Bispectral rational functions and Leonard trios

TL;DR

This paper introduces Leonard trios, an extension of Leonard pairs, and explores their connection with bispectral rational functions, including classification, properties, and specific examples involving Wilson's rational functions and $q$-Hahn type Leonard pairs.

Contribution

It defines and begins classifying Leonard trios, extending Leonard pairs, and connects them with bispectral rational functions, providing new insights and examples.

Findings

Wilson's rational functions are overlap coefficients with proven relations.

Established difference, recurrence, and biorthogonality relations for these functions.

Presented an example with ${}_{4} ext{phi}_3$ functions linked to a dual $q$-Hahn Leonard pair.

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 ${}_{4}\phi_3$ and are associated with a Leonard pair of dual $q$-Hahn type.