The differential equation thus reduces to
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 of the general hypergeometric function are easily obtained from this result. If we write U in the form
- Mathematische Annalen, T. XXV. (1885), p. 213.
- Abhandlungen d. K. Gesell. d. Wissenschaften zu Göttingen, Band VII. (1857), Gesammelte Werke, p. 63.
- See Whittaker's Analysis, p. 240. Forsyth's Theory of Linear Differential Equations, Vol. IV., p. 135.