Orthogonal polynomials and Möbius transformations

Abstract

Given an orthogonal polynomial sequence on the real line, another sequence of polynomials can be found by composing them with a Möbius transformation. In this work, we study the properties of such Möbius-transformed polynomials in a systematically way. We show that these polynomials are orthogonal on a given curve of the complex plane with respect to a particular kind of varying measure, and that they enjoy several properties common to the orthogonal polynomials on the real line. Moreover, many properties of the orthogonal polynomials can be easier derived from this approach, for example, we can show that the Hermite, Laguerre, Jacobi, Bessel and Romanovski polynomials are all related with each other by suitable Möbius transformations; also, the orthogonality relations for Bessel and Romanovski polynomials on the complex plane easily follows.