Some Laguerre-Hahn Orthogonal Polynomials of Class One

Abstract

Let u (β 0) be the regular form fulfilling (u (β 0)) 2n+1 = β 0 (u (β 0)) 2n , n ≥ 0 where β 0 is an arbitrary complex number in such a way that for β 0 = 0 one has the symmetric forms. Recently, the symmetric Laguerre-Hahn forms (when β 0 = 0) of class s ≤ 1 are determined. In this paper, we determine all the Laguerre-Hahn forms u (β 0) of class s = 1, when β 0 = 0, through the resolution of a nonlinear system satisfied by the coefficients of the three-term recurrence relation of their sequences of monic corresponding orthogonal polynomials.