Whipple's Identity

Whipple derived a great many identities for generalized hypergeometric functions, many of which are consequently known as Whipple's identities (transformations, etc.). Among Whipple's identities include

 _3F_2[a,1-a,c; e,1+2c-e;1] 

(Bailey 1935, p. 15; Koepf 1998, p. 32), where _3F_2(a,b,c;d,e;z) is a generalized hypergeometric function and Gamma(z) is a gamma function, and

 _6F_5[a,1+1/2a,b,c,d,e; 1/2a,1+a-b,1-a+c,1+a-d,1+a-e;-1] 
 =(Gamma(1+a-d)Gamma(1+a-e))/(Gamma(1+a)Gamma(1+a-d-e))_3F_2[1+a-b-c,d,e; 1+a-b,1+a-c;1]

(Bailey 1935, p. 28).

See also

Generalized Hypergeometric Function, Watson's Theorem, Whipple's Transformation

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Bailey, W. N. "Whipple's Theorem on the Sum of a _3F_2." §3.4 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 16, 1935.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.Whipple, F. J. W. "Well-Poised Series and Other Generalized Hypergeometric Series." Proc. London Math. Soc. Ser. 2 25, 525-544, 1926.

Referenced on Wolfram|Alpha

Whipple's Identity

Cite this as:

Weisstein, Eric W. "Whipple's Identity." From MathWorld--A Wolfram Web Resource.

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