Definition:Riemann P-symbol

Definition
The Riemann P-symbol, written:


 * $\displaystyle f\left({z}\right) = \operatorname P \left\{ \begin{matrix} a & b & c \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' \end{matrix} \right\}$

denotes the solutions to the hypergeometric differential equation:


 * $\displaystyle \frac {\mathrm d^2 f} {\mathrm d z^2} + \left( \frac {1 - \alpha - \alpha'} {z - a} + \frac {1 - \beta - \beta'} {z - b} +\frac {1 - \gamma - \gamma'} {z - c} \right) \frac {\mathrm d f} {\mathrm d z} + \left( \frac {\alpha \alpha' \left({a - b}\right) \left({a - c}\right)} {z - a} + \frac {\beta \beta' \left({b - c}\right) \left({b - a}\right)} {z - b} + \frac{\gamma \gamma' \left({c-a}\right) \left({c-b}\right)} {z - c}\right) \frac f {\left({z - a}\right) \left({z - b}\right) \left({z - c}\right)} = 0$

where:


 * $\alpha + \alpha' + \beta + \beta' + \gamma + \gamma' = 1$

Also known as
The Riemann P-symbol is also known as the Papperitz symbol.