10 problems

Let $p(n)$ denote the partition function, and for integers $d\ge 1$ and $n\ge 0$ define the Jensen polynomial … A polynomial is hyperbolic if all of its roots are real. Chen–Jia–Wa…

Let $p(n)$ denote the number of unrestricted partitions of $n$. For a sequence $\{a_n\}_{n\geq 0}$, its Turán inequalities of order $d$ mean that the Jensen polynomial … is hyperbo…

Let $k\geq 3$, and let $\Delta_k(n)$ denote the broken $k$-diamond partition function. For a sequence $\{\alpha_n\}_{n\geq 0}$, define its Jensen polynomial of degree $d$ and shift…

Let $\overline{a}(n)$ denote the cubic overpartition function, and define its Jensen polynomials by … For each degree $d$, let $N_{\overline{a}}(d)$ be the minimum integer such tha…

Let $PL(n)$ denote the plane partition function. For a sequence $(\omega_i)_{i=0}^\infty$, the order-$d$ Turán inequality at $n$ is equivalent to hyperbolicity of its degree-$d$, s…

Let $f$ be a real, even entire function of order less than $2$ whose real zeros become equally spaced under repeated differentiation. Since, up to a simple change of variables, the…

For $\alpha\in\mathbb R$, define the fractional partition function $p_\alpha(n)$ by … and let … where $C_\nu$ is the Bessel–Clifford function. For a sequence $a_n$, let…

Let $p(n)$ be the partition function, and let $R(n)$ denote the first term of the Hardy–Ramanujan–Rademacher series for $p(n)$. For a sequence $a_n$, write … for its degree-$d$ Jen…

Jensen polynomial conjecture. For every integer $d\geq4$, there exists a positive integer $N(d)$ such that $J_p^{d,n}(X)$ has only real roots for all $n\geq N(d)$.

Partition-function Jensen-polynomial conjecture. For every positive integer $m\geq 4$, there exists a positive integer $N(m)$ such that, for every $n\geq N(m)$, the polynomial…