A Commutative Family of Integral Transformations and Basic Hypergeometric Series. II. Eigenfunctions and Quasi-Eigenfunctions

A series of conjectures is obtained as further investigation of the integral transformation I(α) introduced in the previous paper. A Macdonald-type difference operator D is introduced. It is conjectured that D and I(α) are commutative with each other. Studying the series for the eigenfunctions under termination conditions, it is observed that a deformed Weyl group action appears as a hidden symmetry. An infinite product formula for the eigenfunction is found for a spacial case of parameters. A one parameter family of hypergeometric-type series F (α) is introduced. The series F (α) is caracterized by a covariant transformation property I(αp−1/2q) · F (α) = F (αp−1/2q) and a certain initial condition given at α = q1/2. We call F (α) the ‘quasieigenfunction’ for short. A class of infinite product-type expressions are conjectured for F (α) at the special points α = −q1/2, α = q, α = ±p1/4q1/2, and α = ±pl/2q1/2 (l = 1, 2, 3, · · ·).