Higher-order differential equations with polynomial solutions associated with classical orthogonal polynomials

Authors

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

DOI:

https://doi.org/10.15407/dopovidi2020.07.003

Keywords:

and Hermite polynomials, classical orthogonal polynomials, higher-order differential equations, Laguerre, Legendre, orthogonality relation, resonance equations, three-term recurrence relation

Abstract

A constructive algorithm for constructing differential equations of higher even orders is found, whose solutions are generalized classical orthogonal polynomials. For these polynomials, an explicit image, a three-term recurrence relation, and the appearance of orthogonality conditions with respect to the corresponding distribution function are obtained. The solutions of the corresponding resonance equations are given.

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References

Hahn, W. (1939). Über Orthogonalpolynome mit drei Parametern. Deutsche Math., 5, pp. 273-278.

Krall, A. M. (1981). Orthogonal polynomials satisfying fourth order differential equations. Pr. Roy. Soc. Edinb., 87A, 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

Makarov, V. L. (1976). Orthogonal polynomials and finite difference schemes with exact spectrum given in closed form (Extended abstract of Doctor thesis). Taras Shevchenko State University of Kyiv, Ukraine (in Russian).

Gavrilyuk, I. & Makarov, V. (2019). Resonant equations with classical orthogonal polynomials. I. Ukr. Mat. Zhurn., 71, No. 2, pp. 190-209.

Gavrilyuk, I. & Makarov, V. (2019). Resonant equations with classical orthogonal polynomials. II. Ukr. Mat. Zhurn., 71, No. 4, pp. 455-470.

Bateman, H. & Erdélyi, A. (1974). Higher trancendental functions. (Vol. 2). Moscow: Nauka (in Russian).

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Published

28.03.2024

How to Cite

Makarov, V. . (2024). Higher-order differential equations with polynomial solutions associated with classical orthogonal polynomials . Reports of the National Academy of Sciences of Ukraine, (7), 3–9. https://doi.org/10.15407/dopovidi2020.07.003

Issue

No. 7 (2020): Reports of the National Academy of Sciences of Ukraine

Section

Mathematics

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Copyright (c) 2023 Reports of the National Academy of Sciences of Ukraine

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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

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