Consider the equation for $w(z)$ :

$$w^{\prime \prime}+p(z) w^{\prime}+q(z) w=0 .$$

State necessary and sufficient conditions on $p(z)$ and $q(z)$ for $z=0$ to be (i) an ordinary point or (ii) a regular singular point. Derive the corresponding conditions for the point $z=\infty$.

Determine the most general equation of the form $(*)$ that has regular singular points at $z=0$ and $z=\infty$, with all other points being ordinary.