7 problems

Let $n$, $\ell$, and $m$ be integers satisfying … For the bounded-multiplicity Gaussian coefficient $\genfrac{[}{]}{0pt}{}{n}{k}_q^{(n+1)}$, define … The preceding result establish…

Let $a$ and $b$ be integers with $0\leq k\leq a<b$. The codicil conjecture. The following equivalent-looking coefficientwise inequalities hold: … or … Here $\binom{n}{k}_q$ denotes…

Let $a$ and $b$ be integers with $0<a<b$, and let $\beta\geq1$. The $\beta$-conjecture. One has … where the inequality is coefficientwise for polynomials in $q$. This is the three-…

Zanello's conjecture. Its coefficients are non-negative and unimodal. The source describes this as a strengthening of the preceding Gaussian inequality and notes that Zanello treat…

Let $a,b,c,d$ be positive integers satisfying … The Gaussian polynomial $\binom{n}{k}_q$ is the $q$-analogue of the binomial coefficient, and $f(q)\geq g(q)$ means that $f(q)-g(q)$…

Let $X\sim N(0,1)$), and let $Y=h(X)$, where $h\in\mathbb{K}[X]$ is a polynomial with $\deg(h)\geq 3$ and ${\rm I\kern-0.16em E}[h(X)]=0$. For $p_0\in\mathbb{K}_0[Y]$, let…

Let $P_n(q)$ be the polynomials defined by the recurrence in the source, with $P_0(q)=1$, $P_1(q)=q^2+1$, $P_2(q)=q^6+q^4+q^2+1$, and, for the displayed initial values, degree…