Regular Article
Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture,☆☆

https://doi.org/10.1006/jath.2001.3586Get rights and content
Under an Elsevier user license
open archive

Abstract

Let τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Qn(x)}n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Qn(x)}n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: LN[y](x)=∑Ni=1 i(x) y(i)(x)=λny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Qn(x)}n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions.

Keywords

differential equations
Bochner–Krall orthogonal polynomials
Magnus' conjecture

Cited by (0)

K.H.K. and G.J.Y. were partially supported by KOSEF (98-0701-03-01-5) and BK-21 project, and L.L.L. thanks the National Science Foundation (DMS-9970478) for partial financial support. All authors thank Professor A. Magnus and the referee for their many valuable comments on the paper.

☆☆

Communicated by Alphonse, P. Magnus

f1

E-mail: [email protected]

f2

E-mail: [email protected]

f3

E-mail: [email protected]