The rational Heun operator and Wilson biorthogonal functions

TL;DR

This paper introduces the rational Heun operator, a specific q-difference operator linked to Wilson biorthogonal functions, and explores its relation to Ruijsenaars-van Diejen Hamiltonians, revealing new solutions in the form of biorthogonal functions.

Contribution

It defines a new class of rational Heun operators on the Askey-Wilson grid and connects them to Wilson biorthogonal functions and Ruijsenaars-van Diejen Hamiltonians.

Findings

Rational Heun operator sends certain rational functions to others of different type.

Wilson biorthogonal functions are solutions to a generalized eigenvalue problem involving these operators.

The operator is related to a degeneration of Ruijsenaars-van Diejen Hamiltonians.

Abstract

We consider the rational Heun operator defined as the most general second-order $q$-difference operator which sends any rational function of type $[(n-1)/n]$ to a rational function of type $[n/(n+1)]$. We shall take the poles to be located on the Askey-Wilson grid. It is shown that this operator is related to the one-dimensional degeneration of the Ruijsenaars-van Diejen Hamiltonians. The Wilson biorthogonal functions of type ${_{10}}\Phi_9$ are found to be solutions of a generalized eigenvalue problem involving rational Heun operators of the special "classical" kind.