56 problems

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 $\mu^{\mathcal{R}(\theta)}_{a_1,a_2}$ denote a solution of the product-formula measure problem for parameters $a_1,a_2\in\mathbb{R}^N$, with $\mathcal{R}$ a finite root system…

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…

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

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 $\eta$ be a positive zero of $J_0$, and consider the monic orthogonal polynomials associated with the weight $x(1-x^2)^{-1/2}\exp(i\eta x)$ on $[-1,1]$. Let $W_j(\eta^2)$ be mo…

Let $\zeta\in\mathbb{R}\setminus\{0\}$ be a zero of $J_{-1}$, and consider the monic orthogonal polynomials associated with the weight $(1-x^2)^{-1/2}\exp(i\zeta x)$ on $[-1,1]$. L…

Let $J_\nu^{(k)}(x)$ denote the $k$-th derivative with respect to $x$ of the Bessel function of the first kind of order $\nu$. For every $n\in\mathbb{N}$, consider its positive zer…

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…

Let $P_n$ denote the relevant space appearing in the critical-length problem, let $C_1$ be the first cycloidal space, and let $P_n\odot C_1$ be their product. For $uinmathbb{R}$,…

Bessel irreducibility conjecture. The $E$-function $\Gamma(\alpha+1)x^{-\alpha}J_\alpha(x)$ is irreducible in the ring of $E$-functions.

Mellin–Barnes integrals conjecture. Certain linear combinations of the $J_w$ functions, in particular the $K_w$ function, can be represented by Mellin–Barnes integrals whose contou…

Direct Integral Representation Conjecture. The integral $\mathcal{I}_w(y,\mu,\delta)$ gives the Bessel function.

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

Endpoint converse conjecture. The same assertion should hold when $\alpha=\frac{1}{2}$: for every non-negative sequence $a_m\to 0$, there should be a function $f$ such that…

Segura's product bound conjecture. It was conjectured that

Modified Bessel ratio conjecture. This ratio is unimodal on $(0,\infty)$. Moreover, if $\nu_1\ge\nu_2>0$, it is log-concave. The paper proves unimodality in the more restricted cas…

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…

Monotonicity conjecture. The sequence $\{K_i\}_{i=1}^{\infty}$ is monotonically increasing and positive, and satisfies

Monotonicity conjecture. The sequence $\{E_i\}_{i=1}^{\infty}$ is negative, decreasing, and satisfies

Let $j_{k}$ denote the positive zeros of the Bessel function $J_1$, and define, for $i\geq 0$, … where the sum is zero for $i=0$. Monotonicity conjecture. The sequence…

Let $j_{\mu,p}$ denote the $p$-th positive zero of the Bessel function of order $\mu$. Quantitative Bessel-zero separation conjecture. There exists an exponent $\kappa>1$ such that…

Global maximizer conjecture. Constant functions are global maximizers, in which case

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…