Czechoslovak Mathematical Journal, first online, pp. 1-12

Criterion of the reality of zeros in a polynomial sequence satisfying a three-term recurrence relation

Innocent Ndikubwayo

Received December 4, 2018.   Published online February 24, 2020.

Abstract:  This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence $\{P_i\}_{i=1}^{\infty}$ generated by a three-term recurrence relation $P_i(x)+ Q_1(x)P_{i-1}(x) +Q_2(x) P_{i-2}(x)=0$ with the standard initial conditions $P_0(x)=1, P_{-1}(x)=0,$ where $Q_1(x)$ and $Q_2(x)$ are arbitrary real polynomials.
Keywords:  recurrence relation; polynomial sequence; support; real zeros
Classification MSC:  12D10, 26C10, 30C15
DOI:  10.21136/CMJ.2020.0535-18

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Affiliations:   Innocent Ndikubwayo, Department of Mathematics, Stockholm University, Kräftriket, SE-106 91 Stockholm, Sweden; Department of Mathematics, Makerere University, P.O Box 7062 Kampala, Uganda, e-mail:,

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