Orthogonal polynomials and Möbius transformations

R. S. Vieira; V. Botta

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arXiv · 1904.10766

Orthogonal polynomials and Möbius transformations

R. S. Vieira·V. Botta

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Abstract

Given an orthogonal polynomial sequence on the real line, another sequence of polynomials can be found by composing these polynomials with a general Möbius transformation. In this work, we study the properties of such Möbius-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 Möbius-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 Möbius transformation. New orthogonality relations for Bessel and Romanovski polynomials are also presented.

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BibTeX RIS

R. S. Vieira, V. Botta. 2019-04-15. Orthogonal polynomials and Möbius transformations. https://arxiv.org/abs/1904.10766

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