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(alpha) introduced in the previous paper. A Macdonald-type difference operator D is introduced. It is conjectured that D and I(alpha) 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(alpha) is introduced. The series F(alpha) is caracterized by a covariant transformation property I(alpha q^{-1} t) F(alpha)=F(alpha q^{-1} t) and a certain initial condition given at alpha=t^{1/2}. We call F(alpha) the `quasi-eigenfunction' for short. A class of infinite product-type expressions are conjectured for F(alpha) at the special points alpha=-t^{1/2}, alpha=q, alpha=pm q^{1/2}t^{1/2}, and alpha=pm q^{ell}t^{1/2} (ell=1,2,3,cdots). 
Jun'ichi Shiraishi