= Odd polynomial solutions of the Hermite differential equation
{title2=$E=4j+2,\quad j\geq0$}
For $-W''+2xW'=EW$, the <power series> coefficients obey $(n+2)(n+1)a_{n+2}=(2n-E)a_n$. The condition $W(0)=0$ makes every even coefficient zero. A nonzero odd series terminates precisely when $E=2d$ with $d$ an odd positive integer. A polynomial of degree $d$ must have this value of $E$ by comparison of leading coefficients, so these are all possibilities. With $W'(0)=1$, the first three <polynomial> solutions are $x$, $x-2x^3/3$, and $x-4x^3/3+4x^5/15$, with $E=2,6,10$. They are scalar multiples of the odd <Hermite polynomials>.
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