An expansion formula for the Askey-Wilson function (Q1604432)

Here an explicit expansion formula for the Askey-Wilson function in terms of Askey-Wilson polynomials is formulated. The proper analogue of the Gaussian for the Askey-Wilson theory is introduced and the image under the polynomial Askey-Wilson transform of an Askey-Wilson polynomial multiplied by the inverse of the Gaussian is computed. A special case leads to the evaluation of a \(q\)-analogue of the (one variable) Macdonald-Mehta integral. Finally two alternative proofs for the evaluation of this important integral are added which do not make use of the properties ot the Askey-Wilson function transform.