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\begin{document}

\title{Approximate substitutions and the normal ordering
problem}

\author{H Cheballah$^{\dag}$,
G H E Duchamp$^{\dag}$, K A Penson$^{\diamondsuit}$ }
%
\address{
$^\dag$ Universit\'e Paris 13\\ Laboratoire d'Informatique Paris
Nord, CNRS UMR 7030\\99 AV.J-B; Cl\'ement 93430 Villetaneuse,
France.\vspace{2mm} }
\address
{$^\diamondsuit$ Laboratoire de Physique Th\'eorique des Liquides,\\
Universit\'e Pierre et Marie Curie, CNRS UMR 7600\\
Tour 24 - 2e \'et., 4 pl. Jussieu, F 75252 Paris Cedex 05, France}

\eads{\linebreak\mailto{hayat.cheballah@lipn-univ.paris13.fr},
\mailto{ghed@lipn-univ.paris13.fr},
\mailto{penson@lptl.jussieu.fr} \linebreak}

\begin{abstract}
In this paper, we show that the infinite generalised Stirling
matrices associated with boson strings with one annihilation are
projective limits of approximate substitutions. The latter being
characterised by a finite set of algebraic equations.
\end{abstract}


\section{Introduction}
The series of papers \cite{GBNOP,CPNOMFG,Gentle_intro,
one_param_group} had two sequels. First one, algebraic, was the
construction of a Hopf algebra  of Feynman-Bender diagrams
\cite{CPNOMFG} arising from the product formula applied to two
exponentials. Second one was the construction and description of
one parameter groups of infinite matrices
\cite{Gentle_intro,one_param_group} and their link with the
combinatorics of so called ``\textit{Sheffer polynomials}''. The
object of this paper is to continue the investigation of those
``\textit{One-parameter groups}'' by highliting the structure of
the group of substitutions.\\ It is showed here how we can see
this group as a projective limit as what will be called
``\textit{approximate substitutions}''

Now, we consider creation and annihilation operators with the
commutation relation
\begin{eqnarray}\label{[a,a+]=1}
% \nonumber to remove numbering (before each equation)
  [a,a^\dag] &=& 1
\end{eqnarray}
Recall that \cite{Blasiak}
\begin{itemize}
    \item Annihilation operator, in second quantification,
    represents an operator which changes each state of Fock space containing $N>1$ particles
for another containing $(N-1)$ particles. In one mode, one has
    $$a|N\rangle=
\sqrt{N}|N-1\rangle$$
    \item The hermitian conjugate of the annihilation operator is the creation
operator which changes each state of Fock space containing $N$
particles to another containing $(N+1)$ particles. Similarly one
has
    $$a^\dag|N\rangle=
\sqrt{N+1}|N+1\rangle$$
\end{itemize}

Starting from the vacuum $|0\rangle$, we can reach all the
normalized states in the Fock space. There are indeed given by:
\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
  |N\rangle &=&
  \prod_k\frac{(a^\dag)^{N}}{\sqrt{N_k}}|0\rangle
\end{eqnarray}

