Noncommutative Catalan numbers revisited

The goal of my second talk on the subject (the first one was given at this seminar two years ago), which is also based on joint work with Vladimir Retakh, is twofold:

First, I will recall motivations, definition and basic properties of noncommutative Catalan numbers C_n, which belong to the free Laurent polynomial algebra in x_0,...,x_n. They admit interesting (commutative and noncommutative) specializations, one of them related to the Garsia-Haiman (q,t)-version Catalan numbers, another -- to solving noncommutative quadratic equations.  We also established total positivity of the corresponding (noncommutative) Hankel matrices H_n and introduced two kinds of noncommutative binomial coefficients which are instrumental in computing the inverse of H_n, its positive factorizations, and other combinatorial identities involving C_n.

Second, I will outline a relationship of the C_n with (noncommutative) orthogonal polynomials, which can be viewed as "characteristic polynomials" of an extended version of the matrices H_n or of the corresponding noncommutative Jacobi matrices J_n.