Analyzing the Spanish Factorial

The Spanish factorial \((2n)_{¡}\) is introduced as a unifying notation for several classical expressions involving ratios of double factorials, central binomial coefficients, and Pochhammer symbols. Asymptotic limits are derived, showing that \(\frac{(2n)_{¡}}{\sqrt{n}} \to \sqrt{\pi}\) and \(\frac{((2n)_{¡})^2}{n} \to \pi\) as \(n\to\infty\). The Spanish factorial is then used to rewrite the series expansions of the complete elliptic integrals \(K(k)\) and \(E(k)\) in compact form. Choosing the singular modulus \(k = 1/\sqrt{2}\) and applying Legendre’s relation yields a new and elegant representation of \(\pi\) expressed entirely in terms of the Spanish factorial and rational numbers. While the underlying mathematics is classical, the Spanish factorial offers a fresh notation.