Proposition 18.41.1.85EDCA2D872C For every $m>0$ and integer $q\geq 0$,
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18.41.2
\begin{equation} K_{m,r}^{(q)}(z,w)=\frac r\pi \sum _{k=q}^\infty \frac{m^{(k+1)/r}(z\bar w)^ k}{\Gamma ((k+1)/r)}. \end{equation}
The series and all its derivatives converge uniformly on compact sets. For $\zeta \in \mathbb C$,
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18.41.3
\begin{equation} m^{-1/r}B_{m,r}^{(q)}(m^{-1/(2r)}\zeta ) =\frac r\pi \mathrm{e}^{-|\zeta |^{2r}} \sum _{k=q}^\infty \frac{|\zeta |^{2k}}{\Gamma ((k+1)/r)}. \end{equation}
Consequently, if $m$ tends to infinity through positive integers and $q=q_ m\to \infty $, the left side tends to zero uniformly on compact $\zeta $-sets.
Proof.
The monomials are orthogonal by rotational invariance. For every integer $k\geq 0$, polar integration with radial variable $\rho $ and the substitution $t=m\rho ^{2r}$ give
\[ \| z^ k\| _{m,r}^2=2\pi \int _0^\infty \rho ^{2k+1}\mathrm{e}^{-m\rho ^{2r}}\, \mathrm{d}\rho =\frac\pi r m^{-(k+1)/r}\Gamma ((k+1)/r). \]
For an entire function $f(z)=\sum _{k\geq 0}a_ kz^ k$, Parseval's identity on circles and Tonelli's theorem yield $\| f\| _{m,r}^2=\sum _{k\geq 0}|a_ k|^2\| z^ k\| _{m,r}^2$. Thus the normalized monomials form an orthonormal basis, and those with $k\geq q$ form an orthonormal basis of the selected subspace. This proves (18.41.2). By Stirling's formula, the reciprocal Gamma coefficients decay faster than every fixed geometric progression, even after multiplication by any fixed power of $k$; this gives uniform convergence on compact sets together with all derivatives. Substituting $z=w=m^{-1/(2r)}\zeta $ gives (18.41.3). On a disk $|\zeta |\leq R$ of fixed radius, the series in (18.41.3) is bounded by the tail, from the index $q$ on, of the convergent series with $|\zeta |$ replaced by $R$. This tail tends to zero as $q\to \infty $, proving the last assertion.
$\square$
Pome
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