PlanetPhysics/Hermite Equation

The linear differential equation ${\frac {d^{2}f}{dz^{2}}}-2z{\frac {df}{dz}}+2nf=0,$ 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.