Dear all,
Tea or Coffee: Please bring your own.
Abstract:
A matrix $M$ of real numbers is called 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.
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
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
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)
coefficientwise 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. In many other
cases it remains an open problem.
One of the more recent advances in this research is perhaps of
independent interest to special-functions workers:
we have found branched continued fractions for ratios of contiguous
hypergeometric series ${}_r \! F_s$ for arbitrary $r$ and $s$,
which generalize Gauss' continued fraction for ratios of contiguous
${}_2 \! F_1$. For the cases $s=0$ we can use these to prove
coefficientwise Hankel-total positivity.
Reference: Mathias P\'etr\'eolle, Alan D.~Sokal and Bao-Xuan Zhu,
arXiv:1807.03271
