59 problems

Closure conjecture. Analyzable functions should be closed under all operations.

Let $L \in \text{\bbold Q}[z,E]$ have leading and trailing coefficients $a_d(z)$ and $a_0(d)$. Let $N_1$ be an integer larger than any integer root of $a_d(z-d)a_0(z)$, and let…

Let ${{\frak g}\,}$ be a simple Lie algebra, and consider the dynamical difference equations and the trigonometric KZ differential equations constructed for ${{\frak g}\,}$. Compat…

Let ${{\frak g}\,}$ be a Lie algebra, let $\la\in{{\frak h\,}}$ with {{\frak h\,}}\subset{{\frak g}\,}} the Cartan subalgebra, and consider the trigonometric KZ differential equa…

Zhang et al.'s conjecture. If $f$ and $\Delta^n f$ share $0$ CM, then $f$ has the form

Liu–Laine's conjecture. If $f$ and $\Delta_c^n f$ share $a\in\mathbb{C}\backslash\{0\}$ CM, then

Let $N$ be a positive integer, let $x_i$, $b_i$, $d_i$, and $\overline{d}_i$ be parameters, and let $\mathcal{H}^{{\widehat{\mathfrak{gl}}_N}}(x_i;b_i,d_i,\overline{d}_i,q,\kappa)$…

Let $(C_n)_{n\in\mathbb{N}}$ be the Catalan numbers, and let $d\in\mathbb{N}\!\setminus\!\{0\}$. A sequence is D-algebraic of order $2$ and degree $2d$ if it satisfies a polynomial…

Consider a single-species Beverton–Holt population model in which only the carrying capacity varies environmentally, and compare the resulting stationary population size with the c…

Consider a single-species Beverton–Holt difference equation in a periodic environment, and compare its long-term average population size with the corresponding nonperiodic or avera…

Consider the Clark equation, the case $\alpha=1$ of the delay difference equation … where $k$ is a fixed integer delay and $f:\mathbb R_+\to\mathbb R_+$ is smooth, decreasing, and…

Let a nonautonomous linear difference system be weakly kinematically similar to a block-diagonal system obtained by the method developed in Chu et al. Block-diagonalization invaria…

Let … Here $n$ is a positive integer. Zhi-Wei Sun's conjecture. The number $a(n)$ is a positive odd integer for every $n>0$. The paper states that this conjecture is proved using t…

Let $\{F_n\}$, $\{G_n\}$ and $\{H_n\}$ be the sequences of real functions and let $g_n$, $h_n$, $g$, $h$ and $\Theta$ be as in the preceding pointwise convergence conjecture. Assum…

Consider the delayed quadratic-rational difference equation referred to as Eq., with parameter $p$, and let $\bar{y}$ denote its equilibrium point. Global stability conjecture. Man…

Grove–Ladas conjecture. No positive non-equilibrium solution has a limit; the question is also to extend and generalize this assertion.

Conjecture. If $\mathcal{L}_c^n(f)\equiv f$, then $f$ assumes the form

Let $q$ be a polynomial, and let $p_1,p_2,\alpha_1,\alpha_2$ be nonzero constants with $\alpha_1\neq\alpha_2$. Consider an entire solution $f$ of … Assume that…

Let $k$ be a positive integer, and consider an additive $2k$th-order difference equation … Here $\mathbf{x}_{n}^{(m,l)}=(x_{n+m},\dots,x_{n+k})$ as defined in the source, and $f$,…

The second canonical form is an additive fourth-order difference equation with a continuum limit given by the fourth-order Painlevé I equation, and its discrete Lagrangian and inva…

The 8-parameter analytic difference operator introduced by van Diejen and the 8-parameter elliptic Painlevé difference equation introduced by Sakai are the two sides of the propose…

Let $\chi$ and $a_i$, $i=1,\ldots,q$, be functions in a Carleman class $C_M(\mathbb{R})$ such that $a_1$ and $a_q$ are nowhere vanishing on $\mathbb{R}$, and let…

Multicomponent Lenard–Magri conjecture. The Lenard–Magri sequence defined by the displayed relations can be infinitely extended; consequently, the resulting differential-difference…

Lenard–Magri integrability conjecture. In both cases, the Lenard–Magri scheme can be infinitely extended, thereby proving integrability of these lattices.

Reduction conjecture. The equations (1.6) and (1.7) can be transformed in this way into an equation of type (2.2a) having the same $r$ and $n$.