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\topmatter
\title
WEIGHTED DERANGEMENTS AND THE LINEARIZATION \\
 COEFFICIENTS OF ORTHOGONAL SHEFFER POLYNOMIALS
\endtitle
\author Jiang Zeng \endauthor
\subjclass{A. M. S. (1980) subject classification : 05A15, 33A65}\endsubjclass
\keywords{Sheffer, Hermite, Charlier, Laguerre,
Meixner, Meixner-Pollaczek polynomials, linearization coefficients, moments, 
derangement, permutation, partition, excedance, maximum.}\endkeywords
\abstract{The present paper is devoted to a systematic study of the 
combinatorial interpretations of the moments
and the linearization coefficients of the orthogonal Sheffer polynomials, 
i.e., Hermite, Charlier, Laguerre, Meixner and Meixner-Pollaczek polynomials.
In particular, we show that Viennot's combinatorial interpretations of the 
moments can be derived directly from their classical analytical expressions
and that the linearization coefficients of Meixner-Pollaczek polynomials 
have an analogous interpretation in the model of derangements as 
those of Laguerre and Meixner polynomials.}\endabstract
\endtopmatter
\document
\heading{1. Introduction}\endheading
Let $\{p_n(x):n=0,1,2, \ldots\}$ be a sequence of polynomials, orthogonal
on the real line with respect to a positive measure $d\alpha (x)$. 
One of the questions concerning the polynomials $p_n(x)$ 
is to determine, or at least say something useful about, the coefficients 
$a(k,n,m)$ in the expansion of 
$$
p_n(x)p_m(x)=\sum_{k=0}^{n+m}a(k,n,m)\,p_k(x).\tag 1.0
$$
This is usually called the linerization problem in the literature \cite{4}.
Taking orthogonality into account, we have then the equivalent equation :
$$\int_{-\infty}^{+\infty} p_n(x)p_m(x)p_k(x)d\alpha (x)=
a(k,n,m)\int_{-\infty}^{+\infty}p_k(x)p_k(x)d\alpha (x).$$
The problem thus turns out to deal with the integrals 
of a product of three polynomials. More generally, we can define
{\sl the linearization coefficients} of $p_n(x)$, abbreviated by {\sl l.c.} in 
what follows, to be the following 
integral of a product of $m$ arbitrary polynomials :
$$\int_{-\infty}^{+\infty}\prod_{i=1}^m p_{n_i}(x)d\alpha (x).\leqno(1.1)$$ 
Any such integral can, of course, be evaluated in finite terms by inserting 
the polynomial expressions for $p_{n_i}(x)$ and integrating term by term.
However, as pointed by several authors \cite{{4, 20, 37}}, 
the interesting problem is not just any representation of (1.1) 
but the one in which the nonnegativity of the integral is apparent for some
range of values of the parameters. Although it is well-known that the integral (1.1)
is positive for most classical orthogonal polynomials \cite{3}, there
is generally no simple expression for (1.1). 
In the past two decades, many authors also considered the refined problem 
of finding combinatorial interpretations of $(1.1)$ 
for the classical orthogonal polynomials 
(See, for example, \cite{{6, 7, 8, 14, 17, 
20, 29, 22, 40, 41}}).
In view of their analytical aspects, it should be interesting 
to find a geometric setup of which the generating polynomial
with {\sl positive weight} is the integral (1.1) 
and which reduces to orthogonality when $m=2$.
 The main object of the present paper is to show that 
such a combinatorial interpretation does exist for 
all the {\sl orthogonal Sheffer polynomials}. 
 
Recall that a sequence of polynomials 
$p_1(x),\, p_2(x),\ldots$ is said to be 
Sheffer type if and only if its exponential generating function  
 has the form :
$$\sum_{n\geq 0}p_n(x){t^n\over n!}=f(t)\exp(xg(t)),\leqno (1.2)$$
where $f(t)$, $g(t)$ are formal power series with $f(0)=1$, $g(0)=0$
and $g'(0)\not=0$.
As proved by Meixner \cite{{12, 34}}, there are only five classes
 of orthogonal Sheffer polynomials, i.e., 
the Hermite, Laguerre, Charlier, Meixner and Meixner-Pollaczek polynomials. 
To be more precise, 
we now give the explicit generating functions and weight functions of these  
polynomials. Throughout this paper, we denote by
$(x)_n$ the shifted factorial $(x)_n =x(x+1)\cdots (x+n-1)$.\newline
(a) Hermite polynomials $(H_n(x))_{n\geq 0}$ :
$$\sum_{n\geq 0}H_n(x){t^n\over n!}=\exp{(2xt-t^2)},\leqno (1.3)$$
orthogonal on ${\Bbb R}$ with respect to the weight function 
$x\mapsto e^{-x^2}$.\newline
(b) Charlier polynomials $(C_n(x;a))_{n\geq 0}$ :
$$\sum_{n\geq 0}C_n(x;a){t^n\over n!}=e^t\bigl(1-{t\over a}\bigr)^x,
\leqno (1.4)$$
orthogonal on ${\Bbb Z_+}$ with respect to the weight function 
$x\mapsto a^x/x!$.\newline
(c) Laguerre polynomials $(L_n^{(\alpha)}(x))_{n\geq 0}$ :
$$\sum_{n\geq 0}L_n^{(\alpha)}(x){t^n\over n!}=(1-t)^{-\alpha-1}
\exp\bigl({xt\over 1-t}\bigr),\leqno (1.5)$$
orthogonal on ${\Bbb R}_+$ with respect to the weight function
$x\mapsto x^\alpha e^{-x}$.\newline 
(d) Meixner polynomials $(M_n(x;\beta,c))_{n\geq 0}$ :
$$\sum_{n\geq 0}M_n(x;\beta,c){t^n\over n!}=\bigl(1-{t\over c}\bigr)^x
(1-t)^{-x-\beta}, \leqno (1.6)$$
orthogonal on ${\Bbb Z}_+$ with respect to the weight function 
$x\mapsto (\beta)_x c^x/x!.$\newline
(e) Meixner-Pollaczek polynomials $(P_n(x;\delta,\eta))_{n\geq 0}$ :
 $$\sum_{n\geq 0}P_n(x;\delta,\eta){t^n\over n!}
=\bigl[(1+\delta t)^2+t^2\bigr]^{-\eta /2}
\exp\Bigl[x \arctan \bigl({t\over 1+\delta t}\bigr)\Bigr],\leqno (1.7)$$
orthogonal on ${\Bbb R}$ with respect to the weight function ( here a 
misprint in \cite{{12, 40}} has been corrected.)
$$x\mapsto [\Gamma (\eta/2)]^{-2}\bigl|\Gamma \bigl((\eta+ix)/2\bigr)
\bigr|^2\exp\bigl(x\arctan\delta\bigr).$$ 
For a fixed {\sl positive} integer $m$, let $\n$ be a sequence 
$\n=(n_1,\ldots,n_m)$ of $m$ {\sl nonnegative} integers.
The linearization coefficients of the five sets of the Sheffer polynomials can 
then be defined, with appropriate normalizations, as follows.
$$\align
{\Cal H}({\n}) &={1\over \sqrt\pi }\int_{-\infty}^{+\infty}
\left(\shave{\prod_{i=1}^m}2^{-n_i/2}H_{n_i}(x)\right)e^{-x^2}dx.\tag 1.8\\
{\Cal C}(\n;a) &=e^{-a}\sum_{x=0}^\infty\left(\shave{\prod_{i=1}^m}
(-a)^{n_i}C_{n_i}(x;a)\right){a^x\over x!}.\tag 1.9\\
{\Cal L}({\n};\alpha) &={1\over \Gamma(\alpha +1)}\int_0^{+\infty}
\left(\shave{\prod_{i=1}^m}(-1)^{n_i}L_{n_i}^{(\alpha)}(x)\right)
x^\alpha e^{-x}dx.\tag 1.10\\ 
{\Cal M}(\n;\beta,c) &=(1-c)^\beta\sum_{x=0}^\infty \left(\shave 
{\prod_{i=1}^m(-c)^{n_i}}M_{n_i}(x;\beta,c)\right)
{c^x(\beta)_x\over x!}.\tag 1.11\\
{\Cal P}(\n;\delta,\eta) &={1\over \int_{-\infty}^{+\infty}w(x)dx}
 \int_{-\infty}^{+\infty}\left(\shave{\prod_{i=1}^m}
 P_{n_i}(x; \delta, \eta)\right)w(x)dx,\tag 1.12
\endalign $$ 
where  $w(x)=[\Gamma (\eta/2)]^{-2}\bigl|\Gamma \bigl((\eta+ix)/2\bigr)
\bigr|^2\exp\bigl(x\arctan\delta\bigr).$ 
 
Although the combinatorial aspects of the {\sl l.c.} of simple Laguerre 
polynomials $L_n^{(0)}(x)$ were first studied
(cf. \cite{{17, 6, 29, 10}}), the interpretation of 
the {\sl l.c.} of the general Laguerre polynomials was found only 
recently by Foata and Zeilberger \cite{20}. 
The interpretation of the {\sl l.c.} of Hermite polynomials 
is due to Godsil \cite{26} and Azor, Gillis and Victor 
\cite{8}. The combinatorial interpretations of the 
{\sl l.c} of Charlier and Meixner polynomials were recently 
found by Zeng \cite{41}. Thus, as far as 
the linerization problems of Sheffer polynomials are concerned,
there is still one interesting question remaining : Is there an analogous
interpretation for the linearization coefficients of Meixner-Pollaczek
polynomials ? 
The object of this paper is to give an affirmative answer to this question, 
but we also present a new approach
to the linearization problem, which is based on generating 
functions and the interpretation of the moments.

