Multivariate Askey–Wilson functions and overlap coefficients

Abstract We study certain overlap coefficients appearing in representation theory of the quantum algebra $$\mathcal {U}_q(\mathfrak {sl}_2(\mathbb {C}))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>sl</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . The overlap coefficients can be identified as products of Askey–Wilson functions, leading to an algebraic interpretation of the multivariate Askey–Wilson functions introduced by Geronimo and Iliev [1]. We use the underlying coalgebra structure to derive q -difference equations satisfied by the multivariate Askey–Wilson functions.