Macdonald polynomials and extended Gelfand–Tsetlin graph

Using Okounkov’s q-integral representation of Macdonald polynomials we construct an infinite sequence $$\Omega _1,\Omega _2,\Omega _3,\dots $$ countable sets linked by transition probabilities from _N$$ to _{N-1}$$ for each $$N=2,3,\dots . The elements the are vertices extended Gelfand–Tsetlin graph, and depend on two parameters, q t. These data determine a family Markov chains, main result is description their entrance boundaries. This work has its origin in asymptotic theory. In subsequent paper, applied large-N limit (q, t)-deformed N-particle beta-ensembles.