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arXiv · 2607.09048

Hypergeometric Series Representations for the Perimeter of Lam\'e Superellipses

R. Omar Rodriguez·Yomber Montilla
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Abstract

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 $0 1$. 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.

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R. Omar Rodriguez, Yomber Montilla. 2026-07-10. Hypergeometric Series Representations for the Perimeter of Lam\'e Superellipses. https://arxiv.org/abs/2607.09048

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