A new approach to the construction of generalized classical polynomials

  • V. L. Makarov Inst. Math. Acad. Sci. Ukraine, Kiev

Abstract

UDC 517.587

In this paper, we develop a new method for constructing generalized classical polynomials, primarily Hermite polynomials in the sense of A. Krall, J. Koekoek, R. Koekoek, H. Bavinck, L. Littlejohn, et al.
We construct a differential operator of infinite order whose eigenfunctions are such polynomials.
For generalized Hermite polynomials, we investigate a number of properties inherent in classical orthogonal polynomials (orthogonality, generalized Rodrigues formula, three-term recurrence relation forming a function).
The versatility of the method is revealed in constructing generalized Legendre and Chebyshev polynomials of the first kind.

References

J. Koekoek, R. Koekoek, H. Bavinck, On differential equations for Sobolev-type Laguerre polynomials, Trans. Amer. Math. Soc., 350, № 1, 347 – 393 (1998), https://doi.org/10.1090/S0002-9947-98-01993-X DOI: https://doi.org/10.1090/S0002-9947-98-01993-X

R. Koekoek, H. G. Meijer, A generalization of Laguerre polynomials, SIAM J. Math. Anal., 24, № 3, 768 – 782 (1993), https://doi.org/10.1137/0524047 DOI: https://doi.org/10.1137/0524047

A. M. Krall, Orthogonal polynomials satisfying fourth order differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 87, № 3-4, 271 – 288 (1981); https://doi.org/10.1017/S0308210500015213 DOI: https://doi.org/10.1017/S0308210500015213

L. L. Littlejohn, The Krall polynomials: a new class of orthogonal polynomials, Quaest. Math., 5, 255 – 265 (1982). DOI: https://doi.org/10.1080/16073606.1982.9632267

V. L. Makarov, Uzagal`neni polinomi Ermita, yikh vlastivosti ta diferenczial`ne rivnyannya, yake voni zadovol`nyayut`, Dop. NAN Ukrayini, № 9, 3 – 9 (2020); https://doi.org/10.15407/dopovidi2020.09.003 DOI: https://doi.org/10.15407/dopovidi2020.09.003

G. Bejtmen, A. E`rdeji, Vy`sshie transczendentny`e funkczii, t. 2, Nauka, Moskva (1974).

The on-line encyclopedia of integer sequences, founded in 1964 by N. J. A. Sloane.

H. L. Krall, Certain differential equations for Tchebysheff polynomials, Duke Math. J., 4, 705 – 718 (1938); https://doi.org/10.1215/S0012-7094-38-00462-4 DOI: https://doi.org/10.1215/S0012-7094-38-00462-4

Published
18.06.2021
How to Cite
MakarovV. L. “A New Approach to the Construction of Generalized Classical Polynomials”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 6, June 2021, pp. 827 -38, doi:10.37863/umzh.v73i6.6256.
Section
Research articles