Orthogonal polynomial solutions of spectral type differential equations: Magnus' conjecture

Let tau = sigma + v be a point mass perturbation of a classical moment func tional a by a distribution v with finite support, We find necessary conditi ons for the polynomials {Q(n)(x)}(n=0)(infinity), orthogonal relative to ta u, to be a Bochner-Krall orthogonal polynomial system (BKOPS); that is, {Q( n)(x)}(n=0)(infinity) are eigenfunctions of a finite order linear different ial operator of spectral type with polynomial coefficients: L-N[y](x) = Sig ma (N)(i=1) l(i)(x) y((i))(x) = lambda (n)y(x). In particular, when v is of order 0 as a distribution, we find necessary and sufficient conditions for {Q(n)(x)}(n=0)(infinity) 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 p oint(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a B KOPS but also explains the phenomena for infinite-order differential equati ons (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions. (C ) 2001 Academic Press.