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

1Makarov, VL
1Institute of Mathematics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2020, 7:3-9
https://doi.org/10.15407/dopovidi2020.07.003
Section: Mathematics
Language: Ukrainian
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.

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

1. Hahn, W. (1939). Über Orthogonalpolynome mit drei Parametern. Deutsche Math., 5, pp. 273-278.
2. 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
3. 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
4. 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).
5. Gavrilyuk, I. & Makarov, V. (2019). Resonant equations with classical orthogonal polynomials. I. Ukr. Mat. Zhurn., 71, No. 2, pp. 190-209.
6. Gavrilyuk, I. & Makarov, V. (2019). Resonant equations with classical orthogonal polynomials. II. Ukr. Mat. Zhurn., 71, No. 4, pp. 455-470.
7. Bateman, H. & Erdélyi, A. (1974). Higher trancendental functions. (Vol. 2). Moscow: Nauka (in Russian).