Generalized Euclidean algorithm and criterion of total non-negativity of generalized Hurwitz matrices

Given a set of real numbers $a_0, \ \ldots, \ a_n$ and a positive integer $M$, $1 \leq M \leq n$, a generalized Hurwitz matrix is defined as follows
$$ H_M = \{h_{ij}\}_{i,j = 1}^{\infty}, $$
where $h_{ij} = a_{Mj -i}, \ i,j = 1, \ 2, \ \ldots$, and $a_i = 0$ for $i < 0$ or $i>n$. We establish a criterion of total nonnegativity (i.e. non-negativity of all the minors) of the infinite-dimensional matrix $H_M$, in terms of positivity of finitely many its "special" minors. Basing on this criterion, we construct a factorization of totally nonnegative matrix $H_M$. The crucial aspect of our results, is the modification of the generalized Euclidean algorithm with step $M$, which is of independent interest. We focus on the connection between the generalized Euclidean algorithm with step $M$ and the Gaussian elimination process, applied to the generalized Hurwitz matrix $H_M$.
This is joint work with Mikhail Tyaglov, Saint Petersburg State University.