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

Abstract

Consider the linear second-order differential equation A n (z)y + B n (z)y + C n y = 0, (1.1) where A n (z) = a 2,n z 2 + a 1,n z + a 0,n with a 2,n = 0, a 2 1,n − 4a 2,n a 0,n = 0, ∀n ∈ N or a 2,n = 0, a 1,n = 0, ∀n ∈ N, B n (z) = b 1,n + b 0,n z are polynomials with complex coefficients and C n ∈ C. Under some assumptions over a certain class of lowering and raising operators, we show that for a sequence of polynomials (φ n) ∞ n=0 orthogonal on the unit circle to satisfy the differential equation (1.1), the polynomial φ n must be of a specific form involving and extension of the Gauss and confluent hypergeometric series.

On Differential Equations Associated with Perturbations of Orthogonal Polynomials on the Unit Circle

“…Differential properties for semi-classical orthogonal polynomials have also been analyzed in the literature (see [21] and references therein). More recently, some examples of OPUC satisfying second-order differential equations with varying polynomial coefficients were studied in [22], and differential equations for para-orthogonal polynomials on the unit circle, along with an electrostatic interpretation of their zeros, were obtained in [23].The study of differential equations for spectral transformations of OPUC remains an open problem. This contribution is oriented in that direction.…”

“…Proof. The expressions follow easily by considering the connection formulas in Proposition 3, and Equations (20), (22), (24), and (28).…”