which is again a Hilbert space of analytic functions, and for which the vectors (\[H3\]) form an orthonormal basis, is a subspace of the Hilbert space $L^2 (\mathbb C , d\mu_s )$, where $$d\mu_s (x,y) = e^{-[(1-s)x^2 + (\frac 1s -1)y^2]}\; dx\; dy\; .
\label{s-measure}$$ Similarly, the polynomials $h_{n,s} (\overline{z})$ would span a reproducing kernel Hilbert space $\mathfrak H_{a-hol} \subset L^2 (\mathbb C , d\mu_s )$ of anti-analytic functions, with reproducing kernel $\overline{K_s(z,\overline{w})}$.

The polynomials $H_{n,m}(z,\overline{z})$ {#subsec-comp-herm}
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A second class of complex Hermite polynomials have been studied in [@Wil]. These are defined, for positive integers $n,m$, as $$\label{E1}
H_{n,m}(z,\overline{z})=(-2)^{n+m}\text{exp}(z\overline{z}/2)\left(\frac{\partial}{\partial z}\right)^n\left(\frac{\partial}{\partial\overline{z}}\right)^m\text{exp}(-z\overline{z}/2).$$ The generating function for this polynomial is $$\label{E2}
\text{exp}[(a\overline{z}+\overline{a}z-a\overline{a})/2]=\sum\frac{(a/2)^n(\overline{a}/2)^m}{n!m!}H_{n,m}(z,\overline{z}).$$ When $m\geq n$ and $n\geq 0$, the complex Hermite polynomials can also be written as $$H_{n,m}(z,\overline{z})=\frac{m!}{(m-n)!}z^{m-n}(-2)^n{}_1F_1(-n,m-n+1,z\overline{z}/2).$$ These polynomials satisfy the orthogonality relation $$\label{E3}
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\text{exp}(-z\overline{z}/2)H_{n,m}(z,
\overline{z})\overline{H_{\nu,\mu}(z,\overline{z})}\; dx\; dy =2\pi \delta_{n \nu}\delta_{m \mu}n!m!2^{n+m}.$$ Now let $$\label{E4}
h_{n,m}(z,\overline{z})=(-1)^{n+m}\text{exp}(|z|^2)\left(\frac{\partial}{\partial z}\right)^n
\left(\frac{\partial}{\partial\overline{z}}\right)^m\text{exp}(-|z|^2)=\sum_{i=0}^{n}
\sum_{j=0}^{m}a_{ij}z^i{\overline{z}}^j.$$ From (\[E3\]) (see also [@Nic; @Ghan] ), the polynomials $h_{n,m}(z,\overline{z})$ form a complete orthogonal system in the Hilbert space $L^2(\mathbb{C}, e^{-|z|^2}d\lambda)$, where $d\lambda=\frac{1}{\pi}d^2z$ is the Lebesque measure on $\mathbb{C}$. For proof of completeness see [@Ghan]. Also from (\[E3\]) we have $$\label{E5}
\int_{\mathbb{C}}e^{-|z|^2}h_{n,m}(z,\overline{z})\overline{h_{n,m}(z,\overline{z})}\; d\lambda
=n!m!$$ and thereby $$\|h_{n,m}\|_{L^2}=\sqrt{n!m!}.$$ The operator $$\label{E6}
\mathfrak{L}=-\frac{1}{4}\left\{4\frac{\partial^2}{\partial z\partial{\overline{z}}}+
