Laguerre equation

Example

Let $\lambda \in \C$. The Laguerre equation is $$z\, y'' + (1 - z)\, y' + \lambda\, y = 0.$$ In normalised form, $p(z) = \frac{1 - z}{z}$ and $q(z) = \frac{\lambda}{z}$. Then $z p(z) = 1 - z$ and $z^2 q(z) = \lambda z$ are analytic at $0$, so $0$ is a regular singular point. The indicial equation is $r (r - 1) + r = r^2 = 0$, a double root $r = 0$. By Fuchs' theorem, there is one solution $y_1 = \sum_{k \ge 0} a_k z^k$, analytic at $0$, and a second solution containing $y_1 \log z$. Inserting the series gives $(k + 1)^2 a_{k+1} = (k - \lambda)\, a_k$. If $\lambda = n$ is a nonnegative integer, the series stops and $y_1$ is a multiple of the Laguerre polynomial $L_n$ of degree $n$.