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The differential equation thus reduces to

\begin{array}{cl}
\frac{d^{2}P}{dz^{2}} & +\left(\frac{1-\alpha-\alpha'}{z-a}+\frac{1-\beta-\beta'}{z-b}+\frac{1-\gamma-\gamma'}{z-c}\right)\frac{dP}{dz}\\
\\ & +\left[\frac{\alpha\alpha'(a-b)(a-c)}{z-a}+\frac{\beta\beta'(b-a)(b-c)}{z-b}+\frac{\gamma\gamma'(c-a)(c-b)}{z-c}+\right]\times\frac{P}{(z-a)(z-b)(z-c)}=0.\end{array}

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

P\ \left\{ \begin{array}{ccccccc}
a &  & b &  & c\\
\\\alpha &  & \beta &  & \gamma &  & z\\
\\\alpha' &  & \beta' &  & \gamma'\end{array}\right\};

hence we have the result that

U=l^{-\alpha}\lambda^{-\alpha'}m^{-\beta}\mu^{-\beta'}n^{-\gamma}\nu^{-\gamma'}P\ \left\{ \begin{array}{ccccccc}
a &  & b &  & c\\
\\\alpha &  & \beta &  & \gamma &  & z\\
\\\alpha' &  & \beta' &  & \gamma'\end{array}\right\}

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

\frac{\partial^{2}U}{\partial l\partial\lambda}+\frac{\partial^{2}U}{\partial m\partial\mu}+\frac{\partial^{2}U}{\partial n\partial\nu}=0.

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

\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{\partial y^{2}}+\frac{\partial^{2}U}{\partial z^{2}}+\frac{\partial^{2}U}{\partial w^{2}}=0.

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

U=l^{-\alpha-\theta}\lambda^{-\alpha'-\theta}m^{-\beta-\phi}\mu^{-\beta'-\phi}n^{-\gamma-\psi}\nu^{-\gamma'-\psi}(l\lambda)^{\theta}(m\mu)^{\phi}(n\nu)^{\psi}P\ \left\{ \begin{array}{ccccccc}
a &  & b &  & c\\
\\\alpha &  & \beta &  & \gamma &  & z\\
\\\alpha' &  & \beta' &  & \gamma'\end{array}\right\},
  1. Mathematische Annalen, T. XXV. (1885), p. 213.
  2. Abhandlungen d. K. Gesell. d. Wissenschaften zu Göttingen, Band VII. (1857), Gesammelte Werke, p. 63.
  3. See Whittaker's Analysis, p. 240. Forsyth's Theory of Linear Differential Equations, Vol. IV., p. 135.