On a Solution of a $q$-Difference Analogue of Laurieella's $D$-Type Hypergeometric Equation with $|q| = 1$

Abstract

By using the double sine function, an integral solution of a (/-difference analogue of Laurieella's £>-type hypergeometric equation is constructed in the case when |#| = 1. Furthermore, its contiguity operators are considered § 1. Introduction g-Analysis with >g|=l is important for the studies on the XXZ model in the gapless regime [8], on the massive field theory [10], [18] and on a representation theory of the quantum group SL q (2, R) [12] . However, it is not satisfactorily explored yet from the viewpoint of the theory of ^-special functions.In [15], the author and K. Ueno presented a method of ^-analysis with \q\-1. By using Kurokawa's double sine function in place of the ^-shifted factorial, they proved that the Euler type integral makes sense and that it gives a solution of a ^-difference analogue of Gauss' hypergeometric equation under some conditions on parameters.In this paper, we apply this method to a ^-difference analogue of Laurieella's D~type hypergeometric equation (We call it "Laurieella's q-HGE" in this paper.), which is a system of ^-difference equations of complex ^-variables