A Commutative Family of Integral Transformations and Basic Hypergeometric Series. I. Eigenfunctions
Abstract
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).
Cite
@article{arxiv.math/0501251,
title = {A Commutative Family of Integral Transformations and Basic Hypergeometric Series. I. Eigenfunctions},
author = {Jun'ichi Shiraishi},
journal= {arXiv preprint arXiv:math/0501251},
year = {2009}
}
Comments
Basic parameters are replaced to make the notation consistent with the standard Macdonald polynomials: q is replaced by t, and p^{1/2} is replaced by q