Dixon's Theorem

Dixon's theorem states that

_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),

(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

_3F_2(a,b,c;d,e;1)=((1/2a)!(a-b)!(a-c)!(1/2a-b-c)!)/(a!(1/2a-b)!(1/2a-c)!(a-b-c)!),

(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.

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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 $1+(p/1)^3+{(p(p+1))/(1·2)}^2+...$ ." 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. https://www2.math.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'