Excess Kurtosis of F-Distribution

Theorem
Let $n, m$ be strictly positive integers.

Let $X \sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\tuple {n, m}$ degrees of freedom.

Then the excess kurtosis $\gamma_2$ of $X$ is given by:


 * $\gamma_2 = \dfrac {12 \paren {5 m n^2 - 22 n^2 + 5 m^2 n - 32 m n + 44 n + m^3 - 8 m^2 + 20 m - 16} } {n \paren {m - 6} \paren {m - 8} \paren {m + n - 2} }$

for $m > 8$, and does not exist otherwise.