# §15.15 Sums

 15.15.1 $\mathop{\mathbf{F}\/}\nolimits\!\left({a,b\atop c};\frac{1}{z}\right)=\left(1-% \frac{z_{0}}{z}\right)^{-a}\sum_{s=0}^{\infty}\frac{(a)_{s}}{s!}\*\mathop{% \mathbf{F}\/}\nolimits\!\left({-s,b\atop c};\frac{1}{z_{0}}\right)\left(1-% \frac{z}{z_{0}}\right)^{-s}.$

Here $z_{0}$ (${\neq 0}$) is an arbitrary complex constant and the expansion converges when $|z-z_{0}|>\max(|z_{0}|,|z_{0}-1|)$. For further information see Bühring (1987a) and Kalla (1992).

For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975).