Duality properties of Macdonald functions and Toda limits

I try to explain the bi-spectral duality formulas of Macdonald functions
in the simplest case of two particles, by only using the q-binomila
Then I consider the q-difference Toda limit (t->0), and find
a certain difference equation. I write a similar equation for
the affine Toda case, from which one can easily define a non-stationally
(heat equation like) counterpart of the affine q-Toda system.