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\begin{document}
\setlength{\baselineskip}{1.0\baselineskip}
\date{\small
Submitted: Apr 2, 2004; Accepted: Jul 20, 2004; Published: Aug 16, 2004}
\title{Orthogonal polynomials represented by $CW$-spheres} 
\author{G\'abor Hetyei\thanks{On leave from the R\'enyi Mathematical
	Institute of the Hungarian Academy of Sciences.\hfil\break 
{\bf 2000 Mathematics Subject Classification:} Primary 05E35; Secondary
06A07, 57Q15 \hfil\break
{\bf Key words and phrases:} partially ordered set, Eulerian, flag,
orthogonal polynomial.}\\ 
	Mathematics Department\\
	UNC Charlotte\\
	Charlotte, NC 28223}
\maketitle

\begin{abstract}
Given a sequence $\{Q_n(x)\}_{n=0}^{\infty}$ of symmetric orthogonal
polynomials, defined by a recurrence formula $Q_n(x)=\nu_n\cdot x\cdot
Q_{n-1}(x)-(\nu_n-1)\cdot Q_{n-2}(x)$ with integer $\nu_i$'s satisfying
$\nu_i\geq 2$, we construct a sequence of nested Eulerian
posets whose $ce$-index is a non-commutative
generalization of these polynomials. Using spherical shellings and direct
calculations of the $cd$-coefficients of the associated Eulerian posets
we obtain two new proofs for a bound on the true interval of orthogonality of
$\{Q_n(x)\}_{n=0}^{\infty}$. Either argument can replace the use of the
theory of chain sequences. Our $cd$-index calculations allow us to represent the
orthogonal polynomials as an explicit positive combination of terms of
the form $x^{n-2r}(x^2-1)^r$. Both proofs may be extended to the
case when the $\nu_i$'s are not integers and the second proof is still valid
when only $\nu_i>1$ is required.
The construction provides a new ``limited testing ground'' for Stanley's
non-negativity conjecture for Gorenstein$^*$ posets, and suggests the
existence of strong links between the theory of orthogonal polynomials
and flag-enumeration in Eulerian posets.  
\end{abstract}


\section*{Introduction}
In a recent paper~\cite{Hetyei-Tch} the present author constructed a
sequence of nested Eulerian partially ordered sets whose $ce$-index
generalizes the Tchebyshev polynomials of the first kind. The main goal
of that paper was to propose a new class of posets to test Stanley's
non-negativity conjecture~\cite[Conjecture 2.1]{Stanley-flag} on the
$cd$-index of Gorenstein$^*$ posets.

In this paper we construct a similar sequence $Q_0,Q_1,\ldots $ of
nested Eulerian posets for any sequence
$\{Q_n(x)\}_{n=0}^{\infty}$ of symmetric orthogonal polynomials
satisfying $Q_{-1}(x)=0$, $Q_0(x)=1$, $Q_1(x)=x$, and a recursion formula 
$Q_n(x)=\nu_n\cdot x\cdot Q_{n-1}(x)-(\nu_n-1)\cdot Q_{n-2}(x)$
for $n\geq 2$ with integers $\nu_n\geq 2$. Since these posets
arise as face posets of a sequence of $CW$-spheres closed under taking
(the boundary complexes of) faces, these sequences of posets may help
testing Stanley's conjecture the same way as the Tchebyshev
posets. (This possibility will be explained in the concluding
Section~\ref{S_cr}.) The study of the structure of these
posets, however, opens up also other, potentially even more
interesting directions of research.

The fact that the true interval of orthogonality of the orthogonal
polynomial systems considered is a subset of $[-1,1]$ is an easy
consequence of the non-negativity of the $cd$-index of the associated
Eulerian posets, which may be shown using spherical shelling. (It is
also fairly easy to extract a proof for non-integer $\nu_i$'s by
inspection of the integral case.) The same result in the classical
theory of orthogonal polynomials seems to depend on the theory of chain
sequences. Both shellings and chain sequences seem to be a tool to
``prove inequalities by induction'' in this context. Moreover, the
recursion formula for the non-commutative generalization of the
orthogonal polynomial systems considered seems to offer a very easy way
to find an explicit non-negative representation, which then may be
``projected down'' to the commutative case. It may be worth finding it
out in the future whether the theory of chain sequences is closer to the
first or second approach to the $cd$-coefficients, if it is close to
any of them. In this process either a new approach to prove
non-negativity results for $cd$-coefficients or new ways to prove
non-negativity results for orthogonal polynomials may be found.

Since it is a goal of the present paper to inspire collaboration between
researchers of orthogonal polynomials and Eulerian posets, experts of
either field will hopefully find some useful information and sufficient
pointers in the preliminary Section~\ref{S_Prelim}. Furthermore, this
section contains a brief (and somewhat ``unorthodox'') introduction to
spherical coordinates, which will be useful in describing our $CW$-spheres.

In Section~\ref{S_Lune} we define complexes of lunes on an
$(n-1)$-dimensional sphere. Every partially ordered set in each sequence
will be the face poset of a lune complex. We introduce a code system for
the faces, and show that each lune complex is a $CW$-sphere. A
fundamental recursion formula for the flag $f$-vector of lune complexes
is shown in Section~\ref{S_ce}. Instances of the $ce$-index form of the same
recursion clearly generalize of the fundamental recursion formula
of the orthogonal polynomials $Q_n(x)$. The fact that the lune complexes
are spherically shellable and thus have a non-negative $cd$-index is
shown in Section~\ref{S_sshell}. 

