23 problems

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

Commutativity conjecture. The family of operators $I(\alpha)$ is commutative on ${\cal F}_n$; equivalently, $[I(\alpha),I(\beta)]=0$.

Let $n$ be a positive integer, let ${\cal F}_n$ be the space of formal power series of degree zero in variables $\zeta_1,\ldots,\zeta_n$ corresponding to the positive cone of the…

Let ${\Bbb D}(X)$ be the algebra of $G$-invariant differential operators on $X$, viewed as holomorphic differential operators on $X_\mathbb{C}$. Let $\mathcal{O}(D_+)_\lambda$ deno…

Let $p,q,t\in\mathbb C$ satisfy $M<|t|^{n+1}<1$. For fixed $w\in\mathbb T^n$, let $C_w$ be a contour inside the annulus … for infinitesimally small positive $\epsilon$, with the po…

Arbitrary-dimensional product formula conjecture. The above product formulae should hold for arbitrary dimensions when $a=\frac{2}{n}$ with $n\in\mathbb{N}_+$.

Let $H(\delta)=\{z\in\mathbb{C}\mid\operatorname{Re}(z)\ge -\delta\}$ for some $0<\delta<1$. Define polynomials by $P_1(x)=1$, $P_2(x)=x$, and … for $m>2$. Let $g(z)$ be analytic o…

Indeterminacy conjecture. The integral $I$ is indeterminate, meaning that its value cannot be uniquely determined from the available information or conditions.

Let $e_r(z)=\displaystyle\sum_{k=0}^{\infty}\frac{z^k}{(k!)^r}$ for $r\geq 1$, and define the generalized Laplace transform … For a real number $\gamma$ in the region of convergenc…

Let … be the integral sequence, where $n$ is a nonnegative integer. The preceding recurrence gives initial values such as $-1+3e^{-1}$, $14-38e^{-1}$, and $-426+1158e^{-1}$. Integr…

Let the $\lambda$-cosine and Funk type transforms be defined on Stiefel and Grassmann manifolds, viewed in relation to the ambient matrix space. Ambient Fourier analysis conjecture…

Let $\mathcal{K}$ denote the class of normalized convex univalent functions, and let $I_\gamma$ be the Hornich scalar multiplication operator. Define … where $\mathcal{S}$ is the c…

Let $L_n^{(\alpha)}$ denote the generalized Laguerre polynomial of degree $n$, and define … Let $S_y^\alpha$ be the associated transform on $L^{2,\alpha}_{\mathbb H}(\mathbb R^+)$,…

Uniqueness conjecture. These generalized Arcsine distributions are the only fixed points of the involution $\mu\mapsto\nu$.

Let $f$ be an even function that is nonincreasing on $(0,\infty)$, and let the sine and cosine transforms be defined in the usual one-dimensional sense. The Fourier-transform relat…

Let $B\subseteq\mathbb R$ be open, and let $\sigma$ be a positive measure such that $\int_B g(t)\,d\sigma$ is finite for every $g(t)\in\mathbb C[t]$. Define … A subspace $V$ of an…

The non-stationary Lamé equation is considered with model parameters $(\kappa,g)$ and the integral transforms constructed in the paper, whose seed functions determine the resulting…

Let ${\mit \Omega}(u)$ be a real-valued function on $[0,+\infty)$ satisfying … and decreasing at infinity more rapidly than every exponential function. Define … where ${\rm ch}$ de…

Let $f$ be a function on the cone of positive definite matrices, and suppose that its M-transform is given by … where $\rho$ lies in the domain of convergence. Uniqueness conjectur…

Let $f$ satisfy … and suppose that its Radon transform $(Rf)(\xi)$ vanishes for almost all hyperplanes in $\mathbb{R}^n$. The additional condition … controls the growth of $f$ near…

The Semyanistyi-type integrals are the analytic family … where … $\operatorname{Re}\alpha>0$, $1\leq k<n$, $x\in\mathbb{R}^n$, $\tau\in\Pi_{n,k}$, and $|x-\tau|$ is the Euclidean d…

Let $u_1,\ldots,u_n$ be coordinates on $(0,\infty)^n$, and let $f\in\mathbb C[u_1,\ldots,u_n]$. Put $B=(0,\infty)^n$. Exponential integral vanishing conjecture. If … for almost all…

Finite-part/logarithmic-coefficient conjecture. For every $n\geq 0$, the coefficients agree: