$3nj$-symbols and identities for $q$-Bessel functions

TL;DR

This paper explores the relationship between $3nj$-symbols and $q$-Bessel functions, deriving new identities and defining multivariate extensions linked to Askey-Wilson polynomials.

Contribution

It introduces multivariate $q$-Bessel functions as $3nj$-symbols and connects them to multivariate Askey-Wilson polynomials, providing new summation identities.

Findings

Derived summation identities for $q$-Bessel functions.

Defined multivariate $q$-Bessel functions as $3nj$-symbols.

Established limits relating multivariate $q$-Bessel functions to Askey-Wilson polynomials.

Abstract

The $6j$-symbols for representations of the $\mathrm{SU}(2)$ quantum group are given by Hahn-Exton $q$-Bessel functions. This interpretation leads to several summation identities for the $q$-Bessel functions. Multivariate $q$-Bessel functions are defined, which are shown to be limit cases of multivariate Askey-Wilson polynomials. The multivariate $q$-Bessel functions occur as $3nj$-symbols.