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Darling's Products

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A generalization of the hypergeometric function identity

_2F_1(alpha,beta;gamma;z)_2F_1(1-alpha,1-beta;2-gamma;z) =_2F_1(alpha+1-gamma,beta+1-gamma;2-gamma;z)_2F_1(gamma-alpha,gamma-beta;gamma;z)
(1)

to the generalized hypergeometric function _3F_2(a,b,c;d,e;x). Darling's products are

_3F_2[alpha,beta,gamma;z; delta,epsilon ]_3F_2[1-alpha,1-beta,1-gamma;z; 2-delta,2-epsilon ] =(epsilon-1)/(epsilon-delta)_3F_2[alpha+1-delta,beta+1-delta,gamma+1-delta;z; 2-delta,epsilon+1-delta ]_3F_2[delta-alpha,delta-beta,delta-gamma;z; delta,delta+1-epsilon ] +(delta-1)/(delta-epsilon)_3F_2[alpha+1-epsilon,beta+1-epsilon,gamma+1-epsilon;z; 2-epsilon,delta+1-epsilon ]_3F_2[epsilon-alpha,epsilon-beta,epsilon-gamma;z; epsilon,epsilon+1-delta ]
(2)

and

(1-z)^(alpha+beta+gamma-delta-epsilon)_3F_2[alpha,beta,gamma;z; delta,epsilon ] =(epsilon-1)/(epsilon-delta)_3F_2[delta-alpha,delta-beta,delta-gamma;z; delta,delta+1-epsilon ]_3F_2[epsilon-alpha,epsilon-beta,epsilon-gamma;z; epsilon-1,epsilon+1-delta ] +(delta-1)/(delta-epsilon)_3F_2[epsilon-alpha,epsilon-beta,epsilon-gamma;z; epsilon,epsilon+1-delta ]_3F_2[delta-alpha,delta-beta,delta-gamma;z; delta-1,delta+1-epsilon ],
(3)

which reduce to (◇) when gamma=epsilon->infty.

See also

Generalized Hypergeometric Function

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References

Bailey, W. N. "Darling's Theorems of Products." §10.3 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 88-92, 1935.

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Darling's Products

Cite this as:

Weisstein, Eric W. "Darling's Products." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DarlingsProducts.html

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