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.
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.
TOPICS
![_3F_2[n,-x,-y; x+n+1,y+n+1]
=Gamma(x+n+1)Gamma(y+n+1)Gamma(1/2n+1)Gamma(x+y+1/2n+1)
×Gamma(n+1)Gamma(x+y+n+1)Gamma(x+1/2n+1)Gamma(y+1/2n+1),](/images/equations/DixonsTheorem/NumberedEquation1.svg)
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
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
." 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.