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

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

2005-01-17

arxiv.org/abs/math/0501251v2

Mathematics

Quantum Algebra

Basic parameters are replaced to make the notation consistent with the standard Macdonald polynomials: q is replaced by t, and

Scientific paper

10.1007/s00220-005-1504-5

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).

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