Bispectral rational functions and Leonard trios

Abstract: It is well-known that Leonard pairs have a close connection with bispectral orthogonal polynomials of the Askey scheme. In this paper, we introduce the notion of a Leonard trio $(V,\oV,Z)$, an algebraic structure extending Leonard pairs, for which the overlap coefficients of eigenfunctions of $V$ and $\oV$ are biorthogonal rational functions satisfying generalized eigenvalue problems. We introduce and start the classification of irreducible Leonard trios by using its connection with Leonard pairs and Heun operators. In particular, we show that Wilson's rational functions appear as overlap coefficients, prove its difference, recurrence and biorthogonality relations, and obtain a summation formula expressing them as a finite sum of products of two $q$-Racah polynomials. We also begin to investigate reduced Leonard trios, for which the general eigenvalue problem simplifies to a $R_I$-type recurrence relation. As an illustration, we present an example of this in which the rational functions appearing as overlap coefficients can be expressed as a ${}_{4}φ_3$ and are associated with a Leonard pair of dual $q$-Hahn type.