The connection between the non-negativity of the $cd$-index of a lune
complex and the statement on the true interval of orthogonality of the
polynomials $Q_n(x)$ is explained in Section~\ref{S_ops}. We also
provide the first proof of the non-negativity of the $cd$-index by the
use of spherical shelling. This approach needs the assumption $\nu_n\geq 2$ for
$n\geq 2$, while in the traditional approach using chain sequences 
only $\nu_n>1$ is needed. This ``gap'' could
probably be filled by using a more general definition for our lune
complexes. The study of this option is omitted, since in Section~\ref{S_expl}
we show how the $cd$-index recursion may be used to obtain an explicit
formula for the $cd$-coefficients of our face posets, and how these
calculations may be ``projected down'' to obtain an explicit
representation of our orthogonal polynomials as a positive combination
of terms of the form $x^{n-2r}(x^2-1)^r$. This proof extends also to the
weakened condition $\nu_n>1$, and does not require constructing Eulerian
posets. However, it seems more difficult to guess the formula found without the
inspiration coming from $cd$-index calculations.

Section~\ref{S_Tch} contains the proof of the fact that for the case
when $\nu_n=2$ for $n\geq 2$, the face posets of $CW$-spheres
constructed in this paper are isomorphic to the duals of the Tchebyshev
posets constructed in~\cite{Hetyei-Tch}. Finally, we present 
our suggestions for future research in Section~\ref{S_cr}.    


\section{Preliminaries}
\label{S_Prelim}

\subsection{Eulerian posets}

A partially ordered set $P$ is {\em graded} if it has a unique minimum
element $\0$, a unique maximum element $\1$, and a rank function $\rho$. 
Here $\rho(\0)=0$, and $\rho(\1)$ is the rank of $P$.
Given a graded partially ordered set $P$ of rank $n+1$ and
$S\subseteq\{1,\ldots,n\}$, $f_S(P)$ denotes the number of saturated
chains of the {\em $S$-rank selected subposet} $P_S=\{x\in P\::\:
\rho(x)\in S\}\cup \{\0,\1\}$. The vector $(f_S(P)\::\: S\subseteq
\{1,\ldots,n\})$ is called the {\em flag $f$-vector} of $P$. 
Equivalent encodings of the flag $f$-vector include the {\em flag
  $h$-vector} $(h_S(P)\::\: S\subseteq
\{1,\ldots,n\})$  (see~\cite{Stanley-flag}) and the flag $\ell$-vector
$(\ell_S(P)\::\: S\subseteq \{1,\ldots,n\})$
(see~\cite{Billera-Hetyei-planar}), given by $h_S(P)=\sum_{T\subseteq S}
(-1)^{|S\setminus T|} f_T(P)$ and
$\ell_S(P)=(-1)^{n-|S|}\sum_{T\supseteq [1,n]\setminus S}
(-1)^{|T|} f_T(P)$ respectively. 
A graded poset is {\em Eulerian} if every interval
$[x,y]$ of positive rank in it satisfies $\sum _{x\leq z\leq y}(-1)^{\rho
  (z)}=0$. All linear relations holding for the flag $f$-vector of an
arbitrary Eulerian poset of rank $n$ were determined by Bayer and
Billera in \cite{Bayer-Billera}. These linear relations were rephrased
by J.\ Fine as follows (see the paper~\cite{Bayer-Klapper} by Bayer and
Klapper). For any $S\subseteq \{1,\ldots,n\}$ define 
the non-commutative monomial $u_S=u_1\ldots u_n$ by setting 
$$
u_i=\left\{
\begin{array}{ll}
b&\mbox{if $i\in S$,}\\
a&\mbox{if $i\not\in S$.}\\
\end{array}
\right.
$$
Then the polynomial $\Psi_{ab}(P)=\sum_S h_S u_S$ in non-commuting
variables $a$ and $b$, called the {\em $ab$-index of $P$},  is a
polynomial of $c=a+b$ and $d=ab+ba$. This form of $\Psi_{ab}(P)$ is
called the {\em $cd$-index of $P$}. Further proofs of the existence of
the $cd$-index may be found in \cite{Bayer-Hetyei-d}, in
\cite{Ehrenborg-Readdy}, and in \cite{Stanley-flag}.
It was noted by Stanley in
\cite{Stanley-flag} that the existence of the $cd$-index is equivalent
to saying that the $ab$-index rewritten as a polynomial of $c=a+b$ and
$e=a-b$ involves only even powers of $e$. It was observed by Bayer and
Hetyei in \cite{Bayer-Hetyei} that the coefficients of the resulting
{\em $ce$-index} may be computed using a formula that is analogous to
the definition of the flag $\ell$-vector. In fact, given a $ce$-word $u_1\cdots
u_n$, let $S$ be the set of positions $i$ satisfying $u_i=e$. Then the
coefficient $L_S(P)$ of the $ce$-word is given by 
\begin{equation}
\label{E_ce}
L_S(P)=(-1)^{n-|S|}\sum_{T\supseteq [1,n]\setminus S} \left(\frac{-1}{
  2}\right)^{|T|} f_T(P).
\end{equation}
 The fact that the $ce$-index is a polynomial
of $c$ and $e^2$ is equivalent to stating that $L_S(P)=0$ unless $S$ is
an {\em even} set, that is, a union of disjoint intervals of even
cardinality.   

