(a) Consider the Papperitz symbol (or P-symbol):

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

Explain in general terms what this $P$-symbol represents.

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

(b) Prove that the action of $[(z-a) /(z-b)]^{\delta}$ on $(\dagger)$ results in the exponential shifting,

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

[Hint: It may prove useful to start by considering the relationship between two solutions, $\omega$ and $\omega_{1}$, which satisfy the $P$-equations described by the respective $P$-symbols ( $\dagger$ ) and ( $)$-]

(c) Explain what is meant by a Möbius transformation of a second order differential equation. By using suitable transformations acting on $(\dagger)$, show how to obtain the $P$ symbol

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

which corresponds to the hypergeometric equation.

(d) The hypergeometric function $F(a, b, c ; z)$ is defined to be the solution of the differential equation corresponding to $(\star)$ that is analytic at $z=0$ with $F(a, b, c ; 0)=1$, which corresponds to the exponent zero. Use exponential shifting to show that the second solution, which corresponds to the exponent $1-c$, is

$$z^{1-c} F(a-c+1, b-c+1,2-c ; z) .$$