International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 643-656
doi:10.1155/S0161171296000919

Generating new classes of orthogonal polynomials

Amílcar Branquinho1 and Francisco Marcellán2

1Departamento de Matemática, FCTUC, Universidade de Coimbra, Apartado 3008, Coimbra 3000, Portugal
2Departamento de Ingeniería, Escuela Politécnica Superior, Universidad Carlos III, C. Butarque, 15, Leganés-Madrid 28911, Spain

Received 28 September 1994; Revised 15 December 1994

Copyright © 1996 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given a sequence of monic orthogonal polynomials (MOPS), {Pn}, with respect to a quasi-definite linear functional u, we find necessary and sufficient conditions on the parameters an and bn for the sequence Pn(x)+anPn−1(x)+bnPn−2(x),   n≥1P0(x)=1,P−1(x)=0 to be orthogonal. In particular, we can find explicitly the linear functional v such that the new sequence is the corresponding family of orthogonal polynomials. Some applications for Hermite and Tchebychev orthogonal polynomials of second kind are obtained.

We also solve a problem of this type for orthogonal polynomials with respect to a Hermitian linear functional.