A Family of Integral Transformations and Basic Hypergeometric Series

Abstract

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).

A Conjecture about Raising Operators for Macdonald Polynomials

“…See Proposition A.6 in Appendix in [8]. Therefore, the following is a consequence of Conjecture 2.1.…”

“…In the paper [8], a modified Macdonald difference operator acting on the space of formal power series…”

“…The commutativity was proved only for the simplest case n = 2 by using the explicit formulas for the eigenfunctions of I(α), and several summation and transformation formulas for the basic hypergeometric series [7]. To prove the commutativity for n ≥ 3 remains as an open problem, since properties of the eigenfunctions have not been studied well.…”

“…The commutativity was proved for the simplest case n = 2. In Appendix of [8], it was shown that the eigenfunction of D 1 (s 1 , · · · , s n , q, t) can be interpreted as a raising operator for the Macdonald polynomials.…”

“…An explicit formula for the eigenfunctions of I(α) or D 1 (s 1 , · · · , s n , q, t) was conjectured for n = 3 in [7]. (See (45) below.)…”