\section{Normally ordered form}
The noncommutativity of annihilation and creation operators may
cause problems in defining an operator function in quantum
mechanics. To solve these problems, we have to find some normal
form which allows computing, reduction and equality decision. One
of the simplest and widely used solution is the \textit{normally
ordered form} of the boson operators in which all $a^\dag$ stand
on the left of the factors $a$ \cite{Blasiak}.\\There are two well
known procedures defined on the boson expressions yielding a
normally ordered form. Namely $\mathcal N$, the \textit{Normal
ordering}
 and  $::$, the \textit{double dot operation}.
\subsection{Normal ordering}
By  normal ordering of a general expression $F(a^\dag,a)$, we mean
 $\mathcal{N}[F(a^\dag,a)]$ which is obtained by moving all the
 annihilation operators $a$ on the right using the commutation
 relation (\ref{[a,a+]=1}).
 \begin{exemple} Let  $w\in\{a,a^\dag\}^*$ a word given by
$w=aa^\dag aaa^\dag a$. The normal ordering of $w$ is
\begin{eqnarray*}
   aa^\dag aaa^\dag a &=&  (1+a^\dag a)a(1+a^\dag a)a=a^2+a^\dag a^3+aa^\dag a^2+a^\dag a^2 a^\dag a^2\\
   &=& a^2+a^\dag a^3+(1+a^\dag a)a^2+a^\dag a(1+a^\dag a)a^2\\
   &=& a^2+a^\dag a^3+a^2+a^\dag a^3+a^\dag a^3+a^\dag aa^\dag
   a^3=2 a^2+3a^\dag a^3+a^\dag(1+a^\dag a)a^3\\
   &=& \underbrace{2a^2+3a^\dag a^3+(a^\dag)^2a^4}
\end{eqnarray*}
 \end{exemple}
 \begin{remark}
Note that all c{\oe}fficients of the normal ordering of a word
(more precisely the c{\oe}fficients of the decomposition of a word
 on the basis $\Big((a^\dag)^j a^l\Big)$)
are positive integers. This suggests that these quantities count
things.
 \end{remark}
\subsection{Double dot operation}
The double dot operation $:F(a^\dag,a):$ is a similar procedure
using another commutation relation i. e. $[a,a^\dag]=0$ instead of
$[a,a^\dag]=1$, \textit{i.e} moving all annihilation operators $a$
to the right as if they were commuting with
the creation operators $a^\dag$.\\
\begin{remark}The double dot operation $: :$ is a linear operator
$$\C<a,a^\dag> \stackrel{::}{\longrightarrow} HW(\C)$$
which can be directly defined by:
$$\quad\quad\quad w\in\{a,a^\dag\}\longmapsto :w:=a^{\dag|w|_{a^\dag}}a^{|w|_{a}}$$
\end{remark}
\begin{exemple} We take the same example as above
 $w=aa^\dag aaa^\dag a$.\\
The double dot operation gives
\begin{eqnarray*}
:aa^\dag aaa^\dag a: &=& \overbrace{a^\dag a^\dag
aaaa}^{\mbox{$a^\dag$ on the left, $a$ on the right}}
\end{eqnarray*}
\end{exemple}

\section{Combinatorics of the normal ordering}
Although Bell and Stirling numbers have a purely combinatorial
origin, in this communication, we will rather consider them as
coefficients of the normal ordering problem.\cite{Blasiak}\\
The general word $w\in\{a,a^\dag\}^*$ with letters in
$\{a,a^\dag\}$ (Boson string) can be described by two sequences of
positive integers $r=(r_1,r_2,\cdots ,r_M)$ and $s=(s_1,s_2,\cdots
,s_M)$, so that we define
\begin{eqnarray}\label{H_rs_def}
% \nonumber to remove numbering (before each equation)
   w_{r,s} &=&
   (a^\dag)^{r_M}a^{s_M}\cdots(a^\dag)^{r_2}a^{s_2}(a^\dag)^{r_1}a^{s_1}
\end{eqnarray}
and $\displaystyle{d=\sum_{m=1}^n(r_m-s_m)\quad ,n=1,\cdots M}$
represents the excess (i. e. the number of creations in excess w.
r. t. the number of annihilations). Then, the normally ordered
form of $w^n$ is
given by\vspace{0.5mm}\\

$\mathcal N(w^n)=$$\left\{%
\begin{array}{ll}
     \displaystyle{(a^\dag)^{nd}\sum_{k=0}^{\infty}
S_{r,s}(n,k)(a^\dag)^ka^k,} & \hbox{if $d>0$;} \\
     \displaystyle{\Big(\sum_{k=0}^{\infty}
S_{r,s}(n,k)(a^\dag)^ka^k\Big) (a^\dag)^{n|d|},} & \hbox{else.} \\
\end{array}%
\right.$ \vspace{0.5mm}

