Wallis-Ramanujan-Schur-Feynman

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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 2π ∫₀∞ dx(x²+1)ⁿ⁺¹ = 12²ⁿ 2nn. 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.