Supersymmetry, shape invariance and the hypergeometric equation

TL;DR

This paper demonstrates that the solvability of hypergeometric and related orthogonal polynomial equations can be explained through supersymmetry and shape invariance, providing a unified theoretical framework.

Contribution

It extends the supersymmetry and shape invariance approach from Legendre equations to hypergeometric equations, revealing a common underlying structure for their solutions.

Findings

Hypergeometric equations exhibit supersymmetry and shape invariance.

Orthogonal polynomials' Rodrigues' formulas are derived from this framework.

Unified understanding of solvability for various special functions.

Abstract

It has been shown earlier that the solubility of the Legendre and the associated Legendre equations can be understood as a consequence of an underlying supersymmetry and shape invariance. We have extended this result to the hypergeometric equation. Since the hypergeometric equation as well as the hypergeometric function reduce to various orthogonal polynomials, this study shows that the solubility of all such systems can also be understood as a consequence of an underlying supersymmetry and shape invariance. Our analysis leads naturally to closed form expressions (Rodrigues' formula) for the orthogonal polynomials.