65–73, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  22. H. E. J. Curzon, “On a connexion between the functions of Herimite and the functions of Legendre,” Proceedings of the London Mathematical Society, vol. 12, no. 1, pp. 236–259, 1913. View at: Google Scholar
  23. S. Fisk, “Hermite polynomials,” Journal of Combinatorial Theory. Series A, vol. 91, no. 1-2, pp. 334–336, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH

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.

3917 Views | 778 Downloads | 15 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder