 Power series solutions of the Hermite differential equation

For $y^{\prime \prime }-2x{y}^{\prime }+2\lambda y=0$, substitute a power series $y={\sum }_{n\ge 0}{a}_n{x}^n$ to obtain the displayed recurrence relation. It separates into an even solution with $a_0=1,a_1=0$ and an odd solution with $a_0=0,a_1=1$: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.