A poset $P$ is called {\em near-Eulerian} if it may be
obtained from an Eulerian poset $\ssp P$, called the
{\em semisuspension} of
$P$, by removing one coatom. The poset $\ssp P$
may be uniquely reconstructed from $P$ by adding a coatom $x$ which
covers all $y \in P$ for which $[y,\1]$ is the three element chain.

\subsection{Spherical shellings}

The posets we consider in this paper may be represented as face posets
of $CW$-spheres. We call a poset~$P$
with $\0$ a {\em $CW$-poset} when for all $x>\0$ in $P$ the geometric
realization $|(\0,x)|$ of the open interval $(\0,x)$ is homeomorphic to
a sphere. By~\cite{Bjorner_CW}, $P$ is a $CW$-poset if and only if it is
the face poset $P(\Omega)$ of a regular $CW$-complex $\Omega$.
When $\Omega$ is a $CW$-sphere, the poset $P_1(\Omega)$,
obtained from $\Omega$ by adding a unique maximum element $\1$, is Eulerian.
Stanley observed the following; see \cite[Lemma 2.1]{Stanley-flag}.
Let $\Omega$ be an $n$-dimensional $CW$-sphere, and $\sigma$ an (open)
facet of $\Omega$. Let $\Omega'$ be obtained from $\Omega$ by
subdividing the closure $\overline{\sigma}$ of $\sigma$ into a regular
$CW$-complex with two facets $\sigma_1$ and $\sigma_2$ such that
the boundary $\partial\sigma$ remains the same and $\overline{\sigma_1}\cap
\overline{\sigma_2}$ is a regular $(n-1)$-dimensional $CW$-ball $\Gamma$. Then 
we have
\begin{equation}
\label{E_Slemma}
   \Phi(P_1(\Omega'))
\: - \:
   \Phi(P_1(\Omega))
\: = \:
   \Phi(\ssp P_1(\Gamma)) \cdot c
\: - \:
   \Phi(P_1(\partial\Gamma))  \cdot (c^2-d) .
\end{equation}
In~\cite{Stanley-flag} Stanley uses the above observation to prove that 
the face poset of a {\em spherically shellable} $CW$-sphere has
non-negative $cd$-index.
A complex $\Omega$ or its face poset $P_{1}(\Omega )$ is called
{\em spherically shellable} (or $S$--shellable)
if either $\Omega = \{ \emptyset \}$ (and so
$P_{1}(\Omega )$ is the two-element chain $\{\0 < \1\}$), or 
else we can linearly order the facets (open $n$-cells)
$F_{1},F_{2},\ldots ,F_{m}$ of $\Omega$ such that for all
$1 \leq i \leq m$ the following two conditions hold:
\begin{itemize}
\item[{\rm ($S$-a)}]
 $\partial \overline{F_{1}}$ is $S$--shellable of dimension $n-1$.

\item[{\rm ($S$-b)}] For $2 \leq i \leq m-1$, let 
$\Gamma_{i} \df \cl [\partial \overline{F_{i}} -
((\overline{F_{1}} \cup \cdots \cup \overline{F_{i-1}}) \cap
\overline{F_{i}})]$. (Here both $\cl$ and $\overline{\ }$ denote
  closure.) Then $P_{1}(\Gamma_{i})$ is near-Eulerian of dimension
$n-1$, and the semisuspension $\ssp\Gamma_{i}$ is $S$--shellable,
with the first facet of the shelling being the facet $\tau =
\tau_{i}$ adjoined to $\Gamma_{i}$ to obtain $\ssp\Gamma_{i}$.
\end{itemize}

\subsection{Orthogonal polynomials}
\label{ss_op}

For fundamental facts on orthogonal polynomials our main reference is
Chihara's book~\cite{Chihara}. A {\em moment functional} ${\mathcal L}$
is a linear map ${\mathbb C}[x]\rightarrow {\mathbb C}$. A sequence of
polynomials $\{P_n (x)\}_{n=0}^{\infty}$ is an {\em orthogonal polynomial
sequence (OPS)} with respect to ${\mathcal L}$ if $P_n(x)$ has degree $n$, 
${\mathcal L}[P_m(x)P_n(x)]=0$ for $m\neq n$, and ${\mathcal
  L}[P_n^2 (x)]\neq 0$ for all $n$. Such a system exists if and only if
${\mathcal L}$ is {\em quasi-definite} (see~\cite[Ch. I, Theorem
  3.1]{Chihara},  the term quasi-definite is introduced in~\cite[Ch. I,
  Definition 3.2]{Chihara}). Whenever an OPS exists, each of its
elements is determined up to a non-zero constant factor (see
\cite[Ch. I, Corollary of Theorem 2.2]{Chihara}).  