Various approaches have been used hitherto to investigate
the combinatorial interpretations of the linearization coefficients.  
In the earliest paper \cite{17}, Even and Gillis showed, by identifying
the recurrence relations, that the {\sl l.c.} of the simple Laguerre polynomials counts the
``generalized derangements'' : rearrangements of a {\sl multiset} of objects
of different ``colours'' with the property that each object goes to an
object of a different colour. Shortly after this, Askey and Ismail \cite{{6}} 
observed that this result can be derived
more directly by identifying the generating function of the integral (1.10) 
($\alpha =0$) with that of the number of derangements, 
which was evaluated by MacMahon's Master theorem. 
Similarly, by refining the counting of derangements, they also
 found a combinatorial interpretation of 
the {\sl l.c.} of Meixner polynomials $M_n(x; c,\beta)$ for $\beta=1$ \cite{6}.
In addition, Carlitz \cite{10} also gave a short proof of Even and Gillis' result
by carrying out the integral (1.10) explicitly, always in $\alpha=0$ case, 
and then applying an inversion formula or the principle of 
inclusion-exclusion. 
One decisive step was made by Foata and Zeilberger \cite{20}
in order to interpret the general $\alpha$ of (1.10).
Instead of multiset, they consider the permutations of 
a total ordered ``colored set'', which permits them to 
introduce the notion of {\sl cycle}.
 Based on this geometric setup, Foata and Zeilberger \cite{20} 
and Zeng \cite{41} successfully generalized 
both Askey-Ismail and Carlitz' methods to
figure out the interpretations of the {\sl l.c.}
of Laguerre, Meixner, Charlier and Krawchouk 
polynomials, by using especially the 
{\sl exponential formula} and the $\beta$-extension of the MacMahon
Master theorem.  

Another different approach was initiated by Jackson \cite{29}. He 
observed that the result of Even and Gillis \cite{17} 
can be naturally derived from the interpretation of the {\sl moments} and 
from the {\sl rook polynomial} interpretations of the simple Laguerre
polynomials \cite{38}. This approach has the advantage of relating naturally
 the three different objects  : the (rook) polynomials, 
the moments, and the linearization coefficients.
Similarly, Godsil \cite{26} has given an interpretation of the 
{\sl l.c.} of Hermite polynomials, but instead of rook polynomials 
he used {\sl matching polynomials}. This method has later been further
developed by De Sainte-Catherine and Viennot \cite{14} 
to provide an interpretation of the {\sl l.c.} of Tchebycheff polynomials
of the second kind; they also found the explicit killing involution
under the principle of inclusion-exclusion involved in the three above 
proofs. More recently, Gessel \cite{22} produced a common
generalization of rook and matching polynomials to give, among others, new 
proofs of the known interpretations of the {\sl l.c.} of the Hermite, 
general Laguerre, and Charlier polynomials. 

Our interpretation of the {\sl l.c.} of the Meixner-Pollaczek polynomials
is again based on the model of derangements. Indeed, we will show that
integral (1.12) counts the derangements according to the three statistics :
{\sl excedances}, {\sl decedance} and {\sl cycles} (cf. Theorem 4). This weight was 
originally inspired by Viennot's combinatorial interpretations for the moments of 
the Sheffer polynomials \cite{40}. 
Moreover, we will propose two new approaches to the interpretation of the linearisationa 
coefficients of Sheffer polynomials based on a {\sl segment decomposition} of derangements.
 
For other combinatorial and group theoretical aspects of 
the orthogonal polynomials discussed in this paper, we refer the reader 
to \cite{{9, 19, 31, 32, 39, 40}}.

This paper is organized as follows.
In {\S} 2, we first evaluate the generating functions of the moments of
Sheffer polynomials and then give their combinatorial interpretations
(cf. Theorem 1), which are essentially
 the combinatorial definitions of the moments of Sheffer
polynomials due to Viennot [40]. In {\S} 3, we use
the generating functions of the moments to evaluate the generating 
functions of their {\sl l.c.}, 
in particular, that of the {\sl l.c.} of Meixner-Pollaczek 
polynomials (cf. Theorem 2).
In {\S} 4, we introduce the notion of c-derangements and study their properties.
In {\S} 5, we evaluate the generating functions of the c-derangements according to 
various weights (cf. Theorem 3). 
In {\S} 6, we deduce the combinatorial 
interpretations of the {\sl l.c.} of Sheffer polynomials from their
generating functions (cf. Theorem 4) and show that 
in the case where $\delta=0$ it has 
another interesting interpretation in the context 
of {\sl general up-down permutations} (cf. Theorem 5).

\heading{2. The moments and combinatorial preliminaries}\endheading
It seems convenient to consider the integral in $(1.1)$ as a formal linear
functional on the vector space of the polynomials. Thus the functionals 
corresponding to the Hermite, Charlier, Laguerre,
Meixner, and Meixner-Pollaczek polynomials can be 
defined by their values on the base $\{x^n\}_{n\geq 0}$ as follows.
$$\align
\nu(x^n) &={1\over \sqrt\pi }\int_{-\infty}^{+\infty}
(\sqrt2 x)^n\, e^{-x^2}dx.\tag 2.1\\
\mu (x^n) &=e^{-a}\sum_{x=0}^\infty
x^n\, {a^x\over x!}.\tag 2.2\\
\psi(x^n) &={1\over \Gamma(\alpha +1)}\int_0^{+\infty}
x^n\,x^\alpha e^{-x}dx.\tag 2.3\\
\rho (x^n) &=(1-c)^\beta\sum_{x=0}^\infty x^n\,
{c^x(\beta)_x\over x!}.\tag 2.4\\
\varphi (x^n) &={1\over \int_{-\infty}^{+\infty}w(x)dx}
 \int_{-\infty}^{+\infty}x^n \,w(x)dx,\tag 2.5
\endalign 
$$
where $w(x)=[\Gamma (\eta/2)]^{-2}\bigl|\Gamma \bigl((\eta+ix)/2\bigr)
\bigr|^2\exp\bigl(x\arctan\delta\bigr).$
One notices that the above quantities are actually
the {\sl moments} of the corresponding polynomials.
\proclaim{Lemma 1} We have
$$
\int_{-\infty}^\infty x^n\, e^{-(\pi-2\phi)x}|\Gamma(a+ix)|^2\,dx
=2^{-n}\,\pi\,\Gamma(2a){d^n\over d\phi^n}\bigl(2\,\sin \phi\bigr)^{-2a}.
\leqno (2.6)
$$
\endproclaim
\demo{Proof} According to Pollaczek \cite{35}, we have 
$$
\int_{-\infty}^\infty e^{-(\pi-2\phi)x}|\Gamma(a+ix)|^2\,dx
=\pi\,\Gamma(2a)\bigl(2\,\sin \phi\bigr)^{-2a}.
$$
Differentiating the above formula $n$ times and then dividing both
sides by $2^n$, we obtain immediately (2.6).
\enddemo

\proclaim{Proposition 1}The following identities hold.
$$
\align
\sum_{n\geq 0}\nu(x^n) {t^n\over n!} &=\exp (t^2/2).\tag 2.7\\
\sum_{n\geq 0}\mu(x^n) {t^n\over n!} &=\exp (a(e^t-1)).\tag 2.8\\ 
\sum_{n\geq 0}\psi(x^n) {t^n\over n!} &=(1-t)^{-\alpha-1}.\tag 2.9\\
\sum_{n\geq 0}\rho (x^n) {t^n\over n!}&=\Bigl({1-c\over 1-ce^t}\Bigr)^\beta.
\tag 2.10\\
\sum_{n\geq 0}\varphi (x^n) {t^n\over n!} &=
(\cos t-\delta \sin t)^{- \eta}.\tag 2.11
\endalign 
$$
\endproclaim
\demo{Proof}Formulae (2.8) and (2.10) can be derived directly by inserting 
the series expressions and exchanging the order of the summations. 
Next, we notice that $\nu(x^{2n+1})=0$ and
$$\align
\nu(x^{2n}) &={1\over \sqrt\pi }\int_{-\infty}^{+\infty}
(\sqrt2\, x)^{2n}\, e^{-x^2}dx=1\cdot 3\cdots (2n-1),\\
\psi(x^n) &={1\over \Gamma(\alpha +1)}\int_0^{+\infty}
x^n\,x^\alpha e^{-x}dx=(\alpha+1)_n.
\endalign
$$
Formulae (2.7) and (2.9) then follow immediately.
Finally, we  will give two proofs of (2.11). 
 The first one
uses only the special case $n=0$ of Lemma 1. By definition (2.5), we have
$$\align
\sum_{n=0}^\infty \varphi(x^n){t^n\over n!}&=
{\int_{-\infty}^{+\infty}e^{xt} \,w(x)dx 
\over \int_{-\infty}^{+\infty}w(x)dx}\\
&={ \int_{-\infty}^{+\infty}\bigl|\Gamma \bigl((\eta+ix)/2\bigr)\bigr|^2 
e^{x(t+\arctan\delta)}dx\over \int_{-\infty}^{+\infty}
\bigl|\Gamma \bigl((\eta+ix)/2\bigr)\bigr|^2 e^{x\arctan\delta}dx}\\
&={\int_{-\infty}^{+\infty}\bigl|\Gamma (\eta/2+iy)\bigr|^2
 e^{y(2t+2\arctan\delta)}dy \over \int_{-\infty}^{+\infty}
\bigl|\Gamma (\eta/2+iy)\bigr|^2 e^{2y\arctan\delta}dy}\\
&=\Bigl [\shave{\sin (\pi/2+t+\arctan \delta)\over 
\sin (\pi/2+\arctan \delta)}\Bigr]^{-\eta}\\
&=\bigl(\cos t-\delta\,\sin t\bigr)^{-\eta}.
\endalign
$$
The second one uses Lemma 1 and Lagrange's inversion formula.
Comparing definition (2.5) and Lemma 1, we see
that $\varphi(x^n)$ is the evaluation of the following function 
$$(\sin \phi)^\eta\,{d^n\over d\phi^n}(\sin \phi)^{-\eta}$$
at $\phi ={\pi\over 2}+\arctan \delta$. Thanks to 
Lagrange's inversion formula or just Taylor formula \cite{36},
we have
$$\align
\sum_{n=0}^\infty (\sin \phi)^\eta\,{d^n\over d\phi^n}(\sin \phi)^{-\eta}
 {t^n\over n!}&=(\sin \phi)^\eta \bigl[\sin(\phi +t)\bigr]^{-\eta}\\
&=\Bigl(\cos\, t+\cot\, \phi\,\sin\,t\Bigr)^{-\eta}.
\endalign
$$
Substituting $\phi={\pi\over 2}+\arctan \delta$ into the last line, we obtain
(2.11). \enddemo 

