Darling's Products

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

${}_3F_2\left[{\matrix{\alpha, \beta, \gamma; z\cr \hfil\delta, \epsilon\hfil\cr... ...x{1-\alpha, 1-\beta, 1-\gamma; z\cr \hfil 2-\delta, 2-\epsilon\hfil\cr}}\right]$
$={\epsilon-1\over\epsilon-\delta}\, {}_3F_2\left[{\matrix{\alpha+1-\delta, \be... ...lta-\beta, \delta-\gamma; z\cr \hfil \delta, \delta+1-\epsilon\hfil\cr}}\right]$
$+{\delta-1\over\delta-\epsilon} \,{}_3F_2\left[{\matrix{\alpha+1-\epsilon, \be... ..., \epsilon-\gamma; z\cr \hfil \epsilon, \epsilon+1-\delta\hfil\cr}}\right]\quad$	(2)

and

$(1-z)^{\alpha+\beta+\gamma-\delta-\epsilon}\,{}_3F_2\left[{\matrix{\alpha, \beta, \gamma; z\cr \hfil \delta, \epsilon\hfil\cr}}\right]$
$={\epsilon-1\over\epsilon-\delta}\,{}_3F_2\left[{\matrix{\delta-\alpha, \delta... ...beta, \epsilon-\gamma; z\cr \hfil\epsilon-1, \epsilon+1-\delta\hfil\cr}}\right]$
$+ {\delta-1\over\delta-\epsilon}\,{}_3F_2\left[{\matrix{\epsilon-\alpha, \epsi... ...a, \delta-\gamma; z\cr \hfil \delta-1, \delta+1-\epsilon\hfil\cr}}\right],\quad$	(3)

which reduce to (1) when $\gamma=\epsilon\to\infty$.

References

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