Regular singular point criterion for a second-order equation

For$z_0$ is a regular singular point when $(z-z_0)P(z)$ and $(z-z_0{)}^{2}Q(z)$ are analytic at $z_0$.

Under $\zeta =1/z$, the point $z=\infty$ is regular singular forexactly when $zP(z)$ and $z^{2}Q(z)$ are holomorphic functions of $1/z$ near $1/z=0$.

When a second-order equation has a repeated indicial root and only one independent Frobenius series, a second solution has the local form