In this paper we consider orthogonal polynomial systems defined
recursively. Every monic OPS $\{P_n (x)\}_{n=0}^{\infty}$ 
may be described by a recurrence formula of the form 
\begin{equation}
\label{E_moprec}
P_n(x)=(x-c_n)P_{n-1}(x)-\lambda_n P_{n-2}(x) \quad\mbox{$n=1,2,3,\ldots$}
\end{equation}
where $P_{-1}(x)=0$, $P_0(x)=1$, the numbers $c_n$ and
$\lambda_n$ are constants, $\lambda_n\neq 0$ for $n\geq 2$, and
$\lambda_1$ is arbitrary (see \cite[Ch. I, Theorem 4.1]{Chihara}).
Conversely, by Favard's theorem~\cite[Ch. I, Theorem 4.4]{Chihara}, for every
sequence of monic polynomials defined in the above way there is a unique
quasi-definite moment functional ${\mathcal L}$ such that ${\mathcal
  L}[1]=\lambda_1$ and $\{P_n (x)\}_{n=0}^{\infty}$  is the monic OPS
with respect to ${\mathcal L}$. 

Due to geometric reasons, the sequences of orthogonal polynomials we
consider are {\em symmetric}, which is equivalent to saying that the
coefficients $c_n$ in (\ref{E_moprec}) are all zero, or that
$P_n(x)=(-1)^n P_n(-x)$ for all $n$ (see \cite[Ch. I, Theorem 4.3]{Chihara}).
We will also assume that the coefficients $\lambda_n$ are {\em real and
  positive}. According to the theorems cited above this is equivalent to
assuming that ${\mathcal L}$ is {\em positive definite}, i.e.,
${\mathcal L}[\pi(x)]>0$ for every polynomial $\pi(x)$ that is not
identically zero and is non-negative for all real $x$~\cite[Ch. I,
  Definition 3.1]{Chihara}. As a consequence of~\cite[Ch. I, Theorem
  5.2]{Chihara} the zeros of the polynomials $P_n(x)$ are all simple and
real. 

The smallest closed interval $[\xi_1,\eta_1]$ containing all zeros of an OPS 
is called the {\em true interval of orthogonality} of the
OPS (see \cite[Ch. I, Definition 5.2]{Chihara}, and the next sentence).
One way to estimate this closed interval is by
the use of {\em chain sequences}. A sequence $\{a_n\}_{n=1}^{\infty}$ is
a chain sequence if there is a sequence $\{g_k\}_{k=0}^{\infty}$
satisfying $0\leq g_0<1$ and $0<g_n<1$ for $n\geq 1$, such that
$a_n=(1-g_{n-1})g_n$ holds for $n\geq 1$ (see \cite[Ch. III, Definition
  5.1]{Chihara}). According  
to~\cite[Ch. III, Exercise 2.1]{Chihara}, for a symmetric OPS given by
(\ref{E_moprec}), the true interval of orthogonality is $[-a,a]$ where
$a$ is the least positive number for which
$\{a^{-2}\lambda_{n+1}\}_{n=1}^{\infty}$ is a chain sequence. (This
exercise an easy consequence of~\cite[Ch. III, Theorem 2.1]{Chihara}.)  

\subsection{Spherical coordinates}

Spherical coordinates are often used in mathematical physics and in the
theory to of group representations to parameterize the points of an
$(n-1)$-dimensional sphere. In this paper we consider the {\em standard
$(n-1)$-sphere} $\{(x_1,\ldots,x_n)\::\: x_1^2+\cdots+x_n^2=1\}$ in an
$n$-dimensional Euclidean space. We parameterize this sphere using the
the set of spherical vectors $\{(\theta_1,\ldots,\theta_{n-1})\::\:
0\leq \theta_1,\ldots,\theta_{n-2}\leq \pi, 0\leq \theta_{n-1}\leq 2\pi
\}$, as given by the system of equations
\begin{equation}
\label{E_spc}
\begin{array}{rcl}
x_1&=&\cos(\theta_1)\\
x_2&=&\sin(\theta_1)\cos(\theta_2)\\
&\vdots&\\
x_i&=&\sin(\theta_1)\sin(\theta_2)\cdots\sin(\theta_{i-1})\cos(\theta_{i})\\
&\vdots&\\ 
x_{n-1}&=&
\sin(\theta_1)\sin(\theta_2)\cdots\sin(\theta_{n-2})\cos(\theta_{n-1})\\
x_n&=&
\sin(\theta_1)\sin(\theta_2)\cdots\sin(\theta_{n-1})
\end{array}
\end{equation}
The classical literature (a sample reference is Vilenkin's~\cite[Chapter
IX, p.\ 435--437]{Vilenkin}) seems to be satisfied stating about this (or a
similar) parameterization that ``for almost all points such a system of
parameters is uniquely defined''. (To be able to make such a  statement,
the restrictions on the spherical coordinates need to be strengthened
somewhat, for example to $\theta_i<\pi$ for $i<n-1$ and $\theta_{n-1}<2\pi$.)

