2010/04/14 by Amdeberhan, Tewodros, Espinosa, Olivier R., Moll, Victor H. +1
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1004.2453
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 \frac2π ∫0^∞ \fracdx(x2+1)n+1 = \frac122n \binom2nn. 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.