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

TL;DR

This paper introduces a new family of orthogonal functions on the unit circle derived from bilateral basic hypergeometric series and develops a multivariate extension of a bilateral matrix inverse and hypergeometric summation.

Contribution

It presents a novel family of orthogonal functions on the unit circle and extends bilateral matrix inverse and hypergeometric summation formulas to multivariate settings.

Findings

Established orthogonality of the new function family.

Derived a bilateral matrix inverse from Dougall's summation.

Extended hypergeometric identities to root system A_r.

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 ${}_2H_2$ summation which we use, in combination with the Pfaff--Saalsch\"utz summation, to derive a summation for a particular bilateral hypergeometric $_3H_3$ series. We finally provide multivariate extensions of the bilateral matrix inverse and the $_3H_3$ summation in the setting of hypergeometric series associated to the root system $A_r$.