# Orthogonal polynomials and M\"obius transformations

## TL;DR

This paper explores how M"obius transformations applied to classical orthogonal polynomials produce new polynomial sequences with similar properties, including orthogonality on complex curves, recurrence relations, and differential equations, revealing interrelations among various classical polynomials.

## Contribution

It introduces a framework for understanding M"obius-transformed orthogonal polynomials, establishing their properties and connections among classical families, and presents new orthogonality relations.

## Key findings

- M"obius-transformed polynomials satisfy orthogonality on complex curves.
- They obey three-term recurrence relations and Christoffel-Darboux identities.
- New orthogonality relations for Bessel and Romanovski polynomials are derived.

## Abstract

Given an orthogonal polynomial sequence on the real line, another sequence of polynomials can be found by composing these polynomials with a general M\"obius transformation. In this work, we study the properties of such M\"obius-transformed polynomials. We show that they satisfy an orthogonality relation in given curve of the complex plane with respect to a varying weight function and that they also enjoy several properties common to the orthogonal polynomial sequences on the real line --- e.g. a three-term recurrence relation, Christoffel-Darboux type identities, their zeros are simple, lie on the support of orthogonality and have the interlacing property, etc. Moreover, we also show that the M\"obius-transformed polynomials obtained from classical orthogonal polynomials also satisfy a second-order differential equation, a Rodrigues' type formula and generating functions. As an application, we show that Hermite, Laguerre, Jacobi, Bessel and Romanovski polynomials are all related to each other by a suitable M\"obius transformation. New orthogonality relations for Bessel and Romanovski polynomials are also presented.