When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

Authors:
Manuel Alfaro;Francisco Marcellán;Ana Peña;M. Luisa Rezola
Affiliations:
Departamento de Matemáticas and IUMA, Univ. de Zaragoza, Spain;Departamento de Matemáticas, Univ. de Carlos III de Madrid, Spain;Departamento de Matemáticas and IUMA, Univ. de Zaragoza, Spain;Departamento de Matemáticas and IUMA, Univ. de Zaragoza, Spain
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

Citing 4
Cited 3

On orthogonal polynomials with perturbed recurrence relations

Journal of Computational and Applied Mathematics
An introduction to difference equations

An introduction to difference equations
Quasi-orthogonality with applications to some families of classical orthogonal polynomials

Applied Numerical Mathematics
The zeros of linear combinations of orthogonal polynomials

Journal of Approximation Theory

Orthogonal polynomials associated with an inverse quadratic spectral transform

Computers & Mathematics with Applications
Higher Order Coherent Pairs

Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
(M,N)-coherent pairs of order (m,k) and Sobolev orthogonal polynomials

Journal of Computational and Applied Mathematics

Quantified Score

Hi-index	7.29

Visualization

Abstract

Given {P"n}"n"="0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e., Q"n(x)=P"n(x)+a"1P"n"-"1(x)+...+a"kP"n"-"k,a"k0,nk. Necessary and sufficient conditions are given for the orthogonality of the sequence {Q"n}"n"="0. An interesting interpretation in terms of the Jacobi matrices associated with {P"n}"n"="0 and {Q"n}"n"="0 is shown. Moreover, in the case k=2, we characterize the families {P"n}"n"="0 such that the corresponding polynomials {Q"n}"n"="0 are also orthogonal.