The generalized Hermite polynomials, their properties, and the differential equation which they satisfy
DOI:
https://doi.org/10.15407/dopovidi2020.09.003Keywords:
differential operator of infinite order, generating function, orthogonality, Rodrigues’ generalized formula, three-term recurrence relationAbstract
The generalizations of the classical orthogonal polynomials satisfying higher-order linear differential equations of a special structure were studied by a number of authors (A. Krall, J. Koekoek, R. Koekoek, H. Bavinck, L. Littlejohn, and several others). The essential requirements were the following. The coefficients of the derivatives must be polynomials of some degree of the independent variable and not dependent on the degree of the polynomials satisfying these differential equations. Such generalizations in the works of the above-mentio ned authors were made for all classical orthogonal polynomials except for the Hermite polynomials. This paper deals with a generalization of the classical Hermite polynomials in the above sense. We construct a differential operator of the infinite order whose eigenfunctions are these polynomials. A number of properties of the generalized Hermite polynomials that are characteristic of classical orthogonal polynomials (orthogonality, generalized Rodrigues’ formula, three-term recurrence relation, generic function) are investigated.
Downloads
References
Koekoek, J., Koekoek, R. & Bavinck, H. (1998). On differential equations for Sobolev-type Laguerre polynomials. Trans. Am. Math. Soc., 350, No. 1, pp. 347-393.
Koekoek, R. & Meijer, H. G. (1993). A generalization of Laguerre polynomials. SIAM J. Math. Anal., 24, Iss. 3, pp. 768-782. https://doi.org/10.1137/0524047
Krall, A. M. (1981). Orthogonal polynomials satisfying fourth order differential equations. Pr. Roy. Soc. Edinb., Sec. A, 87, Iss. 3-4, pp. 271-288. https://doi.org/10.1017/S0308210500015213
Littlejohn, L. L. (1982). The Krall polynomials: a new class of orthogonal polynomials. Quaest. Math., 5, pp. 255-265. https://doi.org/10.1080/16073606.1982.9632267
Bateman, H. & Erdélyi, A. (1974). Higher transcendental functions. Vol. 2. Moscow: Nauka (in Russian).
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Reports of the National Academy of Sciences of Ukraine
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.