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Dougall's Formula

Dougall's formula states that for Re[a+b-c-d]<-1 and a and b not integers,

sum_(n=-infty)^infty(Gamma(a+n)Gamma(b+n))/(Gamma(c+n)Gamma(d+n))=(pi^2csc(pia)csc(pib)Gamma(c+d-a-b-1))/(Gamma(c-a)Gamma(d-a)Gamma(c-b)Gamma(d-b)).

Dividing this identity by Gamma(a)Gamma(b)/[Gamma(c)Gamma(d)] gives Dougall's evaluation of the bilateral hypergeometric function _2H_2(a,b;c,d;1) for generic parameters.

See also

Bilateral Hypergeometric Function, Gamma Function

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References

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 7, 1981.

Referenced on Wolfram|Alpha

Dougall's Formula

Cite this as:

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

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