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

TL;DR

This paper investigates $q^{-1}$-Fibonacci polynomials for 0<q<1, exploring their connection to $q$-exponentials, orthogonal polynomials, and extremal measures, with detailed analysis of special cases involving $q$-trigonometric functions.

Contribution

It provides a novel analysis of $q^{-1}$-Fibonacci polynomials, deriving explicit formulas for extremal measures and their relation to hypergeometric functions.

Findings

Derived a relation between $q^{-1}$-Fibonacci polynomials and $q$-exponentials.

Obtained a formula for the reproducing kernel of related orthogonal polynomials.

Described all N-extremal measures of orthogonality in terms of hypergeometric functions.

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.