The boundary of the Gelfand–Tsetlin graph: new proof of Borodin–Olshanski's formula, and its $q$-analogue

In their recent paper, Borodin and Olshanski have presented a novel proof of the celebrated Edrei–Voiculescu theorem which describes the boundary of the Gelfand–Tsetlin graph as a region in an infinite-dimensional coordinate space. This graph encodes branching of irreducible characters of finite-dimensional unitary groups. Points of the boundary of the Gelfand–Tsetlin graph can be identified with finite indecomposable (= extreme) characters of the infinite-dimensional unitary group. An equivalent description identifies the boundary with the set of doubly infinite totally nonnegative sequences.
A principal ingredient of Borodin–Olshanski's proof is a new explicit determinantal formula for the number of semi-standard Young tableaux of a given skew shape (or of Gelfand–Tsetlin schemes of trapezoidal shape). We present a simpler and more direct derivation of that formula using the Cauchy–Binet summation involving the inverse Vandermonde matrix. We also obtain a $q$-generalization of that formula, namely, a new explicit determinantal formula for arbitrary $q$-specializations of skew Schur polynomials. Its particular case is related to the $q$-Gelfand–Tsetlin graph and $q$-Toeplitz matrices introduced and studied by Gorin.