PlanetPhysics/Hermite Equation

The linear differential equation $$\frac{d^2f}{dz^2}-2z\frac{df}{dz}+2nf = 0,$$ in which $$n$$ is a real constant, is called the Hermite equation .\, Its general solution is\, $$f := Af_1\!+\!Bf_2$$\, with $$A$$ and $$B$$ arbitrary constants and the functions $$f_1$$ and $$f_2$$ presented as\\

\quad $$f_1(z) := z+\frac{2(1-n)}{3!}z^3+\frac{2^2(1-n)(3-n)}{5!}z^5+ \frac{2^3(1-n)(3-n)(5-n)}{7!}z^7+\cdots\!,$$\\

\quad $$f_2(z) := 1+\frac{2(-n)}{2!}z^2+\frac{2^2(-n)(2-n)}{4!}z^4+ \frac{2^3(-n)(2-n)(4-n)}{6!}z^6+\cdots$$\\

It's easy to check that these power series satisfy the differential equation.\, The coefficients $$b_\nu$$ in both series obey the recurrence formula $$b_\nu = \frac{2(\nu\!-\!2\!-\!n)}{\nu(nu\!-\!1)}b_{\nu\!-\!2}.$$ Thus we have the radii of convergence $$R = \lim_{\nu\to\infty}\left|\frac{b_{\nu-2}}{b_\nu}\right| = \lim_{\nu\to\infty}\frac{\nu}{2}\!\cdot\!\frac{1\!-\!1/\nu}{1\!-\!(n\!+\!2)/\nu} = \infty.$$ Therefore the series converge in the whole complex plane and define entire functions.

If the constant $$n$$ is a non-negative integer, then one of $$f_1$$ and $$f_2$$ is simply a polynomial function.\, The polynomial solutions of the Hermite equation are usually normed so that the highest degree term is $$(2z)^n$$ and called the Hermite polynomials.