5 problems

Let $\mu$ be a rigid tile and let $d$ be its associated positive integer. For each $n\geq 1$, let $P_{\mu,d,n}(t)$ denote the tiling polynomial, and write $f(t)\ll g(t)$ when $f$ i…

Let $A=(a_{ij})$ be a real $n\times n$ matrix with non-negative entries, and let $J$ denote the all-ones matrix. Assume that the entries of $A$ are weakly increasing down columns.…

Let $P_n(x;a,b)$ denote the Pseudo-Jacobi polynomials, and let $\{y_{k,n+1}\}_{k=1}^{n+1}$ be the zeros of $P_{n+1}(x;a+1,b)$. Let $E_1$ and $E_2$ be the quantities given by the tw…

Let $P_n^{(\lambda)}(x;\phi)$ be the Meixner–Pollaczek polynomials, and let $\{y_{k,n+1}\}_{k=1}^{n+1}$ denote the zeros of $P_{n+1}^{(\lambda+1)}(x;\phi)$. Let $E_1$ and $E_2$ be…

Let $g\geqslant 0$, $n\geqslant 1$, and let $\mu_1,\ldots,\mu_n\geqslant 1$. The deformed monotone Hurwitz number $\vec{H}^t_{g,n}(\mu_1,\ldots,\mu_n)$ is a polynomial in $t$. Two…