(a) Explain in general terms the meaning of the Papperitz symbol

$$P\left\{\begin{array}{cccc} a & b & c & \\ \alpha & \beta & \gamma & z \\ \alpha^{\prime} & \beta^{\prime} & \gamma^{\prime} & \end{array}\right\}$$

State a condition satisfied by $\alpha, \beta, \gamma, \alpha^{\prime}, \beta^{\prime}$ and $\gamma^{\prime}$. [You need not write down any differential equations explicitly, but should provide explicit explanation of the meaning of $a, b, c, \alpha, \beta, \gamma, \alpha^{\prime}, \beta^{\prime}$ and $\left.\gamma^{\prime} .\right]$

(b) The Papperitz symbol

$$P\left\{\begin{array}{cccc} 1 & -1 & \infty & \\ -m / 2 & m / 2 & n & z \\ m / 2 & -m / 2 & 1-n \end{array}\right\}$$

where $n, m$ are constants, can be transformed into

$$P\left\{\begin{array}{cccc} 0 & 1 & \infty & \\ 0 & 0 & n & \frac{1-z}{2} \\ m & -m & 1-n & \end{array}\right\}$$

(i) Provide an explicit description of the transformations required to obtain ( $*)$ from $(t)$.

(ii) One of the solutions to the $P$-equation that corresponds to $(*)$ is a hypergeometric function $F\left(a, b ; c ; z^{\prime}\right)$. Express $a, b, c$ and $z^{\prime}$ in terms of $n, m$ and $z$.