In this paper we study $CW$-complexes on the unit sphere, whose
combinatorial structure is more transparent if we are allowed to choose 
some vertices to be points with non-unique spherical coordinates. Thus we
need to make our statements little more precise. For
completeness sake, we sketch some of the proofs.  

\begin{definition}
\label{D_sc}
We call the spherical vectors $(\theta_1,\ldots,\theta_{n-1})$ and
$(\theta_1',\ldots,\theta_{n-1}')$ {\em equivalent} if $\theta_i-\theta_i'$
is an integer multiple of $2\pi$ whenever all $j<i$ satisfies
$\theta_j\not\in\{0,\pi\}$.  
\end{definition}
In other words, we read our spherical vectors from left to right, and
stop reading once we find the first $0$ or $\pi$. No matter what
coordinates follow, the spherical vector belongs to the same
equivalence class, and we make no other identification. For example, for
$n=6$, the spherical vector $(\pi/2, 1, \pi,2,3)$ is equivalent 
to $(\pi/2, 1, \pi,1,2\pi)$. We represent the equivalence class of
these spherical vectors by $(\pi/2, 1, \pi,*,*)$, i.e., we replace the
coordinates that ``do not matter'' with a star. If we are forced to read
our vectors till the end, we identify $0$ and $2\pi$ in the last coordinate.
Note also that in this paper we require every $n$-dimensional spherical
vector to  belong to $[0,\pi]^{n-1}\times [0,2\pi]$, other
sources may use different restrictions.
\begin{definition}
\label{D_length}
Assuming $\theta_{\ell}\in\{0,\pi,2\pi\}$ for some ${\ell}\leq n-1$, and
$\theta_i\not\in\{0,\pi,2\pi\}$ for all $i<\ell$, we call the code
$(\theta_1,\ldots,\theta_{\ell},*,\ldots,*)$ the {\em simplified code} of the
corresponding equivalence class of spherical vectors, and $\ell$ the {\em
  length} of the class, denoted by
$\ell(\theta_1,\ldots,\theta_{\ell},*,\ldots,*)$. If $\theta_i\not\in
\{0,\pi,2\pi\}$ for all $i$, we set $\ell(\theta_1,\ldots,\theta_{n-1}):=n$.
\end{definition}

\begin{proposition}
The system of equations (\ref{E_spc}) defines a bijection between
equivalence classes of spherical coordinates and points of the unit sphere.
\end{proposition}
\begin{proof}
The fact that $x_1^2+\cdots+x_n^2=1$ for the $x_i$'s given by
(\ref{E_spc}) is well-known and straightforward. Hence we may consider
the (\ref{E_spc}) as the definition of a map 
$$\Xi: \{(\theta_1,\ldots,\theta_{n-1}):
0\leq \theta_1,\ldots,\theta_{n-2}\leq \pi, 0\leq \theta_{n-1}\leq 2\pi
\}\rightarrow \left\{(x_1,\ldots,x_n): \sum_{i=1}^n x_i^2=1\right\}.$$  
If $\theta_i$ is $0$ or $\pi$ for some $i$ then $x_{i+1}=\cdots=x_n=0$
no matter what the subsequent spherical coordinates are, so $\Phi$ takes
equivalent spherical vectors into the same point. The verification of
the fact that $\Phi$ takes different equivalence
classes into different points is straightforward. Surjectivity
may be shown by and easy induction on $n$.
\end{proof}

Introducing 
$\widetilde{x_k}=\sin(\theta_1)\sin(\theta_2)\cdots\sin(\theta_{k})$
for $k=1,2,\ldots, n-1$, it is easy to show that 
$$
\widetilde{x_k}=
\left\{
\begin{array}{ll}
\sqrt{1-x_1^2-\cdots-x_k^2}&\mbox{if $1\leq k\leq n-2$,}\\
x_n&\mbox{if $k=n-1$.}\\
\end{array}
\right.
$$
Obviously the length of an equivalence class of spherical vectors
determines the length of the ``tail of zeros'' at the end of the
rectangular representation: 
\begin{proposition}
\label{P_tail}
An equivalence class of spherical vectors has length $\ell\leq n-1$ if and only
if the rectangular representation of the same point satisfies
$x_{\ell+1}=\cdots=x_n=0$ and $x_{\ell}\neq 0$. The length is $n$
exactly when $x_n\neq 0$. 
\end{proposition}

