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Families of self-reciprocal polynomials with a three-term recurrence relation


The main purpose of this research project is to show that self-reciprocal polynomials satisfy a three-term recurrence relation and they can be represented by a linear combination of Chebyshev polynomials of first kind. Furthermore, by known results of theory of orthogonal and para-orthogonal polynomials, we intend to explore properties of self-reciprocal polynomials, and to establish relationships between self-reciprocal polynomials and orthogonal polynomials in [- 1,1] and para-orthogonal polynomials, through a change of variable. As a result of this study, we investigate the behavior of zeros of these families of polynomials, focusing on the conditions for these polynomials have unimodular zeros, because there are many applications in mathematics and some areas related to this topic. Although there is a vast literature on the conditions for the zeros of self-reciprocal polynomials are located on the unit circle, none of them deals with the subject in order to explore the properties from the three-term recurrence relation; so, it will be a different item in this research project. Then, we intend to fully characterize families of self-reciprocal polynomials and explore and determine results on the zeros of these families of polynomials. (AU)

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Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
VIEIRA, R. S.; BOTTA, V. Orthogonal polynomials and Mobius transformations. COMPUTATIONAL & APPLIED MATHEMATICS, v. 40, n. 6 SEP 2021. Web of Science Citations: 0.
BOTTA, V.; DA SILVA, J. V. On the behavior of roots of trinomial equations. ACTA MATHEMATICA HUNGARICA, v. 157, n. 1, p. 54-62, FEB 2019. Web of Science Citations: 0.

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