Vol. 11, No. 3, 2018

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Zeros of polynomials with four-term recurrence

Khang Tran and Andres Zumba

Vol. 11 (2018), No. 3, 501–518
Abstract

Given real numbers b,c , we form the sequence of polynomials {Hm(z)}m=0 satisfying the four-term recurrence

Hm(z) + cHm1(z) + bHm2(z) + zHm3(z) = 0,m 1,

with the initial conditions H0(z) = 1 and H1(z) = H2(z) = 0. We find necessary and sufficient conditions on b and c under which the zeros of Hm(z) are real for all m, and provide an explicit real interval on which m=0Z(Hm) is dense, where Z(Hm) is the set of zeros of Hm(z).

Keywords
generating functions, hyperbolic polynomials, recursive sequence
Mathematical Subject Classification 2010
Primary: 30C15, 26C10, 11C08
Milestones
Received: 14 March 2017
Revised: 7 June 2017
Accepted: 19 June 2017
Published: 20 October 2017

Communicated by Kenneth S. Berenhaut
Authors
Khang Tran
Department of Mathematics
California State University
Fresno, CA
United States
Andres Zumba
Department of Mathematics
California State University
Fresno, CA
United States