Orthogonal polynomials and Möbius transformations

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