Orthogonal polynomials and M\'obius transformations
| Autor: | Vieira, R. S., Botta, V. |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: Keywords: Orthogonal polynomials, M\"obius transformations, varying weight functions, classical orthogonal polynomials, Bessel polynomials, Romanovski polynomials |
| Databáze: | arXiv |
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