Symmetries of Differential Equations. The Hypergeometric and Euler–Darboux Equations

Abstract

A general technique is introduced which uses the symmetry group of a linear homogeneous partial differential equation to obtain solutions of the equation and transformation properties of these solutions. As an application it is shown that the Euler–Poisson–Darboux equation $u_{xx} - y_{xx} - ({k / y})u_y = $ admits the symmetry group $SL(2,\mathfrak{C})$ and, if k-changing operators are admitted, the group $SO(5,\mathfrak{C})$. Certain quadratic transformation formulas for hypergeometric functions are related to the $SO(5,\mathfrak{C})$ symmetry. Similarly it is shown that the Euler–Darboux equation $u_{\zeta \mu } + (\zeta - \mu )^{ - 1} (\alpha u_\mu - \beta u_\zeta ) = 0$ admits the symmetry group $SL(2,\mathfrak{C})$ and, if $(\alpha ,\beta )$-changing operators are admitted, the group $SL(4,\mathfrak{C})$. The transformation formulas for the hypergeometric functions and the 24 solutions of Kummer are related to the $SL(4,\mathfrak{C})$ symmetry.