A product formula defined by the Beta function and Gauss's hypergeometric function

Abstract

Let $c$ be a constant in ${\bf R}^t$. For a plane algebraic curve $r^{2m-n} = 2c^n \cos n\theta$, which depends on $m$ and $n$ in ${\bf N}$, we show that the whole length of the curve are given by a value of a product formula defined by the Beta function and Gauss's hypergeometric function depending $m$ and $n$ in ${\bf N}$. Besides, we point out the fact to be a similar model and an expansion for the complete elliptic integral of the second kind. Last, we give a background for the fact explaining the special case $m = n$.