\[\mathbf{F}\left(a,b;c;z\right)=\int_{0}^{1}\frac{t^{d-1}(1-t)^{c-d-1}}{(1-zt)^%
{a+b-\lambda}}\mathbf{F}\left({\lambda-a,\lambda-b\atop d};zt\right)\mathbf{F}%
\left({a+b-\lambda,\lambda-d\atop c-d};\frac{(1-t)z}{1-zt}\right)\,\mathrm{d}t,\]