4 problems

The large-$n$ expansion of the Toeplitz determinant is organized by the remainder $m$ of dividing $n$ by $2r+1$, and let $F^{\{0\}}$ denote the corresponding $O(1)$ coefficient as…

Let $PF_r$ be the class of sequences whose infinite Toeplitz matrix has all minors of order at most $r$ non-negative. For $r\geq2$, define … Here $(x)_k$ is the rising factorial, a…

Let $\{f_k\}_{k=0}^{n}$ be a Pólya frequency sequence of infinite order, denoted $PF_{\infty}$, meaning that all minors of the associated infinite Toeplitz matrix are non-negative.…

Let $\{f_k\}_{k=0}^{n}$ be a non-negative sequence, let $\alpha,\beta>0$, and define … Here $(x)_k$ denotes the rising factorial. Karp–Sitnik's coefficient-positivity conjecture. I…