Note next that subjecting $\theta_{n-1}$ to the same restrictions as the
other coordinates, i.e., 
restricting $\theta_{n-1}$ to $0\leq \theta_{n-1}\leq
\pi$ is equivalent to setting $x_n\geq 0$. In other words:
\begin{proposition}
\label{P_proj}
The restriction of the parameterization (\ref{E_spc}) 
to $\{(\theta_1,\ldots,\theta_{n-1})\::\: 
0\leq \theta_1,\ldots,\theta_{n-1}\leq \pi\}$ yields the
hemisphere $\{(x_1,\ldots,x_n)\::\: x_1^2+\cdots+x_n^2=1, x_n\geq
0\}$ as its surjective image.
\end{proposition}
Finally, the boundary of this hemisphere is again a sphere:
\begin{proposition}
\label{P_hsb}
The set of points that are representable with spherical coordinates\\
$(\theta_1,\ldots,\theta_{n-1})$ satisfying 
$\theta_{n-1}\in\{0,\pi\}$ is the $(n-2)$-sphere  
$\{(x_1,\ldots,x_n)\::\: x_1^2+\cdots+x_n^2=1, x_n=0\}$. The restriction
of the projection $(x_1,\ldots,x_{n})\mapsto 
(x_1,\ldots,x_{n-1})$ to this sphere is a homeomorphism with the
standard $(n-2)$-sphere, which is may be
described at the level of spherical coordinates by
$$
\Pi_{n}: (\theta_1,\ldots,\theta_{n-1})\mapsto
\left\{
\begin{array}{ll}
(\theta_1,\ldots,\theta_{n-3},\theta_{n-2})
&\mbox{if $\theta_{n-1}=0$,}\\
(\theta_1,\ldots,\theta_{n-3},2\pi-\theta_{n-2})
&\mbox{if $\theta_{n-1}=\pi$.}\\ 
\end{array}
\right.
$$ 
\end{proposition}
In fact, $x_n=0$ is equivalent to stating that the length of the
corresponding spherical vector is at most $n-1$, which is 
equivalent to allowing $\theta_{n-1}\in\{0,\pi\}$. Comparing the
parameterization (\ref{E_spc}) for the standard $(n-2)$-sphere and its
embedding into the hyperplane $x_n=0$ yields that the first $n-3$
spherical coordinates may be identified, while the only role of choosing
$\theta_{n-1}\in\{0,\pi\}$ in the embedded version is to set the sign of
$x_{n-2}$ properly: $\theta_{n-1}=0$ corresponds to $x_{n-2}\geq 0$
while $\theta_{n-1}=\pi$ corresponds to $x_{n-2}\leq 0$.
Precisely the same goal may be achieved by replacing 
$\theta_{n-2}\in [0,\pi]$ with $2\pi-\theta_{n-2}\in [\pi,2\pi]$ when
necessary.



\section{The lune complex $L(m_1,\ldots,m_n)$}
\label{S_Lune}

In this section we construct a spherical $CW$-complex
$L(m_1,\ldots,m_n)$ whose $ce$-index we use to 
generalize certain sequences of orthogonal polynomials in Section
\ref{S_ce}.  As a first step, consider the following $r$-dimensional lunes and
hemispheres.

\begin{proposition}
\label{P_hs}
Assume $0\leq r\leq n-2$ is an integer.
Given $\sigma_i\in [0,\pi]$ for $r+1\leq i\leq n-2$ and $\sigma_{n-1}\in
[0,2\pi]$, the set of spherical vectors  
$$
(*,*,\ldots,*,\sigma_{r+1},\ldots,\sigma_{n-1})
:=\{(\theta_1,\ldots,\theta_{n-1})\::\:
0\leq \theta_1,\ldots,\theta_{r}\leq \pi, \quad \theta_i=\sigma_i \quad
\mbox{for $i\geq r$}\}
$$
is an $r$-dimensional hemisphere, i.e., the intersection of an
$r$-dimensional sphere centered at the origin with a half-space whose
boundary contains the origin.
\end{proposition}
\begin{proof}
We proceed by induction on $n-1-r$. If $r=n-2$ then, by (\ref{E_spc}),
there is no restriction on $x_1,\ldots,x_{n-2}$, while the last two
rectangular coordinates are constrained by
$x_{n-1}=\cos(\sigma_{n-1})\cdot \widetilde{x}_{n-2}$
and $x_{n}=\sin(\sigma_{n-1})\cdot \widetilde{x}_{n-2}$. It is easy to show
that these restrictions are equivalent to setting
\begin{equation}
\label{E_hs}
\begin{array}{rcl}
x_1^2+\cdots+x_n^2 &=& 1,\\
-\sin(\sigma_{n-1})\cdot x_{n-1}+\cos(\sigma_{n-1}) \cdot x_n&=&0, \quad
\mbox{and}\\  
\cos(\sigma_{n-1})\cdot x_{n-1}+\sin(\sigma_{n-1})\cdot x_n &\geq &0.\\  
\end{array}
\end{equation}
In fact, the second equation is equivalent to guaranteeing that the vector
$(x_{n-1},x_n)$ is a multiple of
$(\cos(\sigma_{n-1}),\sin(\sigma_{n-1}))$, the first restricts its value
to\\
$\pm(\cos(\sigma_{n-1})\cdot \widetilde{x}_{n-2},
\sin(\sigma_{n-1})\cdot \widetilde{x}_{n-2})$, while the last one picks the
correct sign. 

The resulting hemisphere may be parameterized by
$(\theta_1,\ldots,\theta_{n-2})$ in spherical coordinates and
$(x_1,\ldots,x_{n-2},\widetilde{x}_{n-2})$ in rectangular
coordinates. This parameterization represents the hemisphere as the set of
vectors with non-negative last coordinate. The above argument does not
change if we start with the hemisphere given by $x_n\geq 0$ instead of
the entire sphere. Hence we may repeat it to prove our claim for
$r=n-3$, and so on.
\end{proof}

