# Definition:Riemann P-symbol

Jump to navigation
Jump to search

## 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

- 1920: E.T. Whittaker and G.N. Watson:
*A Course of Modern Analysis*(3rd ed.): $10.7$: Linear differential equations with three singularities