The generalized Hermite polynomials, their properties, and the differential equation which they satisfy

Authors

  • V.L. Makarov Institute of Mathematics of the NAS of Ukraine, Kyiv

DOI:

https://doi.org/10.15407/dopovidi2020.09.003

Keywords:

differential operator of infinite order, generating function, orthogonality, Rodrigues’ generalized formula, three-term recurrence relation

Abstract

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.

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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).

Published

28.03.2024

How to Cite

Makarov, V. . (2024). The generalized Hermite polynomials, their properties, and the differential equation which they satisfy . Reports of the National Academy of Sciences of Ukraine, (9), 3–9. https://doi.org/10.15407/dopovidi2020.09.003

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