Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial

TL;DR

This paper explores the relationship between Heun's differential equation with an apparent singularity, generalized hypergeometric equations, and exceptional Jacobi polynomials, providing new insights and establishing specific cases of the conjectured connection.

Contribution

It conjectures and partially establishes a link between Heun's equation with an apparent singularity and generalized hypergeometric equations, applying findings to exceptional Jacobi polynomials.

Findings

Conjectured a relationship between Heun's and hypergeometric equations.

Established the relationship in specific cases.

Applied results to exceptional Jacobi polynomials.

Abstract

We study Heun's differential equation in the case that one of the singularities is apparent. In particular we conjecture a relationship with generalized hypergeometric differential equation and establish it in some cases. We apply our results to exceptional Jacobi polynomials.