A Commutative Family of Integral Transformations and Basic Hypergeometric Series. I. Eigenfunctions

It is conjectured that a class of n-fold integral transformations {I(alpha)|alpha in {C}} forms a mutually commutative family, namely, we have I(alpha) I(beta)=I(beta) I(alpha) for all alpha, beta in {C}. The commutativity of I(alpha) for the two-fold integral case is proved by using several summation and transformation formulas for the basic hypergeometric series. An explicit formula for the complete system of the eigenfunctions for n=3 is conjectured. In this formula and in a partial result for n=4, it is observed that all the eigenfunctions do not depend on the spectral parameter alpha of I(alpha).