Orthogonal polynomials related to the unit circle and differential-difference equations

Abstract

In this paper we obtain the orthogonal polynomial sequences, related to the unit circle, that verify the following differential-difference equation: (z − α)(z − β)φ'_n(z)/n = (z + αn)φ_n(z) + β_nφ_{n−1}(z) . Since these solutions are the Szegö polynomials and those whose normalized monic kernels are orthogonal, we conclude that on the unit circle the above equation and Hahn’s condition are not equivalent.

Keywords

• orthogonal polynomials
• unit circle
• classical polynomials
• semiclassical polynomials