A product formula defined by the beta function and Gauss's hypergeometric function (Q1957986)

Let \(B(p,q)\) be the Beta function and \(F(\alpha,\beta,\gamma;x)\) the hypergeometric function. Let \(c>0\), \(m,n\in{\mathbb N}\), and let \(L_{m,n}\) be the length of the plane curve in polar coordinates \((r,\theta)\): \[ r^{2m-n}=2c^n\cos n\theta. \] Suppose that \(\frac{n}{2}<m<\frac{n}{2}(1+\sqrt{2})\), and let \(l:=2m-n\). The authors prove that \[ L_{m,n}=\root l\of {2c^n}\frac{n}{l}B\left(\frac{1}{2l},\frac12\right)F\left(\frac{-1}2,\frac{1}{2l},\frac{1}{2l}+\frac12;1-\left(\frac{l}{n}\right)^2\right). \]