Riemann P-Differential Equation

${d^2 u\over dz^2}+\left[{{1-\alpha-\alpha'\over z-a}+{1-\beta-\beta'\over z-b}+{1-\gamma-\gamma'\over z-c}}\right]{du\over dz}$

$+ \left[{{\alpha\alpha'(a-b)(a-c)\over z-a}+{\beta\beta'(b-c)(b-a)\over z-b}+{\gamma\gamma'(c-a)(c-b)\over z-c}}\right]{u\over (z-a)(z-b)(z-c)} = 0,$

where

$\begin{displaymath} \alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1. \end{displaymath}$

Solutions are Riemann P-Series (Abramowitz and Stegun 1972, pp. 564-565).

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Riemann's Differential Equation.'' §15.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 564-565, 1972.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 541-543, 1953.

${d^2 u\over dz^2}+\left[{{1-\alpha-\alpha'\over z-a}+{1-\beta-\beta'\over z-b}+{1-\gamma-\gamma'\over z-c}}\right]{du\over dz}$
$+ \left[{{\alpha\alpha'(a-b)(a-c)\over z-a}+{\beta\beta'(b-c)(b-a)\over z-b}+{\gamma\gamma'(c-a)(c-b)\over z-c}}\right]{u\over (z-a)(z-b)(z-c)} = 0,$