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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] =(2^(1-2c)piGamma(e)Gamma(1+2c-e))/(Gamma[1/2(a+e)]Gamma[1/2(a+1+2c-e)])1/(Gamma[1/2(1-a+e)]Gamma[1/2(2+2c-a-e)])

(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).

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References

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.

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Whipple's Identity

Cite this as:

Weisstein, Eric W. "Whipple's Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WhipplesIdentity.html

URL: https://mathworld.wolfram.com/WhipplesIdentity.html