23 problems

Let $P_\lambda(x)=c_\lambda^{-1}p_\lambda(x)$ be the renormalized multivariable Askey–Wilson polynomials, with $c_\lambda=\hat{\Delta}^+(\rho+\lambda)/\hat{\Delta}^+(\rho)$, and le…

Commutativity conjecture. For general $n\geq 2$, one has

Let $n$ be the rank, let $v$ be a parameter, and let ${\cal D}(v)$ denote the difference operator already constructed in the surrounding text for…

A bispectral rank 1 commutative ring of difference operators is a commutative ring of difference operators whose common eigenfunctions exhibit bispectrality and whose generic joint…

A difference operator has support $[p,q]$ when it is written as … with $b_p(n),b_q(n)\not\equiv 0$, and it is regular when the endpoint coefficients are nowhere vanishing on…

A difference operator is a finite-order expression … acting on functions on $\mathbb Z$, where $T$ is the shift operator, and it is regular when its first and last coefficients…

Unrestricted-type conjecture. There are no restrictions on the possible types $(\mu_i^+\mid\mu_i^-)$ when both multiplicities are positive, that is, when $\mu_i^{\pm}>0$.

The Poisson structures on polygons obtained by reduction from difference operators include lattice analogues such as the Volterra, lattice Virasoro, and Belov–Chaltikian lattice…

A binomial difference operator is a difference operator whose support consists of two elements. Refactorization problems for such operators arise in the step-by-step refactorizatio…

Let $\mathfrak{g}$ be non-twisted, let $D_{a;1}^{(\mathfrak{g})}$ and $D_{a;-1}^{(\mathfrak{g})}$ be the generalized $\mathfrak{g}$-Macdonald operators, and let…

Let $f(z)$ be a transcendental entire function, let $\eta$ be a nonzero complex number, and let $a$ and $b$ be two distinct complex values. Suppose that $f(z)$ and…

Let $f$ be a transcendental meromorphic function, and let $c\in \mathbb{C}\backslash \\{0\\}$ satisfy $\Delta_c f\not\equiv 0$, where $\Delta_c f(z)=f(z+c)-f(z)$. The functions $f$…

Let $\mathrm{Y}(\mathfrak{gl}_{m|n})$ be the super Yangian, and let its Bethe subalgebra be the commutative subalgebra generated by the coefficients of the quantum Berezinian. A di…

Let $I(s_1/s_0,\ldots,s_N/s_0)$ be the integral operator introduced in the cited works, and let Macdonald's difference operator act on the relevant functions. Shiraishi's commutati…

Let $I$ be the set of Dynkin labels, let $I_<\subset I$ be the labels of the short roots in types $B_N$ and $C_N$, and let $t_a=2$ for $a\in I_<$ and $t_a=1$ otherwise. Let…

For a classical root system of type $D_N$, $B_N$, or $C_N$, let $M_{a,k}^{(G)}$ be the corresponding difference operators and let $\Pi_\lambda^{(G)}(q^{-1};\mathbf{x})$ be the dual…

Let $G$ be one of the classical types $D_N$, $B_N$, or $C_N$, and let $M_{a,n}^{(G)}$ be the associated Macdonald difference operators, with the superscript $G$ omitted below. Let…

Let $L$ be an irreducible differential operator of order $k$ solving the generalized Bochner–Krall problem, and let $\Lambda$ be its corresponding difference operator. Order-preser…

The local Mellin transforms $cM^{(0,\infty)}$, $cM^{(x,\infty)}$, and $cM^{(\infty,\infty)}$ apply to invertible connections except for connections in $\mathcal{C}_0$ and…

Let $f$ be a transcendental meromorphic function, let $c\in\mathbb{C}$, and let $e_1,e_2$ be three distinct values together with $\infty$. Suppose that $f$ and … share the three di…

Brändén–Krasikov–Shapiro conjecture. The operator $T$ preserves the set of real-rooted polynomials of degree at most $m$ whose mesh is at least $1$ if and only if $T((x)_m)$ is rea…

Higher-order superfrieze conjecture. These difference equations correspond to a variant of superfriezes analogous to the $2$-friezes. This would extend the relation between supersy…

Let $\rho_{a,N}$ be the Krawtchouk weight. For two finite sets $F_1,F_2$ of positive integers, allowing either set to be empty, consider the measure … If $N$ is a positive integer,…