In order to give combinatorial interpretations of these
moments, we first need to introduce
some basic definitions and notations.
Let $\Bbb N$ be the set of nonnegative integers and let $[n]=\{1,\ldots,n\}$
for $n\in \Bbb N$, by convention $[0]=\emptyset$. 
 Let $\s_n$ be the set of permutations of $[n]$. For $\sigma\in \s_n$, 
we say that $\sigma$ has an
{\obeylines
{\sl excedance} at $i\in [n]$ if $\sigma (i)>i$,  
{\sl decedance} at $i\in [n]$ if $\sigma (i)<i$, 
{\sl maximun} at $i\in [n]$ if $\sigma^{-1}(i)<i > \sigma (i)$,}

\noindent and denote by $\Exc\ \sigma$, $\Dec\,\sigma$, and
$\Max\, \sigma$ respectively the numbers of excedances, decedance and maxima of 
$\sigma$. A {\sl cycle} of ${\sigma}$ is a sequence 
$(a, \sigma(a),\ldots, \sigma^{k-1}(a))$
such that $\sigma^k(a)=a$ and $\sigma^l(a)\not= a$ for $1<l<k$. The 
{\sl length} of this cycle is $k$. In case $k=1$, we say that $a$ is a 
{\sl fixed point} of $\sigma$. 
Clearly, each permutation $\sigma$ can be factorized
as a product of disjoint cycles. 
We denote by $\Cyc\, \sigma$ the number of the cycles of $\sigma$.
A permutation $\sigma$ is said to
be {\sl circular} if $\Cyc\,\pi=1$. 
Furthermore, we call $\sigma$ a {\sl derangement}
of $[n]$ if $\sigma (i)\neq i\text{ for all }i\in [n]$ or equivalently all the cycles of $\sigma$
have lengths $\geq 2$   
 and denote by $\D_n$ the set of derangements of $[n]$.
Finally, we call $\sigma$ an {\sl involution} if $\sigma^2=id.$ and
denote by $\I_n$ the set of {\sl fixed point free} involutions on $[n]$.
 
For a finite set $S$, we denote by $|S|$ the {\sl cardinal} of $S$. A {\sl partition} of $S$ 
is an assemblage of the disjoint subsets (or {\sl blocs}) $\pi=\{E_1,E_2,\ldots,E_l\}$ 
such that $E_1+\cdots+E_l=S$. Let $\Bloc\, \pi$ be the number of
blocs of $\pi$ and $\P_n$ the 
set of partitions of $[n]$. 

Throughout this paper, 
we adopt the following vector notations. 
For $\n=(n_1,\ldots,n_m)$, we define 
$$
\n!=n_1!\cdots n_m!,\quad\x^{\n}=x_1^{n_1}\cdots x^{n_m}.
$$ 
If $f({\bold x}):=f(x_1,\ldots,x_m)$
is a power series in $x_1,\ldots,x_m$, we denote by 
$[{\bold x}^{\bold i}]f({\bold x})$ the coefficient of 
${\bold x}^{\bold i}$ in $f({\bold x})$.
However, for convenience, the coefficient of the monomial $x_1x_2\cdots x_m$ 
in $f({\bold x})$ will be denoted by $[{\bold x}]f({\bold x})$.

\proclaim{Lemma 2} The number of circular permutations on $[n]$ with
$k$ maxima is
$$\Bigl[z^k{x^n\over n!}\Bigr]\log \bigl\{\cosh xy-
y^{-1}\sinh xy\bigr\}^{-1},$$
where $y=(1-z)^{1/2}$.
\endproclaim
\demo{Proof} A proof of Lemma 2 can be found in \cite{27, p. 273}.
Note that this result is usually attributed to Entringer \cite{16}.
 In fact, this formula had been given implicitly by Andr\'e \cite{1} in 1895.
See also Dumont \cite{15} for some related results.
\enddemo
\proclaim{Lemma 3} The number of permutations on $[n]$ with $k$ maxima 
and $l$ cycles is 
$$\Bigl[\gamma^k\eta^l{x^n\over n!}\Bigr]
\bigl(\cosh{xy}-y^{-1}\sinh  xy\bigr)^{-\eta},$$
where $y=\sqrt{1-\gamma}$. 
\endproclaim
\demo{Proof} This is obvious on combining the exponential formula 
\cite{{18, 30}} and Lemma 2. \enddemo

\proclaim {Theorem 1} We have
$$
\align
\nu(x^n) &=|\I_n|.\tag 2.12\\
\mu(x^n) &=\sum_{\pi\in \P_n} a^{\bloc\, \pi}.\tag 2.13\\ 
\psi (x^n) &=\sum_{\sigma \in \s_n}(\alpha +1)^{\cyc\, \sigma}=(\alpha+1)_n.\tag 2.14\\
\rho (x^n) &=(1-c)^{-n}\sum_{\sigma\in \s_n}c^{\dec\, \sigma}\beta^{\cyc\, \sigma}.\tag 2.15\\
\varphi (x^n) &=\delta^n\sum_{\sigma\in \s_n}(1+1/\delta^2)^{\max\, \sigma}\eta^{\cyc\, \sigma}.\tag 2.16
\endalign 
$$
\endproclaim
\demo{Proof}Formula (2.12) is trivial, since the number of 
involutions without fixed point on $[2n+1]$ and $[2n]$ is 
respectively $0$ and $1\cdot 3\cdots (2n-1)$.
 On the other hand, it is well known \cite{13} that 
the number of permutations of $[n]$
with $k$ cycles is $s(n,k)$, the {\sl Stirling number of the first kind},
and the number of partitions
of $[n]$ with $k$ blocs is $S(n,k)$, the {\sl Stirling number
of the second kind}. Moreover, we have (cf. \cite{13} or \cite{ 36, p. 44})
$$\align
\sum_{k=0}^\infty\sum_{k=0}^nS(n,k)a^k{t^n\over n!}&=\exp(a(e^t-1)),\\ 
\sum_{k=0}^\infty\sum_{k=0}^ns(n,k)\beta^k{t^n\over n!}&=(1-t)^{-\beta}.
\endalign
$$
The formulae (2.13) and (2.14) follow then by 
comparing (2.8) and (2.9) with the above identities. In the next place, 
thanks to Foata's fundamental transformation \cite{18},
we know that $\sum_{\sigma\in \s_n}c^{\dec\,\sigma}$ is equal to the
$n$th {\sl Eulerian} polynomial \cite{38}, of which the exponential generating 
function is
$${1-c\over 1-ce^{t(1-c)}}.$$
This and the exponential formula \cite{{18, 30}} imply that
$$\sum_{n\geq 0}\left(\sum_{\sigma\in \s_n}c^{\dec\,\sigma}
\beta^{\cyc\,\sigma}\right){t^n\over n!}
=\left({1-c\over 1-ce^{t(1-c)}}\right)^\beta.$$
 Comparing the above identity with (2.10) yields (2.15).
Finally, it follows from Lemma 3 that
$$\align
\sum_{n\geq 0}\delta^n\sum_{\sigma\in S_n}(1+1/\delta^2)^{\max\,\sigma}
\eta^{\cyc\,\sigma}{t^n\over n!}&=\left(\cosh(it)-{\delta\over i}\sinh(it)
\right)^{-\eta}\\
&=(\cos t-\delta \sin t)^{-\eta}.
\endalign
$$
We obtain (2.16) by comparing the above identity with (2.11).
\enddemo

\demo{Remark} Theorem 1 actually gives Viennot's combinatorial definitions 
of the moments of Sheffer polynomials \cite{40}. 
Note that Viennot's original interpretation 
of $\rho (x^n)$ (respectively $\varphi(x^n)$) uses the number of
{\sl descents} (respectively {\sl pics}) and {\sl saillants}; however 
by applying Foata's fundamental transformation \cite{18}
it is easily seen that
 these statistics have the same distribution as 
$(\Dec,\Cyc)$ (respectively $(\Max,\Cyc)$). 
\enddemo

The following results are useful in the new derivations of
the interpretation of the {\sl l.c.} of Sheffer polynomials. Let 
$$
\align
f_m(\gamma,\eta)&=\sum_{\sigma \in\D_n}\gamma^{\dec\,\sigma}
\eta^{\cyc\,\sigma},\tag 2.17\\
g_m(\gamma,\eta)&=\sum_{\sigma \in\D_n}\gamma^{\max\,\sigma}
\eta^{\cyc\,\sigma}.\tag 2.18
\endalign
$$     
\proclaim{Proposition 2} We have
$$
\align
1+\sum_{m \geq 1}f_m(\gamma,\eta){x^m\over m!} &=
\Bigl\{1-\sum_{k\geq 2}(\gamma+\gamma^2+\cdots +\gamma^{k-1}){x^k\over k!}\Bigr\}^{-\eta},\tag 2.19\\
1+\sum_{m \geq 1}g_m(\gamma,\eta){x^m\over m!} &=
\Bigl\{1-\sum_{k\geq 2}\gamma\sum_{l\geq 0}{k-1\choose 2l+1}
(1-\gamma)^l{x^k\over k!}\Bigr\}^{-\eta}.\tag 2.20\\
\endalign
$$
\endproclaim
\demo{Proof} We just prove (2.20) here since (2.19) can be proved in the same manner, see also \cite{42}. 
For ${\pi}\in \D_n$, the weight function  $w(\pi)=\gamma^{\max \pi}\eta^{\cyc \pi}$
is {\sl multiplicative} with 
respect to the cycle decomposition.
Note that each derangement 
${\pi}$ does not contain any {\sl fixed point} (or cycle of length 1),
whence, by the exponential formula (cf. \cite{18}) and Lemma 2 with 
$y =\sqrt{1-\gamma}$, we have
$$\eqalign {1+\sum_{m\geq 1}d_m(\gamma, \eta){x^m\over m!}&=
\exp\{\eta \log (\cosh xy- y^{-1}\sinh xy)^{-1}-\eta x\}\cr
&=\Bigl\{{e^{(1+y)x}+e^{(1-y)x}\over 2}-
y^{-1}{e^{(1+y)x}-e^{(1-y)x}\over 2}\Bigr\}^{-\eta}\cr
&=\Bigl\{1-\sum_{k\geq 2}(1-y^2)\sum_{l\geq0}{k-1\choose 2l+1}y^{2l}
{x^k\over k!}\Bigr\}^{-\eta}.\cr}$$
The proof is complete by substituting $y^2=1-\gamma$ in the last line. 
\enddemo

