25 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…

Chen's conjecture. The third-order Turán inequality for $p(n)$ is valid for all $n\geqslant94$.

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 $a,b>0$ and $r\in(0,1)$. Let $\Lambda(a,r)$ and $\Lambda_2(a,b_1,b_2,r)$ denote the quantities defined earlier in the source by the inequalities referenced as … , respectively.…

Let $F(a,b;c;r)={}_2F_1(a,b;c;r)$ denote the Gaussian hypergeometric function, defined for the parameters under consideration by … For $a>0$, $b>0$, and $r\in(0,1)$, consider the r…

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…

Craig–Pun's exact threshold conjecture. The thresholds are

Craig–Pun's threshold conjecture. For $k=3,4,5$, the minimal thresholds $N_k$ and $M_k$ with these properties exist.

Heim–Neuhauser–Tröger's conjecture. The function $\mathrm{PL}(n)$ is log-concave for every integer $n\geq 12$.

First Hankel relaxation Riemann hypothesis.

Let $\{\gamma_i\}_{i\geq 0}$ be a shifted multiplier sequence that is eventually increasing, and define … The Turán-transform preservation conjecture. For every $j\in\mathbb{N}$, t…

Let $\{\gamma_i\}_{i\geq 0}$ be a shifted multiplier sequence that is eventually increasing. For $j\in\mathbb{N}$, let $T_j$ denote the order-$j$ Turán operator, and write…

Let $\{\gamma_i\}_{i\geq 0}$ be a multiplier sequence, and let $T_2$ denote the Turán operator, with … A sequence is infinitely-log-concave if $T_2^{(k)}(i)\geq 0$ for every…

Higher-order partition-function inequality. For $n\geq 2$,

Let $f(\mu;x)$ be the generalized hypergeometric function defined in equation, let $\Delta_f(\alpha,\beta;x)$ be its generalized Turánian, and suppose $p_1\le q_1$, $p_2\le q_2$, a…

Let $f(boldsymbol{a}1,boldsymbol{a}2;boldsymbol{b}1,boldsymbol{b}2;boldsymbol{a}1,boldsymbol{a}2;boldsymbol{b}1,boldsymbol{b}2; u;x)$ be the generalized hypergeometric function def…

For $p>0$, define the generalized trigonometric and hyperbolic functions $\sin_p$, $\cos_p$, $\tan_p$, $\sinh_p$, and $\tanh_p$ as the inverse functions of their corresponding inte…

Double Turán inequalities. A necessary condition for the Riemann Hypothesis to hold is that $E_k\geq 0$ for $k=2,3,4,\ldots$. These inequalities would provide another necessary con…

Let $p>1$ and $x\in(0,1)$, and let $\sin_p$, $\cos_p$, $\tan_p$, $\sinh_p$, and $\tanh_p$ denote the generalized trigonometric and hyperbolic functions. Turán inequalities conjectu…

For $x\in(0,1)$ fixed, let $p\mapsto {\rm arcsinh}_p(x)$ denote the generalized inverse hyperbolic sine as a function of $p$. Concavity conjecture. The function…

Bessel-function conjecture. The equation $\Phi_\nu'(x)=0$ has infinitely many roots, with

Monotonicity conjecture. If $\nu>\rho>0$, then $\Phi_{\rho}^{\nu}$ is strictly increasing on $(0,\infty)$, and consequently

Let ${_1F_1}(a;c;x)$ denote the confluent hypergeometric function, defined by … where $(a)_k=a(a+1)\cdots(a+k-1)=\Gamma(a+k)/\Gamma(a)$. Carey–Gordy's conjecture. The Turán-type in…