On orthogonal polynomials with perturbed recurrence relations

摘要

Orthogonal polynomials may be fully characterized by the following recurrence relation: Pn(x) = (x − βn-1)Pn-1(x)-γn-1Pn-2(x), with P0(x)=1, P1(x) = x - β0 and γn ≠ 0. Here we study how the structure and the spectrum of these polynomials get modified by a local perturbation in the β and γ parameters of a co-recursive (βk → βk + μ), co-dilated (γk → λγk and co-modified (βk → βk + μ; γk → λγk) nature for an arbitrary (but fixed) kth element (1 ⩽ k). Specifically, Stieltjes functions, differential equations and distributions of zeros as well as representations of the new perturbed polynomials in terms of the old unperturbed ones are given. This type of problems is strongly related to the boundary value problems of finite-difference equations and to the quantum mechanical study of physical many-body systems (atoms, molecules, nuclei and solid state systems).