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Zeros of a family of hypergeometric para‐orthogonal polynomials on the unit circle
Author(s) -
Dimitrov Dimitar K.,
Ranga A. Sri
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200181
Subject(s) - mathematics , orthogonal polynomials , classical orthogonal polynomials , discrete orthogonal polynomials , wilson polynomials , hahn polynomials , jacobi polynomials , gegenbauer polynomials , monotonic function , difference polynomials , pure mathematics , discrete mathematics , mathematical analysis
Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved.