541 problems

Alternating-sign conjecture. Like the Bernoulli numbers, the signs of the even-indexed Bernoulli-Catalan numbers $\tau_{2n}^*$ alternate.

Let $\Psi(a,c,z)$ be the function used in the paper's Laplace-transform formulas, and let CM mean completely monotone on $(0,\infty)$. The complete-monotonicity criterion. For ever…

Fix $p\in(0,1)$ and $b>1$. For $a>0$, let $q(a)=I^{-1}_p(a,b)$ be the unique value satisfying … and define $\phi(a)=-a\log q(a)$. Concavity conjecture. The function $\phi$ is conca…

Yang's conjecture. The coefficients satisfy

Ismail's conjecture. The sequence $(P_n)_{n\geq 0}$ satisfies

Let $\{P_n(x)\}_{n\geq 0}$ be monic polynomials orthogonal on $\mathbb{R}$ with respect to the weight … where $\rho>-1$ and $m>0$. Write $\beta_n$ for the recurrence coefficients i…

Let $L_{+}$ be the function appearing in the paper, and let $\mathscr{C}$ be its associated constant. Symmetry conjecture. The values $L_{+}(2k)$ satisfy the following symmetry: ……

Let $J_\nu$ and $J_{\nu+m}$ be Bessel functions of the first kind, with $\nu\in\mathbb{R}$ and $m\in\mathbb{N}$. Bourget's hypothesis. The functions $J_\nu$ and $J_{\nu+m}$ have no…

Let $A^{(s)}(n)=\sum_{k=0}^n\binom{n}{k}^s\binom{n+k}{k}^s$ be the generalized Franel numbers, and let $m$ be a positive integer. An Apéry limit is a limit of the quotient of two s…

Let $a_n$ be the coefficient of $t^n/n!$ in $1/(1-\ln(1+t))$, so that for $n\geq 1$, … The first $11$ entries are positive, while $a_{12}$ is negative and the subsequent entries ap…

Let $A_n(\alpha,\beta,\omega)$ be defined by the Maclaurin expansion … where $|\omega|=1$ and $\alpha,\beta\in(0,1]$. Brannan's conjecture. For every positive odd integer $n$, … Th…

Let $n$ be a positive integer and consider the Gamma values … The reflection formula and the Gauss multiplication formula give relations among these values over…

Let $n,k\in\mathbb{N}$ and $m\in\mathbb{N}_0$. Laguerre polynomial divisibility conjecture. If the Laguerre polynomial $L_k$ divides the generalized Laguerre polynomial $L_n^{(m)}$…

Let $p>1$ and let $\omega_p$ denote the optimal discrete $p$-Hardy weight. A function is absolutely monotonic on an interval if all of its derivatives are non-negative there. Absol…

Number-field linear independence conjecture. The numbers

Let $q(x;\alpha,\kappa)$ be a Clarkson–McLeod solution of the fourth Painlevé equation, with $\alpha-\frac{1}{2}\notin\mathbb{Z}$, and let $\kappa^*>0$ denote the critical value se…

Logarithmic convexity conjecture. For every $c\in\mathbb{R}$, $S_{n,c}$ is logarithmically convex; equivalently, $\log S_{n,c}$ is convex on $I_c$.

Morris distribution conjecture.

Ashbaugh–Benguria conjecture. The positive zeros of $\mathbf{\Phi}_\nu$ increase with $\nu$ on $\left[-\frac{1}{2},\infty\right)$. Motivated by the role of these zeros as eigenvalu…

Meijer $G$-function conjecture. The general form of the resolution measure is conjectured to be

Let $\alpha$ be a nonnegative integer, and let $\{L_n^{\alpha,M,N}(x)\}_{n=0}^{\infty}$ be the generalized Laguerre polynomials with parameters $M$ and $N$. Differential-equation c…

Simplicity conjecture. All nonzero roots of $W_N^{[m,k,l]}(z)$ are simple.

Mukhin–Varchenko conjecture. Under these representation-theoretic assumptions on $\alpha_1$ and $\alpha_2$, the corresponding $\mathfrak g$-Selberg integral can be expressed as a r…

Let $N$ be a positive integer, and let $\Phi^{(N)}$ be the finite Fourier transform with matrix entries … For each nonnegative integer $n$, define … where $H_n$ is the Hermite poly…

Macdonald's conjecture. The integral identity