A Commutative Family of Integral Transformations and Basic Hypergeometric Series. I. Eigenfunctions
نویسنده
چکیده
It is conjectured that a class of n-fold integral transformations {I(α)|α ∈ C} forms a mutually commutative family, namely, we have I(α)I(β) = I(β)I(α) for α, β ∈ C. The commutativity of I(α) 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 α of I(α).
منابع مشابه
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 sym...
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