We study some resonant equations related to the classical orthogonal polynomials on infinite intervals, i.e., the Hermite and the Laguerre orthogonal polynomials, and propose an algorithm for finding their particular and general solutions in the closed form. This algorithm is especially suitable for the computer-algebra tools, such as Maple. The resonant equations form an essential part of various applications, e.g., of the efficient functional-discrete method for the solution of operator equations and eigenvalue problems. These equations also appear in the context of supersymmetric Casimir operators for the di-spin algebra and in the solution of square operator equations, such as A2u = f (e.g., of the biharmonic equation).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Bateman and A. Erdélyi, Higher Trancendental Functions, McGraw-Hill, New York, etc. (1953).
N. B. Backhouse, “The resonant Legendre equation,” J. Math. Anal. Appl., 133 (1986).
I. Gavrilyuk and V. Makarov, “Resonant equations with classical orthogonal polynomials. I,” Ukr. Mat. Zh., 71, No. 2, 190–209 (2019); English translation:Ukr. Math. J., 71, No. 2, 215–236 (2019).
A. Krazer and W. Franz, Transzendente Funktionen, Akademie Verlag (1960).
F. Nikiforov and V. Uvarov, Special Functions of Mathematical Physics [in Russian], Nauka, Moscow (1978).
W. C. Parke and L. C. Maximon, “On second solutions to second-order difference equations,” arXiv:1601.04412 [math.CA].
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 4, pp. 455–470, April, 2019.
Rights and permissions
About this article
Cite this article
Gavrilyuk, I., Makarov, V. Resonant Equations with Classical Orthogonal Polynomials. II. Ukr Math J 71, 519–536 (2019). https://doi.org/10.1007/s11253-019-01661-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-019-01661-4