Two families of orthogonal polynomials on the unit circle from basic hypergeometric functions

2016/11/24 by Ranga, A. Sri
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1611.08064

Abstract

The sequence \ ₂ϕ₁(q^-k,q^b+1; q^-b-k+1; q, q^-b+1/2 z)\_{k ≥ 0} of basic hypergeometric polynomials is known to be orthogonal on the unit circle with respect to the weight function |(q^1/2e^iθ; q)_∞/(q^b+1/2e^iθ; q)_∞|². This result, where one must take the parameters q and b to be 0 < q < 1 and \Re(b) > -1/2, is due to P.I. Pastro \citePastro-1985. In the present manuscript we deal with the orthogonal polynomials Φ_n(b;.) and \checkΦ_n(b;.) on the unit circle with respect to the two parametric families of weight functions ω(b; θ) = |(e^iθ; q)_∞/(q^be^iθ; q)_∞|² and \checkω(b;θ) = |(qe^iθ; q)_∞/(q^be^iθ; q)_∞|², where 0 < q < 1 and \Re(b) > 0. With the use of the basic hypergeometric polynomials ₂ϕ₁(q^-k,q^b; q^-b-k+1; q, q^-b+1 z), k ≥ 0, which have zeros on the unit circle when \Re(b) > 0, simple expressions for the (monic) polynomials Φ_n(b;.) and \checkΦ_n(b;.), their norms, the associated Verblunsky coefficients and also the respective Szegő functions are found.

Related

• Orthogonal polynomials on the unit circle: New results
• Self-inversive polynomial and quasi-orthogonality on the unit circle
• On a family of hypergeometric Sobolev orthogonal polynomials on the unit circle
• OPUC on One Foot
• Elliptic polynomials orthogonal on the unit circle with a dense point spectrum