Nevanlinna extremal measures for polynomials related to $q^{-1}$-Fibonacci polynomials

Abstract

The aim of this paper is the study of $q^{-1}$-Fibonacci polynomials with $0<q<1$. First, the $q^{-1}$-Fibonacci polynomials are related to a $q$-exponential function which allows an asymptotic analysis to be worked out. Second, related basic orthogonal polynomials are investigated with the emphasis on their orthogonality properties. In particular, a compact formula for the reproducing kernel is obtained that allows to describe all the N-extremal measures of orthogonality in terms of basic hypergeometric functions and their zeros. Two special cases involving $q$-sine and $q$-cosine are discussed in more detail.

19 pages

References in corpus (2)

• One Parameter Generalizations of the Fibonacci and Lucas Numbers
• The Nevanlinna parametrization for $q$-Lommel polynomials in the indeterminate case

Cited by in corpus (1)

• On Basic Fourier-Bessel Expansions