Ratios of Hahn--Exton $q$-Bessel functions and $q$-Lommel polynomials

TL;DR

This paper proves a conjecture relating ratios of Hahn--Exton $q$-Bessel functions to generating functions for skew shapes and introduces $q$-Lommel polynomials to refine these results, using orthogonal polynomial techniques.

Contribution

It proves Delest and Fédou's conjecture and refines their results by connecting generating functions with $q$-Lommel polynomials and continued fractions.

Findings

Proof of the conjecture confirming rationality of coefficients

Representation of generating functions via $q$-Lommel polynomials

Two continued fraction expressions for the ratio $J_{ u+1}/J_{ u}$

Abstract

In 1993 Delest and F\'edou showed that a generating function for connected skew shapes is given as a ratio $J_{\nu+1}/J_{\nu}$ of the Hahn--Exton $q$-Bessel functions when a parameter $\nu$ is zero. They conjectured that when $\nu$ is a nonnegative integer the coefficients of the generating function are rational functions whose numerator and denominator are polynomials in $q$ with nonnegative integer coefficients, which is a $q$-analog of Kishore's 1963 result on Bessel functions. The first main result of this paper is a proof of the conjecture of Delest and F\'edou. The second main result is a refinement of the result of Delest and F\'edou: a generating function for connected skew shapes with bounded diagonals is given as a ratio of $q$-Lommel polynomials introduced by Koelink and Swarttouw. It is also shown that the ratio $J_{\nu+1}/J_{\nu}$ has two different continued fraction expressions, which give respectively a generating function for moments of orthogonal polynomials of type $R_I$ and a generating function for moments of usual orthogonal polynomials. Orthogonal polynomial techniques due to Flajolet and Viennot are used.