TOPICS
Search

Whipple's Identity

Whipple's identities are identities and transformations for generalized hypergeometric functions. They 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).

For a nonterminating series, the first identity requires Re[c]>0. In the second identity, both defining series converge when Re[a-d-e]>-1 and Re[3a-2(b+c+d+e)]>-4. Elsewhere, the second identity is understood by analytic continuation in the parameters, away from poles.

Arkani-Hamed et al. (2025, p. 31) apply the second transformation to the five-point open-string scattering amplitude, obtaining a closed-form expression in terms of _6F_5(-1) and gamma functions. This representation gives an analytic continuation beyond the region where the original _3F_2(1) series converges, where _6F_5(-1) and _3F_2(1) denote generalized hypergeometric functions with their parameter lists suppressed, evaluated at z=-1 and z=1, respectively.

See also

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

Explore with Wolfram|Alpha

References

Arkani-Hamed, N.; Figueiredo, C.; and Remmen, G. N. "Open String Amplitudes: Singularities, Asymptotics and New Representations." J. High Energy Phys. 2025, 39, 2025. https://doi.org/10.1007/JHEP04(2025)039.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 Resource. https://mathworld.wolfram.com/WhipplesIdentity.html

Subject classifications