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Dougall's Theorem

_5F_4[1/2n+1,n,-x,-y,-z; 1/2n,x+n+1,y+n+1,z+n+1] =(Gamma(x+n+1)Gamma(y+n+1)Gamma(z+n+1)Gamma(x+y+z+n+1))/(Gamma(n+1)Gamma(x+y+n+1)Gamma(y+z+n+1)Gamma(x+z+n+1)),

where _5F_4(a,b,c,d,e;f,g,h,i;z) is a generalized hypergeometric function and Gamma(z) is the gamma function.

Bailey (1935, pp. 25-26) called the Dougall-Ramanujan identity "Dougall's theorem."

See also

Dougall-Ramanujan Identity, Generalized Hypergeometric Function

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References

Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 25-27, 1935.Dougall, J. "On Vandermonde's Theorem and Some More General Expansions." Proc. Edinburgh Math. Soc. 25, 114-132, 1907.Hardy, G. H. "A Chapter from Ramanujan's Note-Book." Proc. Cambridge Philos. Soc. 21, 492-503, 1923.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 84, 1998.Whipple, F. J. W. "On Well-Poised Series, Generalized Hypergeometric Series Having Parameters in Pairs, Each Pair with the Same Sum." Proc. London Math. Soc. 24, 247-263, 1926.

Referenced on Wolfram|Alpha

Dougall's Theorem

Cite this as:

Weisstein, Eric W. "Dougall's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DougallsTheorem.html

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