Orthogonal Polynomials and Their Derivatives, II

Abstract

Let $d\alpha $ and $d\beta $ be nonnegative mass distributions on the real line, with all moments finite, and with infinitely many points of increase. Let $\{ p_n \} $ and $\{ q_n \} $ be the orthonormal polynomials associated with $d\alpha $ and $d\beta $ respectively. We characterize $d\alpha $ and $d\beta $ in the case when there exists a fixed rational function R, a positive integer j and nonnegative integers s and t such that, for $n = 0,1,2,3. \cdots ,Rp_n^{(j)} $ may be expressed as a linear combination of $q_{n - j - t} ,q_{n - j - t + 1} , \cdots ,q_{n - j + s} $.