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.

给定实线上的正交多项式序列，可以通过将它们与莫比乌斯变换组合来找到另一个多项式序列。在这项工作中，我们系统地研究了这种莫比乌斯变换多项式的性质。我们表明，这些多项式在复平面的给定曲线上相对于特定类型的变化测度是正交的，并且它们具有实线上正交多项式的几个共同特性。此外，通过这种方法可以更容易地推导出正交多项式的许多性质，例如，我们可以证明 Hermite、Laguerre、Jacobi、Bessel 和 Romanovski 多项式都通过合适的莫比乌斯变换相互关联；此外，复平面上贝塞尔多项式和罗曼诺夫斯基多​​项式的正交关系很容易遵循。