= Power series solutions of the Hermite differential equation
{title2=$a_{n+2}=2(n-\lambda)a_n/[(n+2)(n+1)]$}
For $y''-2xy'+2\lambda y=0$, substitute a <power series> $y=\sum_{n\geq0}a_nx^n$ to obtain the displayed <recurrence relation>. It separates into an <even> solution with $a_0=1,a_1=0$ and an <odd> solution with $a_0=0,a_1=1$:
$$
y_e=1-\lambda x^2+\frac{\lambda(\lambda-2)}6x^4+\cdots,\qquad y_o=x+\frac{1-\lambda}3x^3+\frac{(1-\lambda)(3-\lambda)}{30}x^5+\cdots.
$$
The ratio of successive terms within each parity tends to zero at every fixed complex $x$, giving infinite <radius of convergence>. The two initial-value vectors are independent, so these solutions span the solution space. A nonzero <polynomial> solution exists precisely when $\lambda=N$ is a nonnegative <integer>: its highest power forces this value, and the recurrence terminates the matching parity. These terminating solutions are nonzero <scalar multiples> of the <Hermite polynomials>. Multiplication by $e^{-x^2/2}$ connects them to <Hermite oscillator quantization>.
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