Infinite-dimensional extension of Leonard trios

Establish whether the notion of a Leonard trio, together with the associated classification, extends from finite-dimensional spaces to infinite-dimensional sequence spaces carrying two-diagonal representations of the meta Hahn algebra.

Background

Leonard trios are introduced in the paper as an axiomatic framework for situations in which overlaps between eigenbases are bispectral and generalized eigenvalue problems produce biorthogonal rational functions. Although Leonard trios are currently defined only on finite-dimensional spaces, the representations constructed in the paper realize the corresponding Hahn-type trio structure on a space of sequences, leaving the infinite-dimensional extension and its classification unresolved.