Orthogonal polynomials on the unit circle satisfying a second-order differential equation with varying polynomial coefficients

Abstract

ABSTRACTConsider the linear second-order differential equation(1.1) where with or , are polynomials with complex coefficients and . Under some assumptions over a certain class of lowering and raising operators, we show that for a sequence of polynomials orthogonal on the unit circle to satisfy the differential equation (1.1), the polynomial must be of a specific form involving and extension of the Gauss and confluent hypergeometric series.