Hermite equationHermiteEquation2005-05-15 11:28:312009-11-07 12:02:47Definitiondifferential equation% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{thmplain}{Theorem}The linear differential equation
$$\frac{d^2f}{dz^2}-2z\frac{df}{dz}+2nf \;=\; 0,$$
in which $n$ is a real \PMlinkescapetext{constant}, is called the {\em Hermite equation}.\, Its general solution is\, $f := Af_1\!+\!Bf_2$\, with $A$ and $B$ arbitrary \PMlinkescapetext{constants} and the functions $f_1$ and $f_2$ presented as\\
\quad $f_1(z) \;:=\; z+\frac{2(1-n)}{3!}z^3+\frac{2^2(1-n)(3-n)}{5!}z^5+
\frac{2^3(1-n)(3-n)(5-n)}{7!}z^7+\ldots\!,$\\
\quad $f_2(z) \;:=\; 1+\frac{2(-n)}{2!}z^2+\frac{2^2(-n)(2-n)}{4!}z^4+
\frac{2^3(-n)(2-n)(4-n)}{6!}z^6+\ldots$\\
It's easy to check that these power series satisfy the differential equation.\, The coefficients $b_\nu$ in both series obey the recurrence \PMlinkescapetext{formula}
$$b_\nu \;=\; \frac{2(\nu\!-\!2\!-\!n)}{\nu(nu\!-\!1)}b_{\nu\!-\!2}.$$
Thus we have the \PMlinkname{radii of convergence}{RadiusOfConvergence}
$$R \;=\; \lim_{\nu\to\infty}\left|\frac{b_{\nu-2}}{b_\nu}\right| \;=\;
\lim_{\nu\to\infty}\frac{\nu}{2}\!\cdot\!\frac{1\!-\!1/\nu}{1\!-\!(n\!+\!2)/\nu} \;=\; \infty.$$
Therefore the series converge in the whole complex plane and define entire functions.
If the \PMlinkescapetext{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 \PMlinkname{degree}{PolynomialRing} \PMlinkescapetext{term} is $(2z)^n$ and called the Hermite polynomials.