\proclaim{Corollary 1} We have
$$
\sum_{\sigma \in\D_n}(\delta+i)^{\dec\,\sigma}(\delta-i)^{\exc\,\sigma}
\eta^{\cyc\,\sigma}
=\delta^n\sum_{\sigma \in\D_n}(1+1/\delta^2)^{\max\,\sigma}
\eta^{\cyc\,\sigma}.
$$     
\endproclaim 
\demo{Proof} It follows from Proposition 2 by noting that ${\Dec\,\sigma}+{\Exc\,\sigma}=n$ for ${\sigma
\in\D_n}$. \enddemo

\heading{3. The generating function of the linearization coefficients}\endheading
In what follows we shall denote by $e_k$ the $k$th $(0\leq k\leq m)$ 
elementary symmetric polynomial of $x_1,\ldots, x_m$, that is, 
$$\sum_{j=0}^m e_j x^j=\prod_{k=1}^m(1+x_k x).\leqno (3.1)$$
By convention, we set $e_k=0$ if $k>m$.
\proclaim{Lemma 4}We have
$$\tan\,\bigl(\arctan x_1+\cdots+\arctan x_m\bigr)
={\sum_{n\geq 0}(-1)^n e_{2n+1}\over\sum_{n\geq 0}(-1)^n e_{2n}}.\leqno (3.2)$$
\endproclaim
\demo{Proof} This is routine, by induction on $m$.
\enddemo
\proclaim{Lemma 5}With $i=\sqrt{-1}$, the following identities hold
$$
\align
\Bigl(\sum_{n\geq 0}(-1)^n e_{2n}\Bigr)^2 &+
\Bigl(\sum_{n\geq 0}(-1)^n e_{2n+1}\Bigr)^2
=\prod_{k=1}^m(1+x_k^2).\tag 3.3\\
\sum_{n\geq 0}(-1)^n e_{2n}&={1\over 2}
\Bigl\{\prod_{k=1}^m(1+ix_k)+\prod_{k=1}^m(1-ix_k)\Bigr\}.\tag 3.4\\
\sum_{n\geq 0}(-1)^n e_{2n+1}&={1\over 2i}
\Bigl\{\prod_{k=1}^m(1+ix_k)-\prod_{k=1}^m(1-ix_k)\Bigr\}.\tag 3.5
\endalign
$$
\endproclaim
\demo{Proof}
Substituting $x$ by $i$ and $-i$ in (3.1), we get
$$\align
\sum_{n\geq 0}(-1)^n e_{2n}+i\sum_{n\geq 0}(-1)^n e_{2n+1}
&=\prod_{k=1}^m(1+ix_k),\tag 3.6\\
\sum_{n\geq 0}(-1)^n e_{2n}-i\sum_{n\geq 0}(-1)^n e_{2n+1}
&=\prod_{k=1}^m(1-ix_k).\tag 3.7
\endalign
$$
Formulae (3.4) and (3.5) are then straightforward. We obtain (3.3) 
by multiplying (3.6) and (3.7). 
\enddemo
\proclaim{Lemma 6}For the linear functional $\varphi$ of (2.5), 
the following identity holds :
$$\varphi\bigl(\exp x\sum_{k=1}^m\arctan x_k\bigr)=
\prod_{k=1}^m(1+x_k^2)^{\eta/2}
\left\{\shave{\sum_{n\geq 0}}(-1)^n e_{2n}-\delta\sum_{n\geq 0}(-1)^n 
e_{2n+1}\right\}^{-\eta}.\leqno (3.8)$$
\endproclaim
\demo{Proof}By virtue of (2.15) of Proposition 2, we have
$$\align
\varphi\bigl(\exp x\sum_{k=1}^m\arctan x_k\bigr)
&=\sum_{n\geq 0}\varphi(x^n)
{(\arctan x_1 +\cdots +\arctan x_m)^n\over n!}\\
&=\bigl\{\cos\bigl( \sum_{k=1}^m\arctan x_k\bigr)-
\delta\sin \bigl(\sum_{k=1}^m\arctan x_k\bigr)\bigr\}^{-\eta}\\
&=\left\{{1-\delta\tan(\arctan x_1+\cdots+\arctan x_m)\over
\sqrt{1+\tan^2(\arctan x_1+\cdots+\arctan x_m)}}\right\}^{-\eta}.
\endalign$$
Substituting (3.2) into the last line, we get
$$\varphi\left(\shave {\exp x\sum_{k=1}^m\arctan x_k}\right)=
\left\{
{\sum_{n\geq 0}(-1)^ne_{2n}-\delta\sum_{n\geq 0}(-1)^ne_{2n+1}
\over \sqrt{\left(\sum_{n\geq 0}(-1)^ne_{2n}\right)^2
+\left(\sum_{n\geq 0}(-1)^ne_{2n+1}\right)^2}}
\right\}^{-\eta},$$
and then (3.8), in view of (3.3).
\enddemo 
We are now in a position to compute the generating function of 
the linearization coefficients of all the Sheffer polynomials
and express them in terms of the elementary symmetric functions 
of $x_1,\ldots,x_m$ with the help of the generating functions of 
the moments and the polynomials.
\proclaim{Theorem 2}We have
$$
\align
\qquad \sum_{{\bold n}\geq {\bold 0}}{\Cal H}(\n)
&{{\bold x}^{\bold n}\over {\bold n}!}=
e^{e_2}.\tag 3.9\\
\sum_{{\bold n}\geq {\bold 0}}{\Cal C}(\n;&a) 
{{\bold x}^{\bold n}\over {\bold n}!}=
e^{ae_2+\cdots +ae_m}.\tag 3.10\\
\sum_{{\bold n}\geq {\bold 0}}{\Cal L}(\n;&\alpha)
{{\bold x}^{\bold n}\over {\bold n}!}=
\Bigl(1-e_2-2\,e_3-\cdots-(m-1)e_m\Bigr)^{-(\alpha+1)}.\tag 3.11\\
\sum_{{\bold n}\geq {\bold 0}}{\Cal M}(\n;\beta,&c)
{{\bold x}^{\bold n}\over {\bold n}!} =
\Bigl(1-\sum_{k=2}^m(c+\cdots +c^{k-1})e_k\Bigr)^{-\beta}.\tag 3.12\\
\sum_{{\bold n}\geq {\bold 0}}{\Cal P}(\n;\delta,&\eta)
{{\bold x}^{\bold n}\over {\bold n}!}
 =\Bigl\{1-\sum_{k=2}^m(\delta^k+\delta^{k-2})\sum_{l\geq 0}
{k-1\choose 2l+1}\bigl({-1\over \delta^2}\bigr)^l\,e_k\Bigr\}^{-\eta}.\tag 3.13
\endalign
$$
\endproclaim
\demo{Proof}
Equations (3.9)--(3.12) are already known \cite{{6, 41}} and also
easily to be established using the generating functions of the polynomials
themselves. So we give here only the proof of (3.13). 
According to (1.7) and (1.12), we have 
$$\split
\sum_{{\bold n}\geq {\bold 0}}{\Cal P}(\n;\delta,\eta)
{{\bold x}^{\bold n}\over {\bold n}!}
&=\varphi \Bigl(\prod_{k=1}^m\sum_{n_k\geq 0}
P_{n_k}(x;\delta,\eta){x_k^{n_k}\over n_k!}\Bigr)\\
&=\prod_{k=1}^m\bigl((1+\delta x_k)^2+x_k^2\bigr)^{-\eta/2}
\varphi \bigl(\exp x\sum_{j=1}^m\arctan {x_j\over 1+\delta x_j}\bigr).
\endsplit $$ 
Now we apply Lemma 6 to the above identity and obtain
$$\sum_{{\bold n}\geq
{\bold 0}}{\Cal P}(\n; \delta,\eta) {{\bold x}^{\bold n}\over {\bold
n}!}=\prod_{k=1}^m(1+\delta x_k)^{-\eta}
 \Bigl\{\sum_{n\geq 0}(-1)^n {\hat e}_{2n}-\delta \sum_{n\geq 0}(-1)^n 
{\hat e}_{2n+1}\Bigr\}^{-\eta},\leqno (3.14)$$
where ${\hat e}_k$ ($k\geq 0$) are the elementary symmetric polynomials of 
the variables :
$${x_1\over 1+\delta x_1},\ldots,{x_m\over 1+\delta x_m}.$$
Next, it follows from (3.3) and (3.4) of Lemma 5 that
$$\align
\sum_{n\geq 0}(-1)^n {\hat e}_{2n}&={1\over 2}
\Bigl\{\prod_{k=1}^m{1+(\delta +i)x_k\over 1+\delta x_k}+
\prod_{k=1}^m{1+(\delta -i)x_k\over 1+\delta x_k}\Bigr\},\tag 3.15\\
\sum_{n\geq 0}(-1)^n {\hat e}_{2n+1}&={1\over 2i}
\Bigl\{\prod_{k=1}^m{1+(\delta +i)x_k\over 1+\delta x_k}-
\prod_{k=1}^m{1+(\delta -i)x_k\over 1+\delta x_k}\Bigr\}.\tag 3.16
\endalign
$$
Finally, putting (3.15) and (3.16) into (3.14), we get
$$\align
\sum_{\bold n}{\Cal P}(\n; \delta,\eta){{\bold x}^{\n}\over {\n}!}&=
\left\{{i-\delta\over 2i}\prod_{k=1}^m(1+(\delta +i)x_k)+
{i+\delta\over 2i}\prod_{k=1}^m(1+(\delta -i)x_k)\right\}^{-\eta}\tag 3.17\\
&=\left\{1+{1+\delta^2\over 2i}\sum_{k=1}^m((\delta-i)^{k-1}-(\delta+i)^{k-1})e_k\right\}^{-\eta},
\endalign
$$
which implies (3.13) after simplification. \enddemo

