Dougall-Ramanujan Identity

A hypergeometric identity discovered by Ramanujan around 1910. From Hardy (1999, pp. 13 and 102-103),

(1)

where

(2)

is the rising factorial (a.k.a. Pochhammer symbol,

(3)

is the falling factorial (Hardy 1999, p. 101),  is a gamma function, and one of

(4)

is a positive integer.

Equation (1) can also be rewritten as

(5)

(Hardy 1999, p. 102). In a more symmetric form, if , , , and  for , 2, ..., 6, then

(6)

where  is the Pochhammer symbol (Petkovšek et al. 1996).

The identity is a special case of Jackson's identity, and gives Dixon's theorem, Saalschütz's theorem, and Morley's formula as special cases.

REFERENCES:

Bailey, W. N. "An Elementary Proof of Dougall's Theorem." §5.1 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 25-26 and 34, 1935.

Dixon, A. C. "Summation of a Certain Series." Proc. London Math. Soc. 35, 285-289, 1903.

Dougall, J. "On Vandermonde's Theorem and Some More General Expansions." Proc. Edinburgh Math. Soc. 25, 114-132, 1907.

Hardy, G. H. "A Chapter from Ramanujan's Note-Book." Proc. Cambridge Philos. Soc. 21, 492-503, 1923.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.

Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, pp. 43, 126-127, and 183-184, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.