===== recycled on Mon 10 May 2021 10:54:59 AM EDT by drorbn on Ubuntu-on-X2 ======


We recall the recipe~\eqref{eq:frakGComposition} for the composition of
generating functions $\xymatrix{ A\ar[r]^\calL & B \ar[r]^\calM & C}$ and
add a third version, the rightmost formula below, which treats $\calL$
and $\calM$ and Greek and Latin letters more symmetrically:
\begin{equation} \label{eq:frakGComposition2}
  \calL\act\calM = \left(\calL|_{z_b\to\partial_{\zeta_b}}\calM\right)_{\zeta_b=0}
  = \left(\calM|_{\zeta_b\to\partial_{z_b}}\calL\right)_{z_b=0}
  = \left.\bbe^{\sum\partial_{z_b}\partial_{\zeta_b}}(\calL\cdot\calM)\right|_{z_b=\zeta_b=0},
\end{equation}
where the indices $b$ run through the set $B$. Here
$\calL\cdot\calM$ stands for the ordinary product
of power series $\bbQ[z_B]\llbracket\zeta_A\rrbracket
\otimes \bbQ[z_C]\llbracket\zeta_B\rrbracket \to \bbQ[z_{A\cup
B}]\llbracket\zeta_{B\cup C}\rrbracket$.\footnote{Strictly speaking this
is valid only if there are no name clashes, namely if $A\cap B=B\cap
C=\emptyset$. That's a non-issue --- if needed the labels in $B$ can be
temporarily renamed before the formula is applied.} Thus we come to the following

===== recycled on Thu 13 May 2021 02:40:51 AM EDT by drorbn on Ubuntu-on-X2 ======


\begin{discussion} Thus for later use, we establish some vocabulary around
Gaussian expressions (the Gaussians themselves, not their integrals). A
{\em Gaussian} over some characteristic-0 ``ring of scalars'' $\Omega$
(which may be $\bbQ$ or $\bbR$ or $\bbC$, but could in itself be a
polynomial ring or something more exotic) is an expression of the
form $\omega\bbe^Q$, where $\omega\in\Omega$ is a scalar and $Q$ is a
quadratic $\sum q_{ij}z_iz_j$ in some finite set of variable $\{z_i\}$
with scalar coefficients $q_{ij}\in\Omega$. A {\em perturbed Gaussian} is
an expression of the form $\bbe^Q\calP$ where $Q$ is as before and $\calP$
is a polynomial in the $z_i$'s with scalar coefficients, or perhaps even
a power series provided it is ``small'' in some algebraic sense (meaning, belongs to some ideal of ``small
things''). Often $\calP$ in itself will be of the form
$\omega\bbe^P$, where $P$ is ``small''. In the special case where $Q$ is of the form $Q=\sum q_{ij}x_iy_j$ (namely,
when the set of variables is decomposed in two, $\{x_i\}\sqcup\{y_j\}$ and the quadratic $Q$ is actually linear in
each of $\{x_i\}$ and $\{y_j\}$ separately) we will say that $\omega\bbe^Q$ is a bipartite Gaussian. In this case
the perturbations $\calP$ can be allowed to be a bit more general: they can be power series in one set of variables
(say the $y_j$'s) provided they remain polynomials in the other set (the $x_i$'s). 

{\red MORE.}
\end{discussion}