Riemann P-symbol in terms of Gaussian Hypergeometric Function

Theorem
Let:


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

where:


 * $\operatorname P$ is the Riemann P-symbol
 * $\alpha + \beta + \gamma + \alpha' + \beta' + \gamma' = 1$
 * $\alpha - \alpha'$ is not a negative integer.

Then:


 * $\displaystyle f\left({z}\right) = \left({ \frac {z - a} {z - b} }\right)^\alpha \left({ \frac {z - c} {z - b} }\right)^\gamma {}_2 \operatorname F_1 \left({ {\alpha + \beta + \gamma, \alpha + \beta' + \gamma} \atop {1 + \alpha - \alpha'} } \, \middle \vert {\, \frac {\left({z - a}\right) \left({c - b}\right)} {\left({z - b} \right) \left({c - a}\right)} }\right)$

where ${}_2 \operatorname F_1$ is the Gaussian hypergeometric function.