Bailey's Theorem

Let  be the gamma function and  denote a double factorial, then

Writing the sums explicitly, Bailey's theorem states

REFERENCES:

Bailey, W. N. "The Partial Sum of the Coefficients of the Hypergeometric Series." J. London Math. Soc. 6, 40-41, 1931.

Bailey, W. N. "On One of Ramanujan's Theorems." J. London Math. Soc. 7, 34-36, 1932.

Darling, H. B. C. "On a Proof of One of Ramanujan's Theorems." J. London Math. Soc. 5, 8-9, 1930.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 106-107 and 112, 1999.

Hodgkinson, J. "Note on One of Ramanujan's Theorems." J. London Math. Soc. 6, 42-43, 1931.

Watson, G. N. "Theorems Stated by Ramanujan (VIII): Theorems on Divergent Series." J. London Math. Soc. 4, 82-86, 1929.

Watson, G. N. "The Constants of Landau and Lebesgue." Quart. J. Math. (Oxford) 1, 310-318, 1930.

Whipple, F. J. W. "The Sum of the Coefficients of a Hypergeometric Series." J. London Math. Soc. 5, 192, 1930.