209 problems

Special-case Umemura-polynomial identity. If $b_1=0$, then

Let $K(x)$ and $E(x)$ denote the complete elliptic integrals of the first and second kinds, respectively, and let $K'(x)$ and $E'(x)$ denote their complementary counterparts. The i…

For every word $w$ in the class defining Hirose's iterated $q$-integrals on the four-punctured projective line, with the prescribed position-dependent $q$-shifts of the parameters,…

For $\nu>0$, let $J_\nu$ denote the Bessel function of the first kind, let $j_{\nu,n}$ be its $n$th positive zero, and define the normalized Turán expression by…

Prove the eighteen explicit conjectural sum–product identities arising in the modulo-nine Kanade–Russell program: ten identities obtained from reflections of finite forms under…

For every partition $\lambda$, specialize the Koornwinder moment indexed by $\lambda$ to $t=q$, normalize it as in Rains' formulation, and write the resulting rational function in…

For $-1<\alpha\leq \tfrac12$, let $J_\alpha$ denote the Bessel function of the first kind, let $j_{\alpha,2}$ be its second positive zero, and define $\beta(\alpha)<\alpha+1$ by…

Let $G_n^{(\ell)}(\mathbf{k})$ denote the generalized Gregory coefficients associated with a multiple polylogarithm of index $\mathbf{k}$ and positive integer order $\ell$, as defi…

For every integer $r>1$, the multiple Apéry-like series $\mathcal{A}_r$ introduced by Genčev and Rucki, involving central binomial coefficients and finite multiple harmonic star s…

Let $\mu:(0,1)\to(0,\infty)$ be the Grötzsch modulus function $\mu(r)=\frac{\pi}{2}\frac{\mathcal{K}(\sqrt{1-r^2})}{\mathcal{K}(r)}$, where $\mathcal{K}$ is the complete elliptic i…

Let $\kappa_r(A,B)\in\mathbb{Q}[A,B]$ be the $r$-th additive phase constant in the asymptotic phase equation for the zeros of Jacobi polynomials, with $A=\alpha^2$ and $B=\beta^2$.…

Let $\Phi$ be the classical Jacobi-theta kernel in the Fourier representation of the Riemann $\Xi$-function. Define $s(t)=\Phi(\sqrt{t})$ and $f(t)=s'(t)^2-s(t)s''(t)$ for $t>0$. T…

Determine all triples $(a,b,c)\in\mathbb{R}_{>0}^{3}$ such that ${}_1F_2\!\left(\begin{matrix}a\\ b,c\end{matrix};-x\right)>0$ for every $x>0$, where…

For $\nu>\mu\geq 0$ and $b>a>0$, define $G:[0,\infty)\to\mathbb{R}$ by…

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: ……

Number-field linear independence conjecture. The numbers

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

Central binomial sum conjecture.

AVV conjecture. The function $H$ is increasing from $(1,\infty)$ into $(1/100,1/30)$. This conjecture is attributed to the authors' earlier work and is motivated by Ramanujan's rec…

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 $m,n\in \mathbb Z$ with $1\le m\le n-1$ and let $\delta\in \mathbb Z_{\ge0}$. For $x\in\mathbb C$ and $q\in\mathbb Z_{\ge0}$, write … Let $\what{n}{\alpha}{s}$ be the Mellin tr…

Monotone subsequences conjecture. There exists a subsequence of the first sequence that is monotone non-decreasing and a subsequence of the second sequence that is monotone non-inc…

Let $l$ be an odd positive integer, and let $a_l^l$ denote the coefficient in the expansion of the solution referred to as Expansion. Classicality criterion. The solution with odd…

Let $\theta$, $\theta_t$, $\beta$, and $s$ be parameters, let $G$ denote the Barnes $G$-function, and let $\Phi$ and the matrix elements appearing below be the conformal-block data…

Continued-fraction conjecture. The positive powers of $q$ follow the sequence $0,3,10,21,\dots$, given by $(k-1)(2k-1)$, while the negative powers follow $0,5,14,27,\dots$, given b…