Irregular singularity

Remarks

Example: $z^2 y'' + y' = 0$ has $p_1 = \frac{1}{z^2}$, and $z\, p_1 = \frac{1}{z}$ is not analytic at $0$, so $0$ is an irregular singular point. Its solutions are the constants and $\int e^{1/z}\, dz$, and the derivative $e^{1/z}$ of the latter has an essential singularity at $0$.

Non-example: $z^2 y'' + z y' - y = 0$ has $z p_1 = 1$ and $z^2 p_2 = -1$, both analytic, so $0$ is a regular singular point and not irregular; its solutions are $z$ and $z^{-1}$.

For $n = 2$, the point $\infty$ is irregular for $y'' + p y' + q y = 0$ exactly when $z\, p(z)$ or $z^2 q(z)$ is not analytic at $\infty$. For example, $\infty$ is irregular for the Bessel equation, the Airy equation $y'' = z y$ and the equation $y'' = y$, whose solutions $e^{\pm z}$ have essential singularities at $\infty$.