On semi-classical weight functions on the unit circle - Bracciali, Cleonice F. [UNESP]; Rampazzi, Karina S. [UNESP]; Silva Ribeiro, Luana L. [UNESP] (Elsevier, 2023)
We consider orthogonal polynomials on the unit circle associated with certain semi-classical weight functions. This means that the Pearson-type differential equations satisfied by these weight functions involve two polynomials of degree at most 2. We determine all such semi-classical weight functions and this also includes an extension of the Jacobi weight function on the unit circle. General structure relations for the orthogonal polynomials and non-linear difference equations for the associated complex Verblunsky coefficients are established. As application, we present several new structure relations and non-linear difference equations associated with some of these semi-classical weight functions.
Keywords: Difference equations; Orthogonal polynomials on the unit circle; Semi-classical weight functions; Structure relations