PlanetPhysics/Hermite Equation

The linear [[../DifferentialEquations/|differential equation]] $\frac{d^{2}f}{d{z}^2}-2z\frac{df}{dz}+2nf=0,$ in which $n$ is a real constant, is called the Hermite equation .\, Its general solution is\, $f:=A{f}_1+B{f}_2$\, with $A$ and $B$ arbitrary constants and the [[../Bijective/|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/|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 $R=\underset{\nu \to \mathrm{\infty }}{\lim }\left|\frac{b_{\nu -2}}{b_{\nu }}\right|=\underset{\nu \to \mathrm{\infty }}{\lim }\frac{\nu }{2}\cdot \frac{1-1/\nu }{1-(n+2)/\nu }=\mathrm{\infty }.$ 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 [[../HermitePolynomials/|Hermite polynomials]].