The differential equation thus reduces to

This is Papperitz's form^{[1]} of the differential equation satisfied by Riemann's general hypergeometric function^{[2]}

; |

hence we have the result that

is a homogeneous function of (*l, m, n, λ, μ, ν*) of degree -1, satisfying the equation

When expressed in terms of *x, y, z* and *w*, it will thus be a solution of the equation

The various transformations^{[3]} of the general hypergeometric function are easily obtained from this result. If we write *U* in the form