$R_I$ biorthogonal polynomials of Hahn type

TL;DR

This paper introduces a new family of $R_I$ biorthogonal polynomials of Hahn type, exploring their properties, eigenvalue problems, and bispectrality, contributing to the understanding of their algebraic and spectral structure.

Contribution

It presents a novel family of $R_I$ polynomials with explicit biorthogonality, recurrence, and difference equations, highlighting their bispectral nature and operator triplet structure.

Findings

Polynomials are biorthogonal to rational functions.

They satisfy generalized eigenvalue problems.

They exhibit bispectrality with a triplet of tridiagonal operators.

Abstract

A finite family of $R_I$ polynomials is introduced and studied. It consists in a set of polynomials of $_{3}F_{2}$ form whose biorthogonality to an ensemble of rational functions is spelled out. These polynomials are shown to satisfy two generalized eigenvalue problems: in addition to their recurrence relation of $R_I$ type, they are also found to obey a difference equation. Underscoring this bispectrality is a triplet of operators with tridiagonal actions.