A 𝑞-Hankel transform associated to the quantum linking groupoid for the quantum 𝑆𝑈(2) and 𝐸(2) groups

De Commer, Kenny; Koelink, Erik

doi:10.1090/S0002-9939-2015-12445-8

A $q$-Hankel transform associated to the quantum linking groupoid for the quantum $SU(2)$ and $E(2)$ groups
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Proc. Amer. Math. Soc. 143 (2015), 2515-2526 Request permission

Abstract:

A $q$-analogue of Erdélyi’s formula for the Hankel transform of the product of Laguerre polynomials is derived using the quantum linking groupoid between the quantum $SU(2)$ and $E(2)$ groups. The kernel of the $q$-Hankel transform is given by the ${}_1\varphi _1$-$q$-Bessel function, and then the transform of a product of two Wall polynomials times a $q$-exponential is calculated as a product of two Wall polynomials times a $q$-exponential.

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