 \vspace{0.5cm} Where the quantities $S_{r,s}(n,k)$ are generalizations of
standard Stirling
numbers.\cite{GBNOP,CPNOMFG}\\
The generalized Bell polynomials $B_{r,s}(n,x)$ and the
generalized Bell numbers $B_{r,s}(n)$ are respectively
\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
  B_{r,s}(n,x) &=& \sum_{k=0}^\infty S_{r,s}(n,k) x^k \\
  B_{r,s}(n,1) &=& B_{r,s}(n)=\sum_{k=0}^\infty S_{r,s}(n,k)
\end{eqnarray}
\begin{exemple}$ $\\
\begin{description}
    \item  [$\bullet\quad w=a^\dag a$]$ $, we get the usual matrix of
\textit{Stirling numbers of second kind}\\

    $$
\left(%
\begin{array}{cccccccc}
  1 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots  \\
  0 & 1 & 0 & 0 & 0 & 0 & 0 & \cdots \\
  0 & 1 & 1 & 0 & 0 & 0 & 0 & \cdots \\
  0 & 1 & 3 & 1 & 0 & 0 & 0 & \cdots \\
  0 & 1 & 7 & 6 & 1 & 0 & 0 & \cdots \\
  0 & 1 & 15 & 25 & 10 & 1 & 0 & \cdots \\
  0 & 1 & 31 & 90 & 65 & 15 & 1 & \cdots \\
 \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & &\ddots \\
\end{array}%
\right)
$$

    \item[$\bullet\quad w=a^\dag aa^\dag$], we have\\
$$
\left(%
\begin{array}{cccccccc}
  1 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots  \\
  1 & 1 & 0 & 0 & 0 & 0 & 0 & \cdots \\
  2 & 4 & 1 & 0 & 0 & 0 & 0 & \cdots \\
  6 & 18 & 9 & 1 & 0 & 0 & 0 & \cdots \\
  24 & 96 & 72 & 16 & 1 & 0 & 0 & \cdots \\
  120 & 600 & 600 & 200 & 25 & 1 & 0 & \cdots \\
  720 & 4320 & 5400 & 2400 & 450 & 36 & 1 & \cdots \\
 \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & &\ddots \\
\end{array}%
\right)
$$


\vspace{1.0cm}

    \item [$\bullet\quad w=a^\dag aaa^\dag a^\dag$], one gets\\
    $$
\left(%
\begin{array}{cccccccccc}
  1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots  \\
  2 & 4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 &  \cdots \\
  12 & 60 & 54 & 14 & 1 & 0 & 0 & 0 & 0 &  \cdots \\
  144 & 1296 & 2232 & 1296 & 306 & 30 & 1 & 0 & 0 &  \cdots \\
  2880 & 40320 & 109440 & 105120 & 45000 & 9504 & 1016 & 52 & 1 &  \cdots \\
 \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & &\ddots \\
\end{array}%
\right)
$$
\end{description}
\end{exemple}

\vspace{0.5cm}

\begin{remark}
In each case, the matrix $S_w$ is of staircase form and the
dimension of the step is the number of $a$'s in the word $w$.
Thus, all the matrices are row-finite and unitriangular iff the
number of its annihilation operators $r=1$. Moreover, the first
column is $(1,0,\cdots ,0,\cdots)$ iff $w$ ends with $a$ (this
means that
$\mathcal N(w^n)$ has no constant term for all $n>0$).\cite{Gentle_intro,one_param_group}.\\

A word $w$ with only one annihilation operator being written We
write $w$ in the unique form $w=(a^\dag)^{r-p}a (a^\dag)^p$, one
has:
\begin{itemize}
    \item if $p=0$, $S_w$ is the matrix of a \textit{unipotent
    substitution}.
    \item if $p>0$, $S_w$ is the matrix of a \textit{unipotent substitution with
    prefunction}.
\end{itemize}
\end{remark}

