Hermite Polynomials and their Applications Associated with Bernoulli and Euler Numbers
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.
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 .
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
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
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
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
Proposition 3.1. One has
Let us take . From (3.4), can be rewritten by where
By integrating by parts, we get
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
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 , it is known that
For , we have
Theorem 3.5. For , one has
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.
S.-H. Rim, A. Bayad, E.-J. Moon, J.-H. Jin, and S.-J. Lee, “A new construction on the -Bernoulli polynomials,” Advances in Difference Equations, vol. 2011, article 34, 2011.View at: Google Scholar
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.View at: Google Scholar
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, Article ID 649248, 10 pages, 2011.View at: Google Scholar
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
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
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
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.View at: Google Scholar
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