## Dougall-Ramanujan Identity

Discovered by Ramanujan around 1910. From Hardy (1959, pp. 102-103),

 (1)
where

 (2)

 (3)

(here, the Pochhammer Symbol has been written ). This can be rewritten as

 (4)

In a more symmetric form, if , , , and for , 2, ..., 6, then

 (5)
where is the Pochhammer Symbol (Petkovsek et al. 1996).

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.

© 1996-9 Eric W. Weisstein
1999-05-24