The hypergeometric equation is represented by the Papperitz symbol

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

and has solution $y_{0}(z)=F(a, b, c ; z)$.

Functions $y_{1}(z)$ and $y_{2}(z)$ are defined by

$$y_{1}(z)=F(a, b, a+b+1-c ; 1-z)$$

and

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

where $c-a-b$ is not an integer.

Show that $y_{1}(z)$ and $y_{2}(z)$ obey the hypergeometric equation $(*)$.

Explain why $y_{0}(z)$ can be written in the form

$$y_{0}(z)=A y_{1}(z)+B y_{2}(z)$$

where $A$ and $B$ are independent of $z$ but depend on $a, b$ and $c$.

Suppose that

$$F(a, b, c ; z)=\frac{\Gamma(c)}{\Gamma(b) \Gamma(c-b)} \int_{0}^{1} t^{b-1}(1-t)^{c-b-1}(1-t z)^{-a} d t$$

with $\operatorname{Re}(c)>\operatorname{Re}(b)>0$ and $|\arg (1-z)|<\pi$. Find expressions for $A$ and $B$.