Riemann P-symbol in terms of Gaussian Hypergeometric Function

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:

$\ds \map f z = \paren {\dfrac {z - a} {z - b} }^\alpha \paren {\dfrac {z - c} {z - b} }^\gamma \map F {\alpha + \beta + \gamma, \alpha + \beta' + \gamma; 1 + \alpha - \alpha'; \dfrac {\paren {z - a} \paren {c - b} } {\paren {z - b} \paren {c - a} } }$

where $F$ is the Gaussian hypergeometric function.

Proof

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Sources

• 1920: E.T. Whittaker and G.N. Watson: A Course of Modern Analysis (3rd ed.): $14.3$: Solutions of Riemann's P-equation by hypergeometric functions