Bispectrality and the ad conditions

TL;DR

This paper explores the role of ad-conditions in the bispectral problem, highlighting their importance in discovering new solutions and their applications in exceptional orthogonal polynomials and non-commutative cases.

Contribution

It demonstrates that adapted ad-conditions can be instrumental in advancing the bispectral problem and related areas like exceptional orthogonal polynomials.

Findings

Ad-conditions are crucial in identifying non-classical bispectral solutions.

Adapted ad-conditions can be applied to exceptional orthogonal polynomials.

Explicit solutions of ad-conditions may lead to new bispectral examples.

Abstract

At the beginning of the study of the bispectral problem, see [18], the ad-conditions played a crucial role in finding non-classical instances. The connection with the ad-conditions has reappeared in several different incarnations of the bispectral problem. Here we show that, properly adapted versions of these conditions, see [44], can play an important role in areas including, for instance, the study of exceptional orthogonal polynomials. This is also true in the non-commutative case. Even at this more advanced stage of the field one may hope that finding explicit solutions of these ad-conditions will provide an additional route to new examples.