\section{Approximate substitutions}
We define here the  transformation matrices (space, topology and
decomposition), then we concentrate on the \textit{Riordan
subgroup} (\textit{i.e} transformations which are substitutions
with prefactor functions).\cite{Gentle_intro,one_param_group}.\\

Let $\C^\N$ be the vector space of all complex sequences, endowed
with the Frechet product topology. The algebra $\mathcal L(\C^\N)$
of all continuous operators $\C^{\N}\longrightarrow \C^{\N}$ is
the space of row-finite matrices with complex coefficients. For a
sequence $A=(a_n)_{n\geq 0}$, the transformed sequence $B=MA$ is
given by $B=(b_n)_{n\geq 0}$ with:
\begin{eqnarray}\label{suite-transform�e-bn}
% \nonumber to remove numbering (before each equation)
  b_n &=& \sum_{k\geq 0} M(n,k)a_k.
\end{eqnarray}
We may associate a univariate series with a given sequence
$(a_n)_{n\in\N}$ using a sequence of denominators $d_n=n!$ as
follows
\begin{eqnarray}\label{suite_transform_serie_univari�e}
% \nonumber to remove numbering (before each equation)
   \sum_{n\geq 0}a_n\frac{z^n}{n!}.
\end{eqnarray}
\subsection{Substitution with prefunction}
We now examine an important class of transformations of $\mathcal
T$ as well as their diagonals: the substitutions with
prefunctions.

We consider, for a generating function $f$ the transformation
\begin{eqnarray}\label{transformation_prefonction}
% \nonumber to remove numbering (before each equation)
  \mathcal T_{g,\phi}[f](x) &=& g(x)f\Big(\phi(x)\Big)
\end{eqnarray}
where $g$ and $\phi$ are generating functions and $\mathcal T$ is
a linear application.\\
The matrix of this transformation $M_{g,\phi}$ is given by the
transforms of the monomials $\frac{x^k}{k!}$ hence
\begin{eqnarray}\label{transformation_matrice}
% \nonumber to remove numbering (before each equation)
  \sum_{n\geq 0}M_{g,\phi}(n,k)\frac{x^n}{n!} =
  \mathcal T_{g,\phi}\Big[\frac{x^k}{k!}\Big]=g(x)\Big[\frac{x^k}{k!}\Big](\phi(x))
  =g(x)\frac{\phi(x)^k}{k!}
\end{eqnarray}
\begin{remark}
Eq. (\ref{transformation_matrice}) is called the \textit{Sheffer
condition} and, for EGF it amounts to state that
\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
  \sum_{n,k\geq 0}T(n,k)\frac{x^n}{n!}y^k &=& g(x)e^{y\phi (x)}
\end{eqnarray}
where $T\in\mathcal L(\C^\N)$ is a matrix with non-zero two first
columns $(T=M_{g,\phi})$.
\end{remark}
\subsection{Approximate substitutions}

We give, in this section, a way to determine whether an unipotent
matrix  is a matrix of substitution.
\begin{definition}
Let $\mathcal M_{[n+1\times n+1]}$ a given unipotente matrix.
$\mathcal M$ is said unipotente matrix of substitution if it
satisfies the following condition:
\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
  c_k &=& c_0\frac{\Big(\frac{c_1}{c_0}\Big)^k}{k!}\label{condition de substitution}
\end{eqnarray}
\begin{figure}[htbp]
\includegraphics[width=12cm]{presentation_calck}
\end{figure}

Where $c_k$ represents the exponential generating series of the
$k^{th}$ column and $c_0$, $c_1$ are respectively the exponential
generating series of the $1^{st}$ and the $2^{nd}$ column.
\end{definition}



