198 problems

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…

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

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…

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…

Continued-fraction conjecture. The positive powers of $q$ in this continued fraction follow the sequence $0,1,6,15,\dots$, given by $k(2k-1)$, while the negative powers follow…

Let $U({\vec{z}},\tau,p)$ be the function defined in Corollary 1, namely … Here $u$ is the function appearing in the qKZB heat equation, $\alpha$ is multiplication by the function…

Let $k,q\in\mathbb{N}$, $v\in\mathbb{C}$, and $\omega,\lambda\in\mathbb{R}_{>0}^2$. Let $H^{q,k}(z,v,\omega,\lambda)$ be the integral defined by … where $J$ and $\mathfrak{g}$ are…

Recurrence-product conjecture. The sequence satisfies

Rational-factor conjecture. At $\alpha=-q^\ell t^{1/2}$,

Product-formula conjecture. The following identities hold:

Transformation conjecture. The function $F(\alpha)$ also satisfies

Homogeneous-limit conjecture. For $\alpha\in Q_+$, $f_\alpha$ remains finite at $s_1=\cdots=s_n=1$ if and only if $\alpha\in C(\Delta)$. Otherwise it diverges in this homogeneous l…

Let $t=\varphi(s)$ be a meromorphic function on $\mathbb{C}$, and let $\zeta_{q,r}(s,t,z;\boldsymbol{\omega})$ and $\zeta_r(s,z;\boldsymbol{\omega})$ denote the corresponding $q$-a…

Let $T_n$ denote the Takeuchi numbers, let $W$ be the Lambert $W$-function, and set $w=W(n)$. Let $C_T$ be a positive real constant. Asymptotic expansion conjecture. As $n$ tends t…

Let $T$ be a positive integer and define the Racah polynomial … Here $R_n(s,T)$ is symmetric in $n$ and $s$. Uniform Racah bound conjecture. For any integers $n,s$ with…

Darboux characterization conjecture. Any polynomial solutions to the bilinear hypergeometric equations of propositions and are given by … , respectively.

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

Let $\Theta_k$ index the multivariate Hermitian Jacobi polynomials $\tilde{P}^{(\alpha,\beta)}_{\tau_1,\dots,\tau_k;\phi}$, defined by triangular expansion in the multivariate Schu…

Let $\mathcal{G}_{\kappa_1,\dots,\kappa_{k+1}}^m$ be the Hermitian Jacobi operator on the simplex, and let the multivariate Schur polynomials be the basis of unitary-conjugation-in…

Let $m$ be a positive integer, let $k$ be the number of matrix variables, let $(\kappa_1,\dots,\kappa_{k+1})$ be the parameters of the Hermitian Jacobi operator, and let…

Let $\Gamma_{\mathbb P}^{\mathrm{cyc}}$ be the prime-cyclotomic gamma function. A prime Hankel representation should consist of an oriented contour…