Hermite equation

Example

Let $\lambda \in \C$. The Hermite equation is $$y'' - 2 z\, y' + 2 \lambda\, y = 0.$$ Its coefficients $-2z$ and $2 \lambda$ are polynomials, so every point of $\C$ is an ordinary point. A power series solution $y = \sum_{k \ge 0} a_k z^k$ gives $$(k + 2)(k + 1)\, a_{k+2} - 2 k\, a_k + 2 \lambda\, a_k = 0, \quad \text{so} \quad a_{k+2} = \frac{2 (k - \lambda)}{(k + 2)(k + 1)}\, a_k.$$ The values $a_0$ and $a_1$ are free, and every solution is an entire function. If $\lambda = n$ is a nonnegative integer, the recurrence stops at $k = n$ for the series of the same parity as $n$. This gives a polynomial solution of degree $n$, a multiple of the Hermite polynomial $H_n$.