If $m=3$, the explicit formulae can be obtained
from Theorem 2 for the linearization coefficients of the Sheffer
polynomials. 
\proclaim{Corollary 2}Let ${\bold n}=(n_1,n_2,n_3)$, we have
the explicit formulae :
$$
\align
{\Cal H}(\n)&=\cases {n_1!n_2!n_3!\over (s-n_1)!(s-n_2)!(s-n_3)!},
&\text{if $n_1+n_2+n_3=2s$};\\
0,&\text{if $n_1+n_2+n_3$ is odd}.\endcases\tag 3.18\\
{\Cal C}(\n;a)&=
\sum_{s\geq 0}{n_1!n_2!n_3!a^s\over
(s-n_1)!(s-n_2)!(s-n_3)!(n_1+n_2+n_3-2s)!}\cdot\tag 3.19\\
{\Cal L}(\n;\alpha)&=
\sum_{s\geq 0}{n_1!n_2!n_3! 2^{n_1+n_2+n_3-2s}(\beta)_s\over
(s-n_1)!(s-n_2)!(s-n_3)!(n_1+n_2+n_3-2s)!}\cdot\tag 3.20\\
{\Cal M}(\n;\beta,c)&=
\sum_{s\geq 0}{n_1!n_2!n_3!(1+c)^{n_1+n_2+n_3-2s}c^s(\beta)_s\over
(s-n_1)!(s-n_2)!(s-n_3)!(n_1+n_2+n_3-2s)!}\cdot\tag 3.21\\
{\Cal P}(\n;\delta,\eta)&=
\sum_{s\geq 0}{n_1!n_2!n_3!(\eta)_s(1+1/\delta^2)^s\delta^{n_1+n_2+n_3}
2^{n_1+n_2+n_3-2s}\over
(s-n_1)!(s-n_2)!(s-n_3)!(n_1+n_2+n_3-2s)!}\cdot\tag 3.22
\endalign
$$
\endproclaim
\demo{Remark} Formulae (3.18)--(3.21) have appeared in 
\cite{{3, 5}}, but, to the author's
knowledge, formula (3.22) seems new. Setting $n_3=0$ in the above corollary, we recover then the 
orthogonalities of the Hermite, Charlier, Laguerre, Meixner, and 
Meixner-Pollaczek polynomials.\enddemo

\heading{4. The combinatorics of $c$-derangements}\endheading
Let us begin with some multianalogues of the notions and notations 
introduced in § 2. For each $i\in [m]$, we define the coloured sets 
${\Bbb N}_i={i}\times {\Bbb N}$ and $[n]_i={i}\times [n]$. Let
$a=(i,j)\in {\Bbb N}_i$, we call the first term $i$ the colour of $a$
and set $c(a)=i$. Let ${\Bbb N}_*=[m]\times{\Bbb N}$ denote the
disjoint union ${\N}_1\cup \cdots\cup{\N}_m$. 
For $\n=(n_1,\ldots, n_m)$, we note also
$[\n]=[n_1]_1\cup\cdots\cup[n_m]_m$. Clearly, the set $\N_*$ is a
{\sl total ordered} set with respect to the lexicographic order.

It is convenient to identify a permutation $\pi$ of a set $A$ with
the {\sl digraph} on $A$ having an edge $a\rightarrow \pi(a)$
for each $a\in A$. We will refer to an edge beginning at a point of
colour $i$ and ending at a point of colour $j$ as an ``$i\rightarrow j$''
edge, calling it {\sl pure} if $i=j$ and {\sl mixed} if $i\ne j$. A 
sequence of consecutive edges $a\rightarrow \pi(a)\rightarrow
\pi^2(a)\rightarrow \cdots \rightarrow \pi^k(a)$ ($k\geq 0$) is
called a {\sl path} of the graph. We say also that this is a $c(a)\f c(\pi(a))\f
c(\pi^2(a))\f \cdots \f c(\pi^k(a))$ path when we just take account of the colours.       

Let $\pi$ be a permutation on $[\n]$. 
 An element $a\in [\n]$ is said to be a {\sl c-fixed point} of $\pi$ if 
$c(a)=c(\pi (a))$. A permutation without c-fixed point is said to be
a {\sl c-derangement} on $[\n]$.
 We denote respectively by ${\D}_{\n}$ and ${\I}_{\n}$ 
the sets of c-derangements and 
 of c-fixed point free involutions on $[\n]$. Similarly, we say that
$\pi$ has a {\sl c-excedance} (respectively {\sl c-decedance}) at $a\in [\n]$ 
if $c(\pi(a))>c(a)$ (respectively $c(\pi(a))<c(a)$)
and  denote by $\Exc \,\pi$ (respectively $\Dec \,\pi$) the number of excedances
 (respectively decedances) of $\pi$. 

\proclaim{Proposition 3} Let $D(\n;\gamma, \eta)=\sum\gamma^{\exc\, \pi}
\eta^{\cyc\,\pi}$\quad $(\pi\in \D_{\n})$ and $\n^*$ be any rearrangement of $\n=(n_1, \ldots ,
n_m)$, then $$D(\n;\gamma,\eta)=D(\n^*; \gamma,\eta).\tag 4.1$$
\endproclaim
\demo{Proof} Without loss of generality we may assume that $\n^*$ is a rearrangement of $\n$ by
just exchanging $n_i$ and $n_{i+1}$, where $1\leq i\leq m-1$. We will prove (4.1) by constructing 
an explicit bijection $\Phi\, :\, \D_{\n}\rightarrow \D_{\n^*}$ such that 
$\Exc\, \pi=\Exc\, \Phi(\pi)$ and $\Cyc\,\pi=\Cyc\,\Phi(\pi)$ for 
each $\pi\in \D_{\n}$.
 
Consider any c-derangement $\pi$ of $[\n]$ identified with its digraph $G$. Recall that 
any vertex of $\pi$ is characterized by a pair of natural numbers $(i,j)$, where $i\;
(1\leq i\leq m)$ is the number of the colour to which it belongs and $j\;(1\leq j\leq n_i)$
is its serial number in that colour. Since there is no pure edge in $G$, each vertex of colour $i$ or 
$i+1$ is necessarily located in either a cycle of which the vertices are only of colour $i$ and $i+1$
or one of the following paths :
$$\align
 &r_1\f i\f l_1, \quad \text{or} \quad r_2\f (i+1)\f l_2, \tag I\\
 &r_3\f (i\f(i+1))^{k_1}\f l_3, \quad \text{or} \quad r_4\f ((i+1)\f i)^{k_2}\f l_4, \tag II\\ 
 &r_5\f (i\f (i+1))^{k_3}\f i\f l_5,\quad \text{or} \quad r_6\f ((i+1)\f i)^{k_4}\f (i+1)\f l_6,\tag III
\endalign
$$
where the $r_j,l_j\notin \{i,i+1\}$ for $1\leq j\leq 6$  denote the colours of 
the corresponding vertices and $k_l\geq 1$ for $1\leq l\leq 4$. 
Apply now the following tranformation $\Phi$ to $G$ :
\roster
\item Exchange the serial numbers of each vertex of colour $i$ (resp. $i+1$) with those of its image in each path of type (II)
 and in each cycle consisting only of vertices of colour $i$ and $i+1$ ;
\item Exchange the colours $i$ and $i+1$ of the vertices in the paths of types (I) and (III).
\endroster
Call the new graph $G'$. It is then readily seen that $G'$ is a graph of a c-derangement
on $[\n^*]$. Denote the corresponding c-derangement by $\Phi(\pi)$. Clearly, the tranformation
$\Phi$ is a bijection between $\D_{\n}$ and $\D_{\n^*}$ with the property that 
$\Cyc\,\pi=\Cyc\,\Phi(\pi)$. On the other hand, each excedance of 
colour $i+1$ (resp. $i$) of $\pi$ in a path of type (II) or a cycle consisting only of colour i and $i+1$
is mapped to an excedance of $\Phi(\pi)$ with the same colour, and 
the number of excedances contained in each path of type (I) or (II) is invariant with respect to this
transformation.  
Therefore the total number of excedances does not 
change after this transformation, that is, $\Exc\,\pi=\Exc\,\Phi(\pi)$. \enddemo

\proclaim{Corollary 3} Let $E(\n;\gamma, \eta)=\sum\gamma^{\dec\, \pi}
\eta^{\cyc\,\pi}$\quad $(\pi\in \D_{\n})$ and $\n^*$ be any rearrangement of 
$\n=(n_1, \ldots, n_m)$, then $D(\n;\gamma,\eta)=D(\n^*; \gamma,\eta).$
\endproclaim
\demo{Proof} 
It follows from the fact that 
$\Dec\, \pi=n_1+\cdots +n_m-\Exc\, \pi$ for each $\pi\in \D_{\n}$. \enddemo

Consider the graph of any c-derangement $\pi$ of $\D_{\n}$. Take any cycle of
$\pi$ and call $X^0$ the hightest ranking vertex of this cycle.
Let $X^i=\pi(X^0)$ for $i\geq 1$. Then this cycle can be linearly presented
as
$$X^0\f X^1\f \cdots \f X^{p-1}(\f X^p=X^0). \tag 4.2$$
Since $\pi$ is a c-derangement, it follows that $p\geq 2$ and 
$c(X^i)\ne c(X^{i+1})$ for $0\leq i\leq p-1$.

A linear arrangement $w=a_1a_2\ldots a_n$ of the elements of a subset
$A\subseteq \N_*$ is said to be a {\sl wave arrangement} of 
 $A$, if no two adjacent elements of $w$ have the same colour, i.e., 
$c(a_i)\ne c(a_{i+1})$ for $1\leq i\leq n-1$.
 A wave arrangement 
 $w=a_1a_2\ldots a_n$ is called  a {\sl wave segment} if 
moreover $n\geq 2$ and $c(a_1)>c(a_i)$ for all $2\leq i\leq n$.
In what follows,
we will identify $w=a_1a_2\ldots a_n$ with the sequence $a_1\f a_2\f \cdots \f a_n$ and 
call it a  $c(a_1)\f c(a_2)\f \cdots \f c(a_n)$ arrangement or segment.
It follows that the cycle (4.2) can be divided up into wave segments
each of which begins with vertices of the same colour as $X^0$. For example, if
the cycle (4.2) is 
$$(7,6)\f(2,5)\f(3,7)\f(7,1)\f(2,4)\f(1,1)\f(4,3)\f(7,2)\f(2,1).$$
\noindent We should have the segments
$$\align
&w_1=(7,6)\f(2,5)\f(3,7),\\
&w_2=(7,1)\f(2,4)\f(1,1)\f(4,3),\\
&w_3=(7,2)\f(2,1).
\endalign
$$
Thus, each cycle of $\pi$ corresponds in this way to a set of wave segments
and a circular permutation acting on it. Let $C_i$ be the set of cycles of $\pi$
of which the highest ranking vertices are of colour $i\quad (2\leq i\leq m)$.
By the above decomposition of cycles, each class $C_i$ $(2\leq i\leq m)$ corresponds to
a pair $(\Omega_i, \sigma_i)$ where $\Omega_i$ is a set of wave segments beginning with a vertex of colour $i$
 and $\sigma_i$ a permutation acting on it. Taking these in count, any c-derangement $\pi$
can be identified with a $(m-1)$-tuple $\left((\Omega_2, \sigma_2),\ldots ,(\Omega_m,\sigma_m)\right)$,
with the product (juxtaposition) of all the wave segments in $\Omega_2\cup\ldots\cup\Omega_m$
 being a rearrangement
of $[\n]$ and the permutations $\sigma_i\;(2\leq i\leq m)$ acting on $\Omega_i$.