We consider now the set of matrices with complex c{\oe}fficients
noted by $\C^{\N\times\N}$ and let $\C^{[0\cdots n]\times [0\cdots
n]}$ all matrices of size $(n+1)$ obtained by the truncation of
$\C^{\N\times\N}$ with $r_n$. Then
$$r_n:\C^{\N\times\N}\longrightarrow \C^{[0\cdots n]\times [0\cdots
n]}$$ $r_n$ is not a morphism.\\We consider now $\mathcal
T(\N,\C)$ the algebra of lower triangular matrices and $T([0\cdots
n],\C)$ the matrices of size $(n+1)$ obtained by the truncation
$\tau_n$. Then
$$\tau_n:\mathcal T(\N ,\C)\longrightarrow T([0\cdots n],\C)$$
$\tau_n$ is a morphism.
$$
\xymatrix@C=50pt{
\C^{\N\times\N}\ar[r]^{r_n}& \C^{[0\cdots n]\times [0\cdots n]} \\
\mathcal T(\N ,\C)\ar[r]^{\tau_n}\ar@{^{(}->}[u]^{\mathcal J_1}
&T([0\cdots n],\C)\ar@{^{(}->}[u]^{\mathcal J_2} }
$$
Where $\mathcal J_1$ and $\mathcal J_2$ are two canonical
injections.
\begin{remark}
We can write
\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
  \mathcal T(\N,\C) &\displaystyle{=\lim_{\longleftarrow}}& (T(\N,\C))
\end{eqnarray}
wich means that $(T(\N,\C))$ is the projective limit of $ \mathcal
T(\N,\C)$
\end{remark}
\subsection{Random generation}
Our motivation in this section consists in approximating the
matrices of infinite substitutions by finite matrices of
(approximate) substitutions.  We are then interested in the
probabilistic study of these matrices. To be done, we randomly
generate unipotent (unitriangular) matrices  and we observe the
number of apparitions of
matrices of substitutions.\\
The construction of unipotent matrices is as follows:
\begin{enumerate}
    \item Create an identity matrix.
    \item Fill randomly the lower part of the identity
    matrix (under the diagonal) using numbers that follow a
    uniform law of $[0,1]$.
    \item  Multiply those numbers by the range previously chosen.
    \item Test if the  built unipotent matrix satisfies Eq.(\ref{condition de substitution})

\end{enumerate}
We start by giving some examples of our experiment which are
summarized in the table below:
\begin{figure}[htbp]
\includegraphics[width=14cm]{presentation_tableau}
\end{figure}

\begin{remark}
According to the results obtained, we observe that the
substitutions matrices are not very frequent. However, in meeting
certain conditions such as size, the numbers of drawing and the
range of variables, we can obtain positive probabilities that
these matrices appear.\\ Let us note that the more the size of the
matrix is small the more one is likely to obtain a matrix of
substitution with a reasonable number of drawings.\\
 We also notice that, if
we vary the range of variables, and this in an increasing way and
by keeping unchanged the number of drawings and size, the
probability tends to 0. We also notice that all the unipotent
matrices of size 3 are all matrices of approximate substitutions.
This is easy to see because the exponential generating series of
the 3$^{rd}$ column
 will always have the form $c_k=\frac{x^2}{2!}$.\\ Thus we can say
that the test actually starts   from the matrices of  size higher
or equal to 4.
\end{remark}
\begin{resultat}
Let $r$ represent the range of variables and $n$ the size of
matrix.\\According to the results obtained, we can say that the
probability $p$ of apparition of the matrices of substitutions
depends on $r$ and $n$ and we have the following upper bound:
\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
  p_{n} &\leq& \frac{r^{2n-3}}{r^{\frac{n(n-1)}{2}}}
\end{eqnarray}
which shows that

\begin{center}
\begin{tabular}{c}

  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  $p_n\longrightarrow 0$ \\
  $n\longrightarrow \infty$ \\
\end{tabular}\end{center}

\end{resultat}

\begin{conjecture}One can conjecture that the effect of the
denominators vanishes at infinity. More precisely that:
\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
  p_{n} &\sim& \frac{r^{2n-3}}{r^{\frac{n(n-1)}{2}}}
\end{eqnarray}
\end{conjecture}

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%
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\end{thebibliography}

\end{document}
