Journal of the Ramanujan Mathematical Society
Volume 39, Issue 4, December 2024 pp. 377–388.
On Ramanujan's formula involving fifth powers of binomial coefficients
Authors:
John M. Campbell and Paul Levrie
Author institution:Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada.
Summary:
Included in Ramanujan's first letter to Hardy was the remarkable formula
1 − 5
(1/2){5} + 9 (1 · 3/2 · 4){5} − 13
(1 · 3 · 5/2 · 4 · 6){5} + · · · =
Γ{4}
(1/4)2π{4} .
Different proofs of this formula have been given by a number of different authors, and this includes a recent proof
due to Cantarini related to the generalized Clebsch–Gordan integral. Series involving fifth powers of binomial coefficients
are known to be very difficult to evaluate, and it is not clear how to generalize the above formula. We introduce
an evaluation technique based on a Fourier–Legendre expansion that was considered by Baranov in 2006, and we
succeed in applying our technique to obtain families of generalizations and variants of Ramanujan's formula.
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