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arXiv · 2006.08120

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

Jang Soo Kim·Dennis Stanton
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Abstract

In 1993 Delest and Fédou showed that a generating function for connected skew shapes is given as a ratio $J_{ν+1}/J_ν$ of the Hahn--Exton $q$-Bessel functions when a parameter $ν$ is zero. They conjectured that when $ν$ 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 Fédou. The second main result is a refinement of the result of Delest and Fédou: 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 $J_{ν+1}/J_ν$ 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.

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Jang Soo Kim, Dennis Stanton. 2021-05-24. Ratios of Hahn--Exton $q$-Bessel functions and $q$-Lommel polynomials. https://arxiv.org/abs/2006.08120

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