Hypergeometric Series Representations for the Perimeter of Lamé Superellipses

We derive exact analytic representations for the perimeter of a Lam\'e superellipse of degree $s>0$. The result is expressed in terms of two branches defined by series whose terms are Gauss hypergeometric functions: a negative branch for $0<s<1$ and a positive branch for $s>1$. For the positive branch, the convergence condition follows from the Leibniz test; the negative branch, although divergent in the ordinary sense, is shown to be Abel-summable. Consistently with the symmetry under interchange of the semi-axes, the formula is invariant under axis permutation. As $s$ varies, the family interpolates between the Lam\'e cross and the rectangle, while the case $s=1$ corresponds to the rhombus, which acts as the transition curve with the shortest perimeter within the family.