To determine the transfer of the weight through this correspondence,
we are led to introduce the corrsponding weight on the set of wave segments.
Let $w=a_1a_2\ldots a_n$ be a wave segment.
By abuse of language, we call $a_i$ ($1\leq i\leq n$) c-{\it decedance} if $c(a_{i-1}<c(a_i)>c(a_{i+1})$ 
with the convention that $a_0=a_n$ and $a_{n+1}=a_1$.
Denote by $\Dec\,w$
the number of c-decedances of $w$ and for 
a set of wave arrangements $\Omega$, 
define $\Dec\, \Omega=\sum_{w\in \Omega}\Dec\,w$.

\proclaim{Proposition 4} The wave segment decomposition, that is,

\centerline{$\pi\longmapsto \left((\Omega_2, \sigma_2), \ldots, (\Omega_m,\sigma_m)\right),$}

\noindent has the following properties :

\centerline{$\Dec\,\pi=\sum_{i=2}^m\Dec\,\Omega_i\qquad \text{and}\qquad \Cyc\,\pi=\sum_{i=2}^m\Cyc\,\sigma_i.$}
\endproclaim

\demo{Proof} Firstly, it is readily seen that $\Cyc\,\pi=\sum_{i=2}^m\Cyc\,w_i$. It then
suffices to consider the case where $\pi$ is a {\sl circular c-derangement}. Let $\pi\in \D_{\n}$ be
a circular derangement with the unique circle $x^0\f x^1\f\cdots \f x^p\f x^0$, where $x^0$ is the
highest ranking vertex and $x^l=\pi^l(x^0)$ for $1\leq l\leq p$.
Let $\Omega=\{w_1, w_2,\ldots, w_k\}$ be the wave segment decomposition of $\pi$ with
$$\align
w_1&=a_0a_1\cdots a_{l_1-1},\\
w_2&=a_{l_1}a_{l_1+1}\cdots a_{l_2-1},\\
&\cdots\cdots\\
w_k&=a_{l_k}a_{l_k+1}\cdots a_p,
\endalign
$$
where $a_l=x^l$ for $0\leq l\leq p$ and $c(x^0)=c(x^{l_1})=\ldots =c(x^{l_k})$. Then the associated
circular permutation $\sigma$ acting on $\Omega$ is $w_1\f w_2\f \cdots\f w_k\f w_1$. 
Clearly, an element $x^l$ is a c-decedance of $\pi$ if and only if $a_l$ is a c-decedance of some $w_i$.
Hence $\Dec\, \pi=\sum_{i=1}^k\Dec\,w_i$.\enddemo

\heading{5. The generating functions of $c$-derangements}\endheading
\proclaim{Theorem 3} The following identities hold :
$$\align
\sum_{\n\geq \0}\left| \I_{\n}\right |{\x^{\n}\over \n!}&=e^{e_2};\tag 5.1\\
\sum_{\n\geq \0}\left ( \sum_{\pi\in \P_{\n}}a^{\bloc \,\pi}\right){\x^{\n}\over \n!}
&=e^{ae_2+\cdots +ae_m};\tag 5.2\\
\sum_{\n\geq \0}\left ( \sum_{\pi\in \D_{\n}}\beta^{\cyc \,\pi}\right){\x^{\n}\over \n!}
&=\left(1-\sum_{k=2}^m(k-1)e_k\right)^{-\beta};\tag 5.3\\
\sum_{\n\geq \0}\left ( \sum_{\pi\in \D_{\n}}c^{\dec \,\pi}\beta^{\cyc\, \pi}\right){\x^{\n}\over \n!}
&=\left(1-\sum_{k=2}^m(c+\cdots +c^{k-1})e_k \right)^{-\beta};\tag 5.4
\endalign
$$
\vskip-10pt
$$
\multline
\sum_{\n\geq \0}\left ( \sum_{\pi\in \D_{\n}}(\delta-i)^{\exc \,\pi}(\delta+i)^{\dec\, \pi}
\eta^{\cyc\, \pi}\right){\x^{\n}\over \n!}\\
=\left\{1-\sum_{k=2}^m (\delta^k+\delta^{k-2})\sum_{l\geq 0}{k-1\choose 2l+1}
\left({-1\over \delta^2}\right)^l\,e_k \right\}^{-\eta}.
\endmultline\tag 5.5
$$
\endproclaim
\demo{Proof} Equations $(5.1)-(5.4)$ are already known \cite{{20, 28, 40}}. In fact,
formulae (5.1) and (5.2) are straightforward consequences of the {\sl multivariable exponential
formula} \cite{30}, and formulae (5.3) and (5.4) were proved in \cite{{20, 41}}
by applying the $\beta$-extension of MacMahon's Master Theorem. Besides, if $c=1$, formula 
(5.4) reduces to (5.3). So in the sequel only the proof of (5.5) will be given. 
First we note that for each $\pi\in \D_{\n}$, we have
$$
\Exc\,\pi+\Dec\,\pi=n_1+\cdots+n_m.
$$
Hence, in (5.4) if we substitute $c$ by $(\delta+i)/(\delta-i)$, $\beta$ by $\eta$ and $x_k$ by $(\delta-i)^k$, 
we get immediately
$$\multline
\sum_{\n\geq \0}\left ( \sum_{\pi\in \D_{\n}}(\delta-i)^{\exc \,\pi}(\delta+i)^{\dec\, \pi}
\eta^{\cyc\, \pi}\right){\x^{\n}\over \n!}\\
=\left\{1-{\delta^2+1\over 2i}\sum_{k=2}^m\left[(\delta+i)^{k-1}-(\delta-i)^{k-1}\right]\,e_k\right\}^{-\eta},
\endmultline
$$
which yields (5.5) after simplification. \enddemo

In view of the importance of (5.4), we give here two more proofs of this formula.
\proclaim{Lemma 7}
Let $\Phi(x_1, x_2, \ldots)$ be a power series in $x_1, x_2, \ldots$, then 
$$[\x]\Phi(e_1, e_2,\ldots)=\left[{x^m\over m!}\right]\Phi({x^1\over 1!}, {x^2\over 2!}, \ldots,).$$
\endproclaim

For a proof of this result, we refer the reader to \cite{27, p. 233}.
\demo{First New Proof of (5.4)}
We will count the $c$-derangements through the wave segment decomposition given in Proposition 4.
For $1\leq i \leq m-1$, let $\n_i=(n_1,\ldots, n_i)$ and $W_i$ be the set of wave arrangements of 
$[n_1]\cup\ldots \cup [n_i]$. Let $\pi=a_1\ldots a_n$ be a wave arrangement of $W_i$ and 
$\alpha\in \N_*$ be an element with colour $c(\alpha)=i+1$, it
is then clear that $\alpha \pi=\alpha\, a_1\ldots a_n$ is a wave segment beginning with colour $i+1$.
Define the generating function for $W_i$ by
$$
f_i(x_1,\ldots, x_i;\gamma)=\sum_{\n_i}\left(\sum_{\pi\in W_i}\gamma^{\dec\,\pi}\right)
{x_1^{n_1}\over n_i!}\cdots {x_m^{n_m}\over n_m!}.\tag 5.6
$$
It follows that the generating function of the wave segments beginning with colour $i+1$ should be 
$\gamma\,x_{i+1}f_i(x_1,\ldots, x_i;\gamma)$. On the other hand, since each $c$-derangement $\pi$ corresponds 
bijectively to a $(m-1)$-tuple $\left((\Omega_2, \sigma_2),\ldots ,(\Omega_m, \sigma_m)\right)$ and
the generating function of permutations with each cycle being weighted by $\eta$ is 
$(1-x)^{-\eta}$, the generating function of $(\Omega_{i+1},\sigma_{i+1})$ should be 
$(1-\gamma\,x_{i+1}f_i(x_1,\ldots, x_i;\gamma))^{-\eta}$ where $1\leq i \leq m-1$. 
Thanks to Proposition 4, we obtain
$$
\sum_{\n\geq \0}D(\n; \gamma,\eta){\x^{\n}\over \n!}=
\prod_{i=1}^{m-1}(1-\gamma \,x_{i+1}f_i(x_1,\ldots, x_i;\gamma))^{-\eta}.\tag 5.7
$$
\noindent Moreover, by Corollary 3, both sides of (5.7) are symmetric functions of 
$x_1,\ldots, x_m$, which implies that the function
$$
\prod_{i=1}^{m-1}(1-\gamma\,x_{i+1}f_i(x_1,\ldots, x_i;\gamma)) \tag 5.8
$$
is also symmetric. Furthermore, we note that (5.8) is actually {\sl linear} in $x_m$. By symmetry,
it is a {\sl linear symmetric function} of $x_1,\ldots, x_m$. We can then expand (5.8) as a linear
combination of the elementary symmetric functions $e_i$ ($0\leq i\leq m$) (cf. \cite{33})
$$
\prod_{i=1}^{m-1}(1-\gamma\,x_{i+1}f_i(x_1,\ldots, x_i;\gamma))=a_0+a_1e_1+\cdots +a_me_m,\tag 5.9
$$
where the coefficients $a_i\;(0\leq i\leq m)$ are to be determined. Using (5.9) we can rewrite (5.7)
as
$$
\sum_{\n\geq \0}D(\n; \gamma,\eta){\x^{\n}\over \n!}=(a_0+a_1e_1+\cdots +a_me_m)^{-\eta}.\tag 5.10
$$
Applying Lemma 7 to (5.10) and noticing that the coefficient of $x_1\ldots x_m$ in the 
left-hand side of (5.10) is the generating polynomial 
$d_m(\gamma,\eta)$ of the ordinary derangements of $\D_m$ (cf. Proposition 2), 
 we obtain the following equation  :
$$
\left\{1-\sum_{k\geq 2}(\gamma+\gamma^2+\cdots +\gamma^{k-1}){x^k\over k!}\right\}^{-\eta}=
\left(\sum_{k\geq 0}a_k{x^k\over k!}\right)^{-\eta}.
$$
Equating the coefficients of $x^n$ gives $a_0=1$, $a_1=0$ and for $k\geq 2$
$$
a_k=-(\gamma+\gamma^2+\cdots +\gamma^{k-1}).
$$
Finally, substituting these values into (5.10) yields (5.4). 
\enddemo

