Discovered by Ramanujan around 1910. From Hardy (1959, pp. 102-103),
(1) |
(2) |
(3) |
(4) |
In a more symmetric form, if , , , and for , 2, ..., 6, then
(5) |
The identity is a special case of Jackson's Identity.
See also Dixon's Theorem, Dougall's Theorem, Generalized Hypergeometric Function, Hypergeometric Function, Jackson's Identity, Saalschütz's Theorem
References
Dixon, A. C. ``Summation of a Certain Series.'' Proc. London Math. Soc. 35, 285-289, 1903.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1959.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 43, 126-127, and
183-184, 1996.