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 πcscπα(0 < α < 1) both
of which reduce for α = 1∕2 to one of the known refinements of the Wallis
estimate.
