From Conformal Group to Symmetries of Hypergeometric Type Equations

TL;DR

This paper reveals how hypergeometric type equations' symmetries can be understood through their derivation from higher-dimensional PDEs with constant coefficients, linking classical equations to conformal and Schrödinger symmetries.

Contribution

It demonstrates a unified approach to deriving symmetries of hypergeometric equations from fundamental PDE symmetries, clarifying their underlying geometric structures.

Findings

Symmetries of hypergeometric and Gegenbauer equations derived from conformal symmetries of Laplace equations.

Symmetries of confluent and Hermite equations obtained from Schrödinger symmetries of heat equations.

Properties of the ${}_0F_1$ equation linked to the Helmholtz equation in 2D.

Abstract

We show that properties of hypergeometric type equations become transparent if they are derived from appropriate 2nd order partial differential equations with constant coefficients. In particular, we deduce the symmetries of the hypergeometric and Gegenbauer equation from conformal symmetries of the 4- and 3-dimensional Laplace equation. We also derive the symmetries of the confluent and Hermite equation from the so-called Schr\"odinger symmetries of the heat equation in 2 and 1 dimension. Finally, we also describe how properties of the ${}_0F_1$ equation follow from the Helmholtz equation in 2 dimensions.