Riemann P-symbol in terms of Gaussian Hypergeometric Function

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Theorem

Let:

$\map f z = \operatorname P set {\begin{matrix} a & b & c \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' \end{matrix} }$

where:

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

Then:

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

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


Proof




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