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May 14, 2026Open Access

A Structurally Symmetric Reformulation of the Classical Legendre Relation

Key Points

The aim is to reformulate the classical Legendre relation for complete elliptic integrals, providing a clearer structure.
Developed a dual-hypergeometric representation of π using {}_2F_1 series at z = 1/2.
Provided rigorous truncation error bounds with explicit constants.
Proved the formula from first principles and contextualized it within classical elliptic integral theory.
The new representation exhibits parameter-sign symmetry between companion series without needing large modular invariants.
While the convergence speed is not as rapid as Ramanujan–Sato series, the structural clarity is significantly improved.
All numerical computations supporting the findings are reproducible using the provided algorithm.

Abstract

We present a structurally symmetric reformulation of the classical Legendre relation for complete elliptic integrals, specialized at the lemniscate modulus k = 1/2. The result is a dual-hypergeometric representation of expressed as a rational function of two convergent ₂F₁ series evaluated at the elementary point z = 1/2. The formula does not achieve the convergence speed of Ramanujan–Sato series, but offers complete structural transparency: it requires no large modular invariants or obscure integer constants and exhibits a natural parameter-sign symmetry between the two companion series. We provide rigorous truncation error bounds with explicit constants, prove the formula from first principles, and contextualize it within the classical theory of elliptic integrals. All numerical computations are reproducible using the algorithm provided in the Appendix.

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