Definition:Riemann P-symbol

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Definition

The Riemann P-symbol, written:

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

denotes the solutions to the hypergeometric differential equation:

\(\displaystyle \) \(\) \(\displaystyle \frac {\d^2 f} {\d z^2}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \paren {\frac {1 - \alpha - \alpha'} {z - a} + \frac {1 - \beta - \beta'} {z - b} + \frac {1 - \gamma - \gamma'} {z - c} } \frac {\d f} {\d z}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \paren {\frac {\alpha \alpha' \paren {a - b} \paren {a - c} } {z - a} + \frac {\beta \beta' \paren {b - c} \paren {b - a} } {z - b} + \frac {\gamma \gamma' \paren {c - a} \paren {c - b} } {z - c} } \frac f {\paren {z - a} \paren {z - b} \paren {z - c} }\)
\(\displaystyle \) \(=\) \(\displaystyle 0\)


where:

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


Also known as

The Riemann P-symbol is also known as the Papperitz symbol, for Erwin Papperitz.


Also see

  • Results about Riemann P-symbol can be found here.


Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.


Sources