§15.16 Products

 15.16.1 $\mathop{F\/}\nolimits\!\left({a,b\atop c-\frac{1}{2}};z\right)\mathop{F\/}% \nolimits\!\left({c-a,c-b\atop c+\frac{1}{2}};z\right)=\sum_{s=0}^{\infty}% \frac{\left(c\right)_{s}}{\left(c+\frac{1}{2}\right)_{s}}A_{s}z^{s},$ $|z|<1$,

where $A_{0}=1$ and $A_{s}$, $s=1,2,\dots$, are defined by the generating function

 15.16.2 $(1-z)^{a+b-c}\mathop{F\/}\nolimits\!\left(2a,2b;2c-1;z\right)=\sum_{s=0}^{% \infty}A_{s}z^{s},$ $|z|<1$.

Also,

 15.16.3 $\mathop{F\/}\nolimits\!\left({a,b\atop c};z\right)\mathop{F\/}\nolimits\!\left% ({a,b\atop c};\zeta\right)=\sum_{s=0}^{\infty}\frac{\left(a\right)_{s}\left(b% \right)_{s}\left(c-a\right)_{s}\left(c-b\right)_{s}}{\left(c\right)_{s}\left(c% \right)_{2s}s!}\left(z\zeta\right)^{s}\mathop{F\/}\nolimits\!\left({a+s,b+s% \atop c+2s};z+\zeta-z\zeta\right),$ $|z|<1$, $|\zeta|<1$, $|z+\zeta-z\zeta|<1$.
 15.16.4 $\mathop{F\/}\nolimits\!\left({a,b\atop c};z\right)\mathop{F\/}\nolimits\!\left% ({-a,-b\atop-c};z\right)+\frac{ab(a-c)(b-c)}{c^{2}(1-c^{2})}z^{2}\mathop{F\/}% \nolimits\!\left({1+a,1+b\atop 2+c};z\right)\mathop{F\/}\nolimits\!\left({1-a,% 1-b\atop 2-c};z\right)=1.$

Generalized Legendre’s Relation

 15.16.5 $\mathop{F\/}\nolimits\!\left({\frac{1}{2}+\lambda,-\frac{1}{2}-\nu\atop 1+% \lambda+\mu};z\right)\mathop{F\/}\nolimits\!\left({\frac{1}{2}-\lambda,\frac{1% }{2}+\nu\atop 1+\nu+\mu};1-z\right)+\mathop{F\/}\nolimits\!\left({\frac{1}{2}+% \lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)\mathop{F\/}\nolimits\!% \left({-\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)-\mathop% {F\/}\nolimits\!\left({\frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu}% ;z\right)\mathop{F\/}\nolimits\!\left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu% \atop 1+\nu+\mu};1-z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1+\lambda+% \mu\right)\mathop{\Gamma\/}\nolimits\!\left(1+\nu+\mu\right)}{\mathop{\Gamma\/% }\nolimits\!\left(\lambda+\mu+\nu+\frac{3}{2}\right)\mathop{\Gamma\/}\nolimits% \!\left(\frac{1}{2}+\nu\right)},$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$, $|\mathop{\mathrm{ph}\/}\nolimits\!\left(1-z\right)|<\pi$.

For further results of this kind, and also series of products of hypergeometric functions, see Erdélyi et al. (1953a, §2.5.2).