Dixon's Theorem

(1)

where  is a generalized hypergeometric function and  is the gamma function. It can be derived from the Dougall-Ramanujan identity. It can be written more symmetrically as

(2)

where  has a positive real part, , and  (Bailey 1935, p. 13; Petkovšek et al. 1996; Koepf 1998, p. 32). The identity can also be written as the beautiful symmetric sum

(3)

(Petkovšek et al. 1996). In this form, it closely resembles Dixon's identity.

REFERENCES:

Bailey, W. N. "Dixon's Theorem." §3.1 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 13-14, 1935.

Cartier, P. and Foata, D. Problèmes combinatoires de commutation et réarrangements. New York: Springer-Verlag, 1969.

Dixon, A. C. "On the Sum of the Cubes of the Coefficients in Certain Expansion by the Binomial Theorem." Messenger Math. 20, 79-80, 1891.

Dixon, A. C. "Summation of 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, pp. 104 and 111, 1999.

Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1997.

Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 18-19, 1998.

MacMahon P. A. "The Sums of the Powers of the Binomial Coefficients." Quart. J. Math. 33, 274-288, 1902.

Morley, F. "On the Series ." Proc. London Math. Soc. 34, 397-402, 1902.

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

Richmond, H. W. "The Sum of the Cubes of the Coefficients in ." Messenger Math. 21, 77-78, 1892.

Watson, G. N. "Dixon's Theorem on Generalized Hypergeometric Functions." Proc. London Math. Soc. 22, xxxii-xxxiii (Records for 17 May, 1923), 1924.

Zeilberger, D. and Bressoud, D. "A Proof of Andrews' -Dyson Conjecture." Disc. Math. 54, 201-224, 1985.