\demo{Second New Proof of (5.4)} Let $f_i(\gamma)=f_i(x_1,\ldots, x_i; \gamma)$,
 it is then well-known \cite{27, p.78} that
$$
f_i(\gamma)={\prod_{j=1}^i(1+\gamma\,x_j)-\prod_{j=1}^i(1+x_j)\over
\gamma \,\prod_{j=1}^i(1+x_j)-\prod_{j=1}^i(1+\gamma\,x_j)}
$$
where $1\leq i\leq m-1$. It follows that
$$
\left(1-\gamma\,x_{i+1}f_i(\gamma) \right)^{-1}=
{\gamma\,\prod_{j=1}^i(1+x_j)-\prod_{j=1}^i(1+\gamma\,x_j)\over 
\gamma\,\prod_{j=1}^{i+1}(1+x_j)-\prod_{j=1}^{i+1}(1+\gamma\,x_j)}.\tag 5.11
$$
Substituting this in (5.7) we get
$$\align
\sum_{\n\geq \0}D(\n; \gamma,\eta){\x^{\n}\over \n!}&=
\prod_{i=1}^{m-1}(1-\gamma\,x_{i+1}f_i(\gamma))^{-\beta}\\
&=\prod_{i=1}^{m-1}\left({\gamma\,\prod_{j=1}^i(1+x_j)-\prod_{j=1}^i(1+\gamma\,x_j)\over 
\gamma\,\prod_{j=1}^{i+1}(1+x_j)-\prod_{j=1}^{i+1}(1+\gamma\,x_j)}\right)^{-\beta}\\
&=\left\{{\gamma(1+x_1)-(1+\gamma\,x_1)\over
\gamma\,\prod_{j=1}^m(1+x_j)-\prod_{j=1}^m(1+\gamma\,x_j)}\right\}^{-\beta}\\
&=\bigl(1-\gamma\,e_2-(\gamma+\gamma^2)\,e_3-\ldots-(\gamma+\cdots+ \gamma^{m-1})\,e_m\bigr)^{-\beta},
\endalign
$$
as desired.\enddemo 

\heading{6. Combinatorial interpretations of the linearization coefficients}
\endheading
Combining Theorems 2 and 3, we immediately derive the following 
combinatorial interpretations of the linearization coefficients
of the orthogonal Sheffer polynomials.
\proclaim{Thoerem 4} The following identities hold :
$$\align
{\Cal H}(\n)&=|{\I}_{\n}|;\tag 6.1\\
{\Cal C}(\n;a)& =\sum_{\pi\in {\P}_{\n}}a^{\bloc\,\pi};\tag 6.2\\
{\Cal L}(\n;\alpha)&=\sum_{\pi\in {\D}_{\n}}(1+\alpha)^{\cyc\,\pi};\tag 6.3\\
 {\Cal M}(\n;\beta,c)&=
\sum_{\pi\in {\D}_{\n}}c^{\dec\,\pi}\beta^{\cyc\,\pi};\tag
6.4\\ {\Cal P}(\n;\delta,\eta)&=
\sum_{\pi\in
{\D}_{\n}}(\delta+i)^{\dec\,\pi}(\delta-i)^{\exc\,\pi}\eta^{\cyc\,\pi};\tag 6.5
\endalign $$
\endproclaim

As mentioned in Section 1, formula (6.1)--(6.4) have been established by other
methods in \cite{8, 14, 20, 41}. One notices that Theorem 4 shows that
the linearization coefficients of Hermite, Charlier, Laguerre
and Meixner polynomials are
actually the polynomials of some appropriate variables with nonnegative integral
coefficients. Therefore, the {\sl classical positivities} (cf. \cite{4}) of these coefficients become {\sl obvious}.
However, although it is evident from (6.5) that ${\Cal P}(\n;\delta,\eta)$ is a polynomial
of $\delta$ and $\eta$,  the fact that the coefficients are {\sl nonnegative integers} is not
obvious.

\proclaim{Proposition 5} The coefficients of the polynomial ${\Cal P}(\n;\delta,\eta)$ of 
$\delta$ and $\eta$ are nonnegative integers.
\endproclaim 
\demo{proof} We first prove that it is a polynomial with integral coefficients.
Cearly, it is sufficient to prove that in the
summation of (6.5) each term appears with its conjugate. For this end, we construct an involution $T$ on ${\D}_{\n}$ as 
follows. For each $\pi\in{\D}_{\n}$, we define $T(\pi)=\pi'$ by $\pi'(a)=b$
if and only if $\pi(b)=a$, where $a,\;b$ are elements of $[\n]$.
It is easy to see that $T$ is an involution without fixed point on ${\D}_{\n}$
such that 
$$
\Cyc\,\pi=\Cyc\, \pi',\quad\Dec\,\pi=\Exc\, \pi', \quad \Exc\,\pi=\Dec\, \pi'.\tag 6.6
$$ 
Therefore, in the summation of (6.5), the summand
corresponding to $\pi'$ is the {\sl conjugate} of that corresponding to $\pi'$.
It follows that ${\Cal P}(\n;\delta,\eta)$ is a polynomial in $\delta$ and $\eta$ with integral coefficients.
It remains to prove that the coefficients are positive. We shall prove this for $m\leq 4$, for the general cases
are similar to the case $m=4$. Set $P_n :=P_n(x; \delta, \eta)$. By (3.22) we have
$$
\varphi (P_{n_1}P_{n_2}P_{n_3})=
\sum_{s\geq 0}{n_1!n_2!n_3!(\eta)_s(\delta^2 +1)^s(2\,\delta)^{n_1+n_2+n_3-2s}\over
(s-n_1)!(s-n_2)!(s-n_3)!(n_1+n_2+n_3-2s)!}\cdot\tag 6.7
$$
Therefore, the proposition is true for $m\leq 3$. 
For $m=4$ we derive from the linearization formula (1.0) that
$$
\varphi (P_{n_1}P_{n_2}P_{n_3}P_{n_4})=\sum_{k\geq 0}{\varphi (P_kP_{n_1}P_{n_2})
\varphi (P_kP_{n_3}P_{n_4})\over \varphi (P_kP_k)}\cdot\tag 6.8
$$
By (6.7) and noting that $\varphi (P_kP_k)=(k!)^2(\eta)_k(\delta^2+1)^k$, we may conclude that
(6.8) is a polynomial with positive coefficients.
\enddemo

When $\delta =0$, the generating function (1.7) reduces to
$$
\sum_{n\geq 0}M_n(x;0,\eta){t^n\over n!}=
\bigl(1+t^2\bigr)^{-\eta /2}\exp \bigl( x \arctan t\bigr). \tag 6.9
$$

Note that the polynomials $i^n\,M_n(-ix; 0,\eta)$ were also introduced by Carlitz [11]. 
We first derive some consequences from Theorem 4 for ${\Cal P}(\n; 0,\eta)$ 
before we proceede to give another interesting interpretation in a different context. 

\proclaim{Proposition 6}
If $n=n_1+\cdots +n_m$ is odd, then ${\Cal P}(\n; 0,\eta)=0$.
\endproclaim
\demo{Proof}
Recall that the involution $T$ defined in the proof of Proposition 5 has the following properties :
if $T(\pi)=\pi'$, then
$$
\Cyc\,\pi=\Cyc\, \pi', \quad \Exc\,\pi=n-\Exc\, \pi'.\tag 6.10
$$ 
Whence, if $n=n_1+\cdots +n_m$ is odd, in the sum of (6.5) the weight of $\pi$ 
is killed by that of $\pi'$, i.e.,
$$
(-1)^{\exc\,\pi}i^n\eta^{\cyc\,\pi}
+(-1)^{\exc\,\pi'}i^n\eta^{\cyc\,\pi'}=0.
$$
Thus the total sum (6.5) is clearly zero.
\enddemo

Let ${\E}_{\n}$ be the set of permutations of ${\D}_{\n}$
of which all the cycles are of even length. 
It follows that ${\E}_{\n}=\emptyset$ if $n=n_1+\cdots +n_m$ is odd. 

