Abstract:    

A matrix $M$ of real numbers is called {\em totally positive}\/ if every minor of $M$ is nonnegative.  Gantmakher and Krein showed in 1937 that a Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$ of real numbers is totally positive if and only if the underlying sequence $(a_n)_{n \ge 0}$ is a Stieltjes moment sequence, i.e.~the moments of a positive measure on $[0,\infty)$. Moreover, this holds if and only if the ordinary generating function $\sum_{n=0}^\infty a_n t^n$ can be expanded as a Stieltjes-type continued fraction with nonnegative coefficients:

  $$

  \sum_{n=0}^{\infty} a_n t^n

  \;=\;

  \cfrac{\alpha_0}{1 - \cfrac{\alpha_1 t}{1 - \cfrac{\alpha_2 t}{1 -  \cfrac{\alpha_3 t}{1- \cdots}}}}

  $$

  (in the sense of formal power series) with all $\alpha_i \ge 0$. So totally positive Hankel matrices are closely connected with the Stieltjes moment problem and with continued fractions.

Here I will introduce a generalization:  a matrix $M$ of polynomials (in some set of indeterminates) will be called {\em coefficientwise totally positive}\/ if every minor of $M$ is a polynomial with nonnegative coefficients. And a sequence $(a_n)_{n \ge 0}$ of polynomials will be called  {\em coefficientwise Hankel-totally positive}\/ if the Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$  associated to $(a_n)$ is coefficientwise totally positive. It turns out that many sequences of polynomials arising naturally in enumerative combinatorics are (empirically) coefficient wise Hankel-totally positive. In some cases this can be proven using continued fractions, by either combinatorial or algebraic methods;  I will sketch how this is done.  There is also a more general algebraic method, called {\em production matrices}\/. In a vast number of cases, however, the conjectured coefficientwise Hankel-total positivity remains an open problem.