\section{Notation for the VD distribution table}\label{app:vd_notation}
This appendix defines the symbols used in Table~\ref{tab:vd_distributions}.
For $\nu>-1/2$, the normalized Bessel functions are
\[
\mathcal J_\nu(t)=\Gamma(\nu+1)\left(\frac2t\right)^\nu J_\nu(t),
\qquad
\mathcal I_\nu(t)=\Gamma(\nu+1)\left(\frac2t\right)^\nu I_\nu(t),
\]
interpreted by their even power series, with value $1$ at zero.
The symmetric beta-type density in row 9 is proportional to
$(1-x^2)^{\nu-1/2}\mathbf1_{\{|x|<1\}}$.
The Student characteristic function is
\[
T_\nu(t)=
\frac{(\sqrt\nu|t|)^{\nu/2}K_{\nu/2}(\sqrt\nu|t|)}
{2^{\nu/2-1}\Gamma(\nu/2)},\qquad T_\nu(0)=1.
\]
Here $J_\nu$, $I_\nu$, and $K_\nu$ denote the ordinary Bessel,
modified Bessel, and modified Bessel functions of the second kind,
respectively. For the generalized normal family,
\[
f_p(x)=\frac{p}{2\Gamma(1/p)}e^{-|x|^p},\qquad
F_p(t)=\frac{p}{\Gamma(1/p)}
\int_0^\infty e^{-x^p}\cos(tx)\,dx,\qquad p>0.
\]
The generalized L\'evy--Bessel functions, whenever defined under the
construction hypotheses, are
\[
\mathcal J_\Psi(t)=\sum_{n=0}^\infty
\frac{(-1)^nt^{2n}}{\prod_{k=1}^n\Psi(k)},\qquad
\mathcal I_\Psi(t)=\sum_{n=0}^\infty
\frac{t^{2n}}{\prod_{k=1}^n\Psi(k)},
\]
where the empty product is $1$.