Series of Lectures and Exercises: Continued Fractions and Hankel-Total Positivity; Prof. Alan Sokal
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The expansion of power series into continued fractions goes back at least to Euler in 1746, but it gained impetus following Flajolet's seminal 1980 discovery that any Stieltjes-type (resp. Jacobi-type) continued fraction can be interpreted combinatorially as a generating function for Dyck (resp. Motzkin) paths with specified height-dependent weights. There are now literally dozens of sequences (an)n?0 of combinatorial numbers or polynomials for which a continued-fraction expansion is explicitly known. I will discuss some of these, as well as some methods for proving them. I will also discuss a very recent generalization of these ideas to branched continued fractions. A matrix of real numbers is called totally positive if all its minors are nonnegative; total positivity has applications to many areas of pure and applied mathematics. In particular, a Hankel matrix (ai+j)i,j?0 is totally positive if and only if its underlying sequence (an)n?0 is a Stieltjes moment sequence. I will extend these ideas to discuss the coefficientwise total positivity of matrices of polynomials. It turns out that many sequences of combinatorial polynomials are (or at least appear empirically to be) coefficientwise Hankel-totally positive; and in some cases this can be proven using continued fractions.