19 problems

Let $N\geq 1$, let $\mathcal{R}\subset\mathbb{R}^N$ be a finite root system, and let $\theta\in\theta(\mathcal{R})$ be nonnegative. For $a_1,a_2\in\mathbb{R}^N$, consider the Dunkl…

Bessel common-zero conjecture. If $f$ and $J_\alpha$ share a common zero in $\mathbb C^*$, then

Nonnegativity conjecture. For all $0\leqslant\alpha\leqslant\frac{1}{2}$, $M\in\mathbb{N}$, and $x\in[0,1]$,

Let $p$ be an integer, let $\beta,\gamma,\nu\in\mathbb{C}$, and let $u,\varw\in\mathbb{C}$ satisfy … For the Bessel function $\boldsymbol{B}_{\nu,p}$ and the associated function…

Let $A_e^{w_i}$ and $A_e^{w_{G_n}}$ be the domains associated with the relative Shalika germs $K_e^{w_i}$ and $K_e^{w_{G_n}}$, respectively, and let $\Delta$ be the discriminant fu…

Let $j_{\nu,\kappa}$ denote the $\kappa$th positive zero of the Bessel function $J_\nu$, and let McMahon's expansion be the asymptotic expansion for these zeros. For…

Dirichlet disk growth conjecture. If $1\leq\gamma<3$, then

Convexity conjecture. For every $a,b>0$, the set ${\mathcal{D}}_{a,b}$ is convex.

Let $n<1$, let $ugeq 1-n+\sqrt{2(1-n)}$, and define … For the modified Bessel function of the second kind $K_\mu$, the author conjectures that, for every $x>0$, … The exponent shif…

Bessel normalization conjecture. For every $p\in\mathbb{N}^*$,

Let $r\in\{1,2,3\}$, let $\zeta^{*}(s,r)$ be the completed zeta function defined in the paper, and let $a_r(n)$ and $b_r(n)$ be the coefficient sequences introduced by the theta-fu…

B$^3$G sum rule. For every such pair $(n,k)$, the sum vanishes identically:

Redheffer-type bounds conjecture. For $|x|<j{ u,1}$,

Asymptotic distribution conjecture. The asymptotic location of the $n$-th pure imaginary zero is determined by

For $n\in\mathbb{N}_0$ and $m\in\mathbb{N}$, let $j_{\nu,m}^{(n)}$ denote the $m$th positive zero of the $n$th derivative $J_{\nu}^{(n)}$ of the Bessel function of the first kind.…

Let $n\in\mathbb{N}_0$, let $\nu\in\mathbb{R}$, and define the real entire function … A Fourier critical point of a real entire function is a point at which some derivative has a c…

Let $n\in\mathbb{N}_0$. For a real parameter $\nu$, write $J_{\nu}^{(n)}$ for the $n$th derivative of the Bessel function of the first kind with respect to its argument. Generalize…

The Bessel-zero quotient monotonicity conjecture. The quotients $\hat q_k$ decrease monotonically as $k\in\mathbb{N}$ increases. The conjecture is supported by numerical experiment…

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