There are known expression for Toeplitz minors in terms of skew Schur polynomials, see the paper entitled ''Toeplitz minors'' by Bump and Diaconis, or e.g. 1705.08067 and 1706.02574
In particular, take $a(t) = \sum_k a_k t^k$ a Laurent polynomial with roots $(z_1, z_2, \dots)$ and define the $N$ by $N$ Toeplitz matrix
$$T_N(a) = [a_{j-k}]_{j,k=1}^N ~. $$
Taking $N \to \infty$, under certain conditions, the minors $T_N(a)$ can be expressed as skew Schur polynomials $s_{\lambda/\mu}(z_1, z_2, \dots)$, where the skew partition $\lambda/\mu$ is determined by the precise striking of rows and columns done to construct the Toeplitz minor.
My question is: does a similar expression exist for Hankel minors?
Simple Hankel minors (where one only deletes a single row or column) occur naturally as expansion coefficients in the theory of orthogonal polynomials, but I am not aware of expression for more complicated minors where one deletes multiple rows and columns, such as exist for Toeplitz matrices.
Of course, one can write a Hankel minor as a Toeplitz minor simply by re-ordering the rows (or columns), but it is not clear to me that the results from the aforementioned sources simply carry over to the case of Hankel minors. For example, if one has a Hankel matrix $H_N(w) = [w_{j+k-2}]_{j,k=1}^N$ where $w_k$ are the moments of some probability measure and one replaces $j$ by $N-j$ to turn this into a Toeplitz matrix, for large $N$ all the finite entries will be concentrated in the lower left corner of the matrix, and it is not evident that e.g. Szegö's limit theorem applies to such a Toeplitz matrix or its minors.
Any help is much appreciated.