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Whipple's Transformation

_7F_6[a,1+1/2a,b,c,d,e,-m; 1/2a,1+a-b,1+a-c, ; 1+a-d,1+a-e,1+a+m] =((1+a)_m(1+a-d-e)_m)/((1+a-d)_m(1+a-e)_m)_4F_3[1+a-b-c,d,e,-m; 1+a-b,1+a-c,d+e-a-m]

(Bailey 1935, p. 25), where _7F_6(a_1,...,a_7;b_1,...,b_6) and _4F_3(a_1,...,a_4;b_1,b_2,b_3) are generalized hypergeometric functions with argument z=1 and Gamma(z) is the gamma function.

Another transformation due to Whipple (1926ab) is given by

_4F_3[a,b,-z,-n; u,v,w;1] =(Gamma(u+z+n)Gamma(w+z+n)Gamma(v)Gamma(w))/(Gamma(v+z)Gamma(v+n)Gamma(w+n)Gamma(w+z))_4F_3[u-a,u-b,-z,-n; 1-v-z-n,1-w-z-n,u;1]

for one of z and n a nonnegative integer (Andrews and Burge 1993).

See also

Generalized Hypergeometric Function, Watson-Whipple Transformation, Whipple's Identity

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References

Andrews, G. E. and Burge, W. H. "Determinant Identities." Pacific J. Math. 158, 1-14, 1993.Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 25 and 29, 1935.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, 1926a.Whipple, F. J. W. "Well-Poised Series and Other Generalized Hypergeometric Series." Proc. London Math. Soc. Ser. 2 25, 525-544, 1926b.Whipple, F. J. W. "A Fundamental Relation Between Generalized Hypergeometric Series." Proc. London Math. Soc. 26, 257-272, 1927.

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Whipple's Transformation

Cite this as:

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

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