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Nonreal Zeros of Polynomials in a Polynomial Sequence Satisfying a Three-Term Recurrence Relation

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Abstract

This paper discusses the location of zeros of polynomials in a polynomial sequence \(\{P_{n}(z)\}_{n=1}^{\infty}\) generated by a three-term recurrence relation of the form \(P_{n}(z)+B(z)P_{n-1}(z)+A(z)P_{n-k}(z)=0\) with \(k>2\) and the standard initial conditions \(P_{0}(z)=1\), \(P_{-1}(z)=P_{-k+1}(z)=0,\) where \(A(z)\) and \(B(z)\) are arbitrary coprime real polynomials. We show that there always exist polynomials in \(\{P_{n}(z)\}_{n=1}^{\infty}\) with nonreal zeros.

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ACKNOWLEDGMENTS

I am indebted to my advisor Professor Boris Shapiro for the discussions and comments. I am grateful to Dr. Alex Samuel Bamunoba for the fruitful discussions and guidance. Thanks go to the anonymous referee for careful reading this work and giving insightful comments and suggestions.

Funding

The work is supported by the Sida Phase-IV bilateral program with Makerere University.

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Authors and Affiliations

  1. Stockholm University, Stockholm, Sweden

    I. Ndikubwayo

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  1. I. Ndikubwayo
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Correspondence to I. Ndikubwayo.

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Ndikubwayo, I. Nonreal Zeros of Polynomials in a Polynomial Sequence Satisfying a Three-Term Recurrence Relation. J. Contemp. Mathemat. Anal. 56, 87–93 (2021). https://doi.org/10.3103/S1068362321020072

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  • DOI: https://doi.org/10.3103/S1068362321020072

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