The hypergeometric function $F(a, b ; c ; z)$ is defined as the particular solution of the second order linear ODE characterised by the Papperitz symbol

$$\mathrm{P}\left\{\begin{array}{cccc} 0 & 1 & \infty \\ 0 & 0 & a & z \\ 1-c & c-a-b & b \end{array}\right\}$$

that is analytic at $z=0$ and satisfies $F(a, b ; c ; 0)=1$.

Using the fact that a second solution $w(z)$ of the above ODE is of the form

$$w(z)=z^{1-c} u(z)$$

where $u(z)$ is analytic in the neighbourhood of the origin, express $w(z)$ in terms of $F$.