Dear all,

**Talk Announcement:**

**Title:**Coefficientwise Hankel-total positivity

**Speaker**: Alan Sokal (University College London and New York)

**When:**Thursday, January 27, 2022 - 4:00 PM - 5:00 PM (IST)

**Where:**Zoom

**:**Write to sfandnt@gmail.com for the link

**Live Link: https://youtu.be/wUo4QcAS3mQ**

**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

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