\begin{proposition}
\label{P_lune}
Assume $1\leq r\leq n-2$ is an integer, and $0\leq \alpha<\beta\leq \pi$ are
such that $[\alpha,\beta]\neq [0,\pi]$. Given
$\sigma_i\in [0,\pi]$ for $r+1\leq i\leq n-2$, and $\sigma_{n-1}\in
[0,2\pi]$, the set of spherical vectors  
$(*,\ldots,*,[\alpha,\beta],\sigma_{r+1},\ldots,\sigma_{n-1})$,
satisfying $\alpha\leq \theta_{r}\leq \beta$ and $\theta_i=\sigma_i$ for
$i\geq r+1$, 
is an $r$-dimensional closed region. The boundary of this
region is the union of the $(r-1)$-dimensional hemispheres
$(*,\ldots,*,\alpha,\sigma_{r+1},\ldots,\sigma_{n-1})$ and
$(*,\ldots,*,\beta,\sigma_{r+1},\ldots,\sigma_{n-1})$.
Similarly, for $r=n-1$, given $0\leq \alpha<\beta\leq 2\pi$, where
$[\alpha,\beta]\neq [0,2\pi]$, the set of
spherical vectors  $(*,\ldots,*,[\alpha,\beta])$ defined by $\alpha\leq
\theta_{n-1}\leq \beta$ 
is an $(n-1)$-dimensional closed region
with boundary $(*,\ldots,*,\alpha)\cup(*,\ldots,*,\beta)$. 
\end{proposition}
\begin{proof}
In analogy to the proof of Proposition~\ref{P_hs} we may proceed by
induction on $n-1-r$ and the only interesting case is the induction
basis $r=n-1$, since the lower dimensional cases may be obtained
by reparameterizing the hemispheres obtained along the way. 
Again, there is no essential restriction on $x_1,\ldots, x_{n-2}$. Let
us fix these coordinates. Then $\theta_{n-1}\in[\alpha,\beta]$ is equivalent
to stating that the vector 
$(x_{n-1},x_n)$ is on an arc of radius $\widetilde{x}_{n-2}$ with endpoints 
corresponding $\widetilde{x}_{n-2}\cdot (\cos(\alpha),\sin(\alpha))$ and
$\widetilde{x}_{n-2}\cdot(\cos(\beta),\sin(\beta))$. Equivalently,
$(x_{n-1},x_n)$ is either $(0,0)$ , or it is on the same side of
the line connecting $(0,0)$ with $(\cos(\alpha),\sin(\alpha))$ as
$(\cos(\beta),\sin(\beta))$, and vice versa.  
In analogy to (\ref{E_hs}) we may obtain the following equivalent
description of $(*,\ldots,*,[\alpha,\beta])$:
\begin{equation}
\label{E_lune}
\begin{array}{rcl}
x_1^2+\cdots+x_n^2 &=& 1,\\
\sin(\beta-\alpha)\cdot \left(-\sin(\alpha)\cdot x_{n-1}+\cos(\alpha) \cdot x_n\right)&\geq&0, \quad
\mbox{and}\\  
\sin(\alpha-\beta)\cdot \left(-\sin(\beta)\cdot x_{n-1}+\cos(\beta) \cdot x_n\right)&\geq&0.\\  
\end{array}
\end{equation}
Hence $(*,\ldots,*,[\alpha,\beta])$ is the intersection of two
half-spaces, containing the origin on their boundary, and of the unit
$(n-1)$-sphere. The boundary of the resulting region is the intersection
of the $(n-1)$-sphere with either of the hyperplanes defining the two
half-spaces.
\end{proof}
Generalizing the $3$-dimensional terminology, we call a region
$$(*,\ldots,*,[\alpha,\beta],\sigma_{r+1},\ldots,\sigma_{n-1})$$
an {\em $r$-dimensional lune}. 
Obviously, each equivalence class of spherical vectors is either
completely contained in a lune
$(*,\ldots,*,[\alpha,\beta],\sigma_{r+1},\ldots,\sigma_{n-1})$ or it is
disjoint from it. Hence we may extend our equivalence relation to the
code of the lunes considered in the obvious way.
\begin{corollary}
\label{C_Lequiv}
The $r$-dimensional lunes
$$(*,\ldots,*,[\alpha,\beta],\sigma_{r+1},\ldots,\sigma_{n-1})\mbox{
  and }
(*,\ldots,*,[\alpha',\beta'],\sigma'_{r+1},\ldots,\sigma'_{n-1})$$
are equal if and only if $\alpha=\alpha'$, $\beta=\beta'$, and
$\sigma_i=\sigma'_i$ whenever $\sigma_j\not\in\{0,\pi\}$ holds for
$r+1\leq j<i$.
\end{corollary}
Thus we may extend our simplified notation for equivalence classes of
spherical vectors to lunes. For example, for
$n=6$, the lune $(*, [1,2], 2, \pi,3)$ is equal to $(*, [1,2], 2,
\pi,\sqrt{2})$. Both codes of this same $2$-dimensional lune 
may be simplified to $(*, [1,2], 2, \pi,*)$. Using this simplified
notation, every lune considered has a unique code of the form
$$(*,\ldots,*,[\alpha,\beta], \sigma_{r+1},\ldots,\sigma_{\ell},*,\ldots,*)$$
where $\sigma_i\not\in \{0,\pi,2\pi\}$ for $r+1\leq i\leq \min ({\ell}-1,n-1)$,
and $\sigma_{\ell}\in \{0,\pi\}$ if ${\ell}\leq n-2$.
\begin{definition}
Extending Definition~\ref{D_length}, 
we call $(*,\ldots,*,[\alpha,\beta],
\sigma_{r+1},\ldots,\sigma_{\ell},*,\ldots,*)$ above the {\em simplified
  code} of the lune, and $\ell$ its {\em length} if
$\sigma_{\ell}\in\{0,\pi,2\pi\}$. We set the length to be 
$n$ if $r=n-1$ or the simplified code is 
$(*,\ldots,*,[\alpha,\beta], \sigma_{r+1},\ldots,\sigma_{n-1})$ where
$\sigma_{n-1}\not\in\{0,\pi,2\pi\}$. 
\end{definition}
It is easy to verify the length of a lune is greater or equal to the
length of any lune or spherical vector contained in it.

