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Complementary Romanovski-Routh polynomials: From orthogonal polynomials on the unit circle to Coulomb wave functions

Authors: A. Martínez-Finkelshtein, L. L. Silva Ribeiro, A. Sri Ranga and M. Tyaglov
Journal: Proc. Amer. Math. Soc. 147 (2019), 2625-2640
MSC (2010): Primary 42C05, 33C45; Secondary 33C47
Published electronically: March 7, 2019
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Abstract: We consider properties and applications of a sequence of polynomials known as complementary Romanovski-Routh polynomials (CRR polynomials for short). These polynomials, which follow from the Romanovski-Routh polynomials or complexified Jacobi polynomials, are known to be useful objects in the studies of the one-dimensional Schrödinger equation and also the wave functions of quarks. One of the main results of this paper is to show how the CRR-polynomials are related to a special class of orthogonal polynomials on the unit circle. As another main result, we have established their connection to a class of functions which are related to a subfamily of Whittaker functions that includes those associated with the Bessel functions and the regular Coulomb wave functions. An electrostatic interpretation for the zeros of CRR-polynomials is also considered.

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Additional Information

A. Martínez-Finkelshtein
Affiliation: Department of Mathematics, Baylor University, 76798 Waco, Texas; and Departamento de Matemáticas, Universidad de Almería, 04120 Alamería, Spain

L. L. Silva Ribeiro
Affiliation: Pós-Graduação em Matemática, UNESP-Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil

A. Sri Ranga
Affiliation: Departamento de Matemática Aplicada, IBILCE, UNESP-Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil

M. Tyaglov
Affiliation: School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, People’s Republic of China

Keywords: Romanovski-Routh polynomials, second order differential equations, para-orthogonal polynomials on the unit circle, Coulomb wave functions.
Received by editor(s): June 7, 2018
Received by editor(s) in revised form: June 12, 2018, and October 1, 2018
Published electronically: March 7, 2019
Additional Notes: The first author was partially supported by the Spanish government together with the European Regional Development Fund (ERDF) under grant MTM2017-89941-P (from MINECO), by Junta de Andalucía (the research group FQM-229), and by Campus de Excelencia Internacional del Mar (CEIMAR) of the University of Almería.
This work was started while the first and the third authors were visiting the fourth author at Shanghai Jiao Tong University in the fall of 2016. Subsequently, this work was developed as part of the Ph.D. thesis of the second author, partially supported by the grant 2017/04358-8 from FAPESP of Brazil.
The recent collaboration of the third author in this work was also supported by the grants 2016/09906-0 and 2017/12324-6 of FAPES Brazil and 305073/2014-1 of CNPq Brazil. The third author is the corresponding author
The fourth author’s work was supported by the Joint NSFC-ISF Research Program, jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation (No. 11561141001).
Communicated by: Mourad Ismail
Article copyright: © Copyright 2019 American Mathematical Society