and two similar equations obtained by permuting in (19.18.10).
and also a system of Euler–Poisson differential equations (of which only are independent):
Here and . For group-theoretical aspects of this system see Carlson (1963, §VI). If , then elimination of between (19.18.11) and (19.18.12), followed by the substitution , produces the Gauss hypergeometric equation (15.10.1).
The function satisfies an Euler–Poisson–Darboux equation:
Also , with , satisfies a wave equation:
Similarly, the function satisfies an equation of axially symmetric potential theory:
and , with , satisfies Laplace’s equation: