= Solution
For the normalized second-order <linear differential equation> $y''+P(z)y'+Q(z)y=0$, an <ordinary point> $z_0$ has analytic $P,Q$ near $z_0$. At a <regular singular point>, which is not ordinary, $(z-z_0)P(z)$ and $(z-z_0)^2Q(z)$ are analytic. Ordinary points admit two independent convergent <power series> solutions. Regular singular points instead call for the <Frobenius method>, which permits leading powers and, in some cases, <logarithms>.
Here zero is an <ordinary point>. Substitute $y=\sum_{n\geq0}a_nz^n$ into the <Hermite differential equation> and equate coefficients:
$$
(n+2)(n+1)a_{n+2}+(\lambda-2n)a_n=0,\qquad\boxed{a_{n+2}=\frac{2n-\lambda}{(n+2)(n+1)}a_n.}
$$
The even and odd initial coefficients are independent, giving
$$
y=a_0\left[1-\frac\lambda2z^2+\frac{\lambda(\lambda-4)}{24}z^4+\cdots\right]+a_1\left[z+\frac{2-\lambda}{6}z^3+\frac{(2-\lambda)(6-\lambda)}{120}z^5+\cdots\right].
$$
These are the <power series solutions of the Hermite differential equation>, with the present parameter equal to twice the parameter in that article. The ratio of successive terms within each nonterminating parity is asymptotically $2|z|^2/n$, so both series have infinite <radius of convergence>.
For a nonzero <polynomial> of degree $m$, its highest coefficient in $-2zy'+\lambda y$ forces $\lambda=2m$. Conversely at that value the matching parity terminates. Hence
$$
\boxed{\lambda=2m,\qquad m=0,1,2,\ldots.}
$$
The <polynomial> solutions are <scalar multiples> of the <Hermite polynomials>. Two examples are $\boxed{\lambda=0:\ y=1;\qquad\lambda=2:\ y=z}$; direct substitution verifies both. The other independent solution generally remains nonpolynomial.
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