Wallis-Ramanujan-Schur-Feynman

This article by Tewodros Amdeberhan, Olivier R. Espinosa, Victor H. Moll and Armin Straub has been published in American Mathematical Monthly (August/September 2010).

You can also find this article on the arXiv as 1004.2453v1 [math.CA].

Abstract

One of the earliest examples of analytic representations for π 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.

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