![_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) |
(1)
|
其中 
是一個 廣義超幾何函式，
是 伽瑪函式。它可以透過 Dougall-Ramanujan 恆等式推匯出來。它可以更對稱地寫成
 |
(2)
|
其中 
具有正實部，
，並且 
(Bailey 1935, p. 13; Petkovšek et al. 1996; Koepf 1998, p. 32)。這個恆等式也可以寫成優美的對稱和
 |
(3)
|
(Petkovšek et al. 1996)。在這種形式下，它與 狄克遜恆等式非常相似。
狄克遜定理
另請參閱
狄克遜恆等式, Dougall-Ramanujan 恆等式, 廣義超幾何函式, Zeilberger-Bressoud 定理使用 探索
參考文獻
Bailey, W. N. "狄克遜定理。" §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. "關於二項式定理展開式中係數立方和。" Messenger Math. 20, 79-80, 1891.Dixon, A. C. "某些級數的求和。" 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. "二項式係數的冪和。" Quart. J. Math. 33, 274-288, 1902.Morley, F. "關於級數
。" 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. "係數 
的立方和。" Messenger Math. 21, 77-78, 1892.Watson, G. N. "關於廣義超幾何函式的狄克遜定理。" Proc. London Math. Soc. 22, xxxii-xxxiii (Records for 17 May, 1923), 1924.Zeilberger, D. and Bressoud, D. "安德魯斯 q-戴森猜想的證明。" Disc. Math. 54, 201-224, 1985.在 中被引用
狄克遜定理引用為
Weisstein, Eric W. "狄克遜定理。" 來自 —— Wolfram 網路資源。 https://mathworld.tw/DixonsTheorem.html