Wallis-Ramanujan-Schur-Feynman

TL;DR

This paper explores generalizations of Wallis' pi integral using products of quadratic factors, revealing connections to Ramanujan formulas, Schur functions, and Feynman diagram sums.

Contribution

It introduces new integral representations involving products of quadratic factors and links them to Ramanujan, Schur functions, and Feynman diagram sums.

Findings

Derived new integral formulas related to pi and quadratic products

Established connections between these formulas and Ramanujan's identities

Linked the integral representations to Feynman diagram sums

Abstract

One of the earliest examples of analytic representations for $\pi$ is given by an infinite product provided by Wallis in 1655. The modern literature often presents this evaluation based on the integral formula $$ \frac{2}{\pi} \int_0^\infty \frac{dx}{(x^2+1)^{n+1}} = \frac{1}{2^{2n}} \binom{2n}{n}. $$ In trying to understand the behavior of this integral when the integrand is replaced by the inverse of a product of distinct quadratic factors, the authors encounter relations to some formulas of Ramanujan, expressions involving Schur functions, and Matsubara sums that have appeared in the context of Feynman diagrams.