\proclaim{Proposition 7} We have
$$
{\Cal P}(\n; 0,\eta)=\sum_{\pi\in {\E}_{\n}}(-1)^{\exc\,\pi}i^n\eta^{\cyc\,\pi}.\tag 6.11
$$
\endproclaim
\demo{Proof} Clearly, if we can define an involution 
$\theta\, :\, {\D}_{\n}\setminus{{\E}_{\n}}\longrightarrow {\D}_{\n}\setminus{{\E}_{\n}}$ 
such that $\theta (\pi)=\pi'$ with
$$
\Cyc\,\pi=\Cyc\,\pi'\quad\hbox{and}\quad \Exc\,\pi+\Exc\,\pi'\quad\hbox{is odd}.\tag 6.12
$$
The identity (6.10) would follow immediately. For each $\pi\in {\D}_{\n}\setminus{{\E}_{\n}}$, 
we may assume that the set of cycles of $\pi$ is totaly ordered (for example, according to 
the order of the highest ranking element of each cycle). Among the cycles of $\pi$, let 
$a_1\rightarrow a_2\rightarrow \cdots \rightarrow a_r$ 
be the least cycle with odd length $r$, then 
$\pi'=\theta(\pi)$ is defined by the same cycles as $\pi$, except that the least cycle of odd length is 
$a_r\rightarrow a_{r-1}\rightarrow \cdots \rightarrow a_1$. Clearly $\theta$ is an involution on 
${\D}_{\n}\setminus{{\E}_{\n}}$ and satisfies (6.12), for $\Exc\,\pi+\Exc\,\pi'\equiv r\equiv 1 \pmod {2}$.
\enddemo

The new interpretation of ${\Cal P}(\n; 0,\eta)$ relies on the model
of {\sl generalized up-down permutations}. Hereafter,
we consider a {\sl permutation} of any finite subset $A$ of 
${\Bbb N}_*$ as a linear arrangement $a_1a_2\cdots a_n$ of its elements.
 A permutation $\pi=a_1a_2\cdots a_n$ is said to be
{\sl up-down} if $c(a_1)<c(a_2)\geq c(a_3)< \cdots {}_{\geq}^<c(a_n)$, and 
{\sl basic} if it begins with its smallest element.
Also, we say that the {\sl length} of the permutation 
$\pi=a_1a_2\cdots a_n$ is $n$. 
Let ${\s}_{\bold n}$ be the set of all 
 permutations on $[\bold n]$ and ${\U}_{\bold n}$ the set of up-down
permutations on $[\bold n]$.
A {\sl record } of a permutation $\pi=a_1a_2\cdots a_n$
is an $a_j$ such that $i<j$ implies $a_i>a_j$. Denote by $\rec\,\pi$ the
number of records of $\pi$ and by $F(\pi)$ the first element of $\pi$.
A permutation $\pi=a_1a_2\cdots a_n$ 
is {\sl basic} if it begins with its 
smallest element. Let ${\B}_{\bold n}$ be the set of basic 
permutations of ${\s}_{\bold n}$. 

As noted by Gessel \cite{24}, every permutation $\pi=a_1\,a_2\cdots a_n$ has a 
unique factorization $\pi=\beta_1\beta_2\cdots \beta_k$ such that each 
$\beta_i$ is basic and $F(\beta_1)>F(\beta_2)>\cdots>F(\beta_k)$. 
Such a factorization
is called {\sl basic decomposition} of $\pi$ and the $\beta_i$
the {\sl basic components} of $\pi$. We call permutation {\sl reduced }
if it is a permutation of $[\bold n]$. To any permutation 
$\pi=a_1a_2\cdots a_n$ we may associate a reduced permutation, 
$\red(\pi)$, by replacing in $\pi$ the $j$th smallest element of color $i$
by $(i,\, j)$. Thus 
$$
\red\bigl((1,7)(2,9)(1,2)(2,6)\bigr)=(1,2)(2,2)(1,1)(2,1).
$$
The {\sl content} of the permutation $\pi=a_1a_2\cdots a_n$ is the set 
$\con(\pi)=\{a_1,\,a_2,\ldots,a_n\}$. We note that a permutation is determined by its reduction and its content.

A function $\omega$ defined on permutations (with values in some commutative
algebra over the rationals) is {\sl multiplicative} if  for all permutations
$\pi$ :
\roster
\item $\omega(\pi)=\omega(\red(\pi))$.
\item If $\beta_1 \beta_2\cdots \beta_k$ is the basic decomposition of $\pi$, 
then
$$\omega(\pi)=\omega(\beta_1)\,\omega(\beta_2)\cdots\omega(\beta_k).$$
\endroster
Thus a multiplicative function is determined by its values on reduced basic
permutations, and these may be chosen arbitrarily. One of the fundamental
properties of a multiplicative function is the so-called exponential
formula. The proof is similar to the case of one variable.
\proclaim{Lemma 8}
Let $\omega$ be a multiplicative function on permutations. For
any sequence ${\bold n}\geq {\bold 0}$,
set $g_{\bold n}=\sum_{\pi \in {\s}_{\bold n}}\omega (\pi)$ and 
$f_{\bold n}=\sum_{\pi \in {\B}_{\bold n}}\omega (\pi)$, then
$$
\sum_{{\bold n}\geq {\bold 0}}g_{\bold n}
{{\bold x}^{\bold n}\over {\bold n}!}
=\exp\sum_{{\bold n}>{\bold 0}}f_{\bold n}
{{\bold x}^{\bold n}\over {\bold n}!}.$$ 
\endproclaim
 
To each permutation $\pi$ of $[\bold n]$ we associate the weight
$w(\pi)=\eta^{\rec(\pi)}$ 
if $\pi$ is up-down and of even length or $0$ otherwise.
Clearly $S_{\bold n}(\eta)=\sum_{\pi \in {\S}_{\bold n}} w(\pi)$
and $B_{\bold n}(\eta)=\sum_{\pi \in {\B}_{\bold n}} w(\pi)$ 
count respectively the up-down  and 
basic up-down permutations of even length according to the records.
\proclaim{Lemma 9} We have
$$\sum_{{\bold n}\geq {\bold 0}}S_{\bold n}(\eta)
{{\bold x}^{\bold n}\over {\bold n}!}
=\Bigl(\sum_{n\geq 0}(-1)^n e_{2n}\Bigr)^{-\eta}.\leqno (6.13)$$
\endproclaim
\demo{Proof}As the weight $w(\pi)$ is multiplicative, by Lemma 8, we have 
$$\sum_{{\bold n}\geq {\bold 0}}S_{\bold n}(\eta)
{{\bold x}^{\bold n}\over {\bold n}!}
=\exp \sum_{{\bold n}> {\bold 0}}B_{\bold n}(\eta )  
{{\bold x}^{\bold n}\over {\bold n}!}
=\exp \eta\, \sum_{{\bold n}> {\bold 0}}B_{\bold n}(1)  
{{\bold x}^{\bold n}\over {\bold n}!}.\leqno (6.14)$$
But it is well-known \cite{Go-Ja, p. 234} that 
$$\exp \sum_{{\bold n} > {\bold 0}}B_{\bold n}(1)
{{\bold x}^{\bold n}\over {\bold n}}=
\sum_{{\bold n}\geq{\bold 0}}S_{\bold n}(1)
{{\bold x}^{\bold n}\over {\bold n}!}=\Bigl (\sum_{n\geq 0}(-1)^n 
e_{2n}\Bigr)^{-1}.$$
The proof is then complete by putting the last identity into (6.14).
\enddemo

\proclaim{Theorem 5}For any $\n=(n_1,\ldots, n_m)$, the polynomials ${\Cal P}(\n;0,\eta)$ are  
the generating polynomials of up-down permutations of even length on 
$[\bold n]$ weighted by records, viz., 
$$
 {\Cal P}(\n;0,\eta)=\cases \sum_{\pi\in {\U}_{\bold n}}
\eta^{\rec\,\pi},&\text{if $n_1+\cdots +n_m$ is even,}\\
0,&\text{if $n_1+\cdots +n_m$ is odd}.\endcases
$$   
\endproclaim
\demo{Proof}If $\delta=0$, the right-hand side of (3.13) reduces to
that of (6.13). 
The Theorem follows then from Lemma 9 and Theorem 2.
\enddemo

By comparison of Proposition 6 with Theorem 5, we derive the following
\proclaim{Corollary 4}We have
$$
\sum_{\pi\in {\E}_{\n}}(-1)^{\exc\,\pi}i^n\eta^{\cyc\,\pi}=\sum_{\pi\in {\U}_{\bold n}}\eta^{\rec\,\pi}.\tag 6.15
$$
\endproclaim
\demo{Remark} When $\n=(1, 1, \ldots, 1)$ and $\eta=1$,  identity (6.15) reduces to a well-known result \cite{43, Th. 4.7}.
\enddemo
In the case where $\eta=0$, the polynomials $p_n(x):=p_n(x;0,0)$ are no 
longer orthogonal with respect to $\varphi$. However, there is another 
linear functional $\zeta$ of some interest, which
may be defined by the generating function of 
the sequence $\zeta(x^n)\,(n\geq 0)$ :
$$\sum_{n\geq 0}\zeta(x^n){x^n\over n!}=\tan x.\tag 6.16$$
It then follows from Lemma 5 that
$$\zeta\bigl(\exp x\sum_{k=1}^m\arctan x_k\bigr)=
{\sum_{n\geq 0}(-1)^n e_{2n+1}\over \sum_{n\geq 0}(-1)^n e_{2n}}.$$
In other words, the following identity holds :
$$\sum_{\bold n}\zeta\Bigl(\prod_{k=1}^m p_{n_k}(x)\Bigr)
{{\bold x}^{\bold n}\over {\bold n}!}=
{\sum_{n\geq 0}(-1)^n e_{2n+1}\over \sum_{n\geq 0}(-1)^n e_{2n}}.
\leqno(6.17)$$
However, it is well-known that the right hand side of (6.17) is the 
generating function
of the numbers of up-down permutations of {\sl odd} length on $[\bold n]$
(cf. \cite{27, p. 196}). 
So, $\zeta\Bigl(\prod_{k=1}^m p_n(x)\Bigr)$ counts the
up-down permutations of {\sl odd} length on $[\bold n]$.
Especially, if $m=2$, we obtain the quasi-orthogonality :
$$\zeta\bigl(p_{n_1}(x)p_{n_2}(x)\bigr)=\cases {n_1}!\,{n_2}!, &
\text{if $|n_1-n_2|=1$};\\
0,&\text{otherwise}.\endcases $$
These results may be of some interest in the theory of {\sl coefficients 
extraction} of symmetric functions (cf. \cite{23}).

\heading{Acknowledgements}\endheading
This work was done while the author was visiting
Institute for Advanced Study as a member in 1989-1990 and
supported by NSF Grant DMS-8610730. The author is also indebted to
Professor Jet Wimp for helpful discussions 
and correspondence regarding Meixner-Pollaczek polynomials, which led 
particularly to the discovery of Lemma 1.
\bigskip

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\vskip 20pt
Running Head--{\sl  Combinatorics of Sheffer polynomials}.
\vskip 30pt
{\obeylines
D\'epartement de math\'ematique,\hfill
Universit\'e Louis-Pasteur,\hfill
7, rue Ren\'e-Descartes,\hfill
F-67084 Strasbourg, France\hfill}

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