Frobenius method

Remarks

The rules come from inserting the series into $(z - z_0)^2 y'' + (z - z_0) \cdot (z - z_0) p\, y' + (z - z_0)^2 q\, y = 0$ and comparing the coefficients of $(z - z_0)^{r + k}$. The roots of the indicial equation are the exponents of the equation at $z_0$. Fuchs' theorem says when the series converges and how to get a second solution.

Example: for $z^2 y'' + z y' + (z^2 - \frac{1}{4}) y = 0$ at $z_0 = 0$, one has $p_0 = 1$, $q_0 = -\frac{1}{4}$, $q_2 = 1$, and all other $p_k, q_k$ vanish. So $F(s) = s^2 - \frac{1}{4}$, with roots $r = \pm \frac{1}{2}$. For $r = \frac{1}{2}$ the recurrence is $k (k + 1) a_k = - a_{k-2}$, with $a_1 = 0$, which gives $y = z^{-1/2} \sin z$ when $a_0 = 1$.

Non-example: at an irregular singular point the method can fail. For $z^2 y'' + y' = 0$ at $0$, the only solutions of the form above are constants, while $y = \int e^{1/z}\, dz$ is a second solution whose derivative has an essential singularity at $0$.