A family of orthogonal functions on the unit circle and a new multilateral matrix inverse

Abstract

Using Bailey’s very-well-poised ${}_6{\psi }_6$ summation, we show that a specific sequence of well-poised bilateral basic hypergeometric ${}_3{\psi }_3$ series form a family of orthogonal functions on the unit circle. We further extract a bilateral matrix inverse from Dougall’s ${}_2{H}_2$ summation which we use, in combination with the Pfaff–Saalschütz summation, to derive a summation for a particular bilateral hypergeometric ${}_3{H}_3$ series. We finally provide multivariate extensions of the bilateral matrix inverse and the ${}_3{H}_3$ summation in the setting of hypergeometric series associated to the root system $A_r$.