Vol. 39, No. 3, 1971

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Refinements of Wallis’s estimate and their generalizations

T. S. Nanjundiah

Vol. 39 (1971), No. 3, 745–749

Some refinements of Wallis’s estimate for π noticed in the recent literature are pointed out as already contained in a certain continued fraction expansion due to Stieltjes. A property of the approximants to this continued fraction is established which yields a simple proof of the expansion and furnishes, in particular, interesting monotone sequences of rational numbers with limit π. Two estimates of the Wallis type involving quotients of gamma functions are derived. They include estimates for Γ(α) and π cscπα(0 < α < 1) both of which reduce for α = 12 to one of the known refinements of the Wallis estimate.

Mathematical Subject Classification 2000
Primary: 40A15
Received: 25 January 1971
Published: 1 December 1971
T. S. Nanjundiah