6 problems

Three-term recurrence conjecture. There exist matrices $A_p$, $B_p$, and $C_p$, independent of $t$, such that

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…

Let a differential operator be given as in \, and let \{P_n\}\ be its sequence of eigenpolynomials. The coefficients \{\alpha_{n,k}\}\ are those occurring in the recurrence r…

Rank-two classification conjecture. The maximally commutative rank $2$ bispectral meromorphic algebras are precisely the algebras $4\mathcal{B}_x(\psi)4$ corresponding to all bispe…

Conjecture. Under these assumptions, a solution of the integrable hierarchy is topological if and only if its matrix resolvent possesses bispectrality of Type I. The same assertion…

A bispectral triple consists of operators and a common eigenfunction; in the scalar setting considered here, the relevant operator in the spectral variable $z$ also acts by transla…