\begin{remark}
\label{R_hsl}
{\em 
A hemisphere
$(*,*,\ldots,*,\sigma_{r+1},\ldots,\sigma_{n-1})$  may be considered as
a generalized lune\hfill\\
$(*,\ldots,*,[\alpha,\beta],\sigma_{r+1},\ldots,\sigma_{n-1})$, 
satisfying $\alpha=0$ and $\beta=\pi$. (This was exactly
the excluded choice of $\alpha$ and $\beta$ above.)
In the study of lune complex $L(m_1,\ldots,m_n)$ we will use
the obvious homeomorphism 
$$
\rest{\phi^r_{\alpha,\beta}}{(*,\ldots,*,[\alpha,\beta],\sigma_{r+1},\ldots,\sigma_{n-1})} :
(*,\ldots,*,[\alpha,\beta],\sigma_{r+1},\ldots,\sigma_{n-1})
\rightarrow (*,\ldots,*,\sigma_{r+1},\ldots,\sigma_{n-1})
$$
given by
$$
\phi^r_{\alpha,\beta}\left( (\theta_1,\ldots,\theta_{n-1})\right):=
\left(\theta_1,\ldots,\theta_{r-1},(\theta_r-\alpha)\cdot 
\frac{\pi}{\beta-\alpha}
,\theta_{r+1},\ldots,\theta_{n-1}\right).
$$
}
\end{remark}

\begin{definition}
Given a vector of positive integers $(m_1,\ldots,m_{n-1})$ satisfying
$m_i\geq 2$, we define the {\em lune complex $L(m_1,\ldots,m_{n-1})$} as the
following $CW$-complex on the $(n-1)$-sphere:
\begin{itemize}
\item[(i)] Its vertices are all points with spherical coordinates 
$(t_1\cdot\frac{\pi}{m_1},\ldots,t_{n-2}\cdot\frac{\pi}{m_{n-2}},
  t_{n-1}\cdot\frac{2\pi}{m_{n-1}})$, where each $t_j\in [0,m_j]$ is 
  an integer.
\item[(ii)] For $1\leq r\leq n-2$, its $r$-dimensional faces are lunes 
$$\left(*,\ldots,*,\left[s_r\cdot \frac{\pi}{m_r},(s_r+1)\cdot
  \frac{\pi}{m_r}\right], s_{r+1}\cdot \frac{\pi}{m_{r+1}},
  \ldots,s_{n-2}\cdot \frac{\pi}{m_{n-2}},s_{n-1}\cdot
  \frac{2\pi}{m_{n-1}}\right),$$  
where each $s_j$ is an integer, $s_r\in [0,m_r-1]$, and
$s_j\in [0,m_j]$ for $j>r$.  
\item[(iii)] Its $(n-1)$-faces (facets) are lunes 
$
\left(*,\ldots,*,\left[s_{n-1}\cdot \frac{2\pi}{m_{n-1}},(s_{n-1}+1)\cdot
  \frac{2\pi}{m_{n-1}}\right]\right)$, where $0\leq s_{n-1}\leq
m_{n-1}-1$ is an integer.
\end{itemize}
\end{definition}
\begin{theorem}
\label{T_Lsphere}
Assuming $m_1,\ldots,m_{n-1}\geq 2$, the lune complex
$L(m_1,\ldots,m_{n-1})$ is a $CW$-complex, homeomorphic to an
$(n-1)$-sphere.  
\end{theorem}
\begin{proof}
Observe first that the union of the facets indeed covers the
$(n-1)$-sphere, and that the intersection of any two facets is either
empty or a hemisphere of the form $(*,\ldots,*,t_{n-1}\cdot
\frac{2\pi}{m_{n-1}})$, where $t_{n-1}$ is any integer from
  $[0,m_{n-1}]$. This set is a union of $(n-2)$-faces: 
$$
\left(*,\ldots,*,t_{n-1}\cdot \frac{2\pi}{m_{n-1}}\right)
\!\!=\!\!\!\!
\bigcup_{s_{n-1}=0}^{m_{n-1}-1} \left(*,\ldots,*,\left[s_{n-2}\cdot
  \frac{\pi}{m_{n-1}},(s_{n-2}+1)\cdot
  \frac{\pi}{m_{n-1}}\right],t_{n-1}\cdot \frac{2\pi}{m_{n-1}}\right)\!\!.  
$$  
Rather than repeating a similar argument in lower dimensions, let us
observe that the faces contained in any facet replicate the face
structure of $L(m_1,\ldots,m_{n-2},2\cdot m_{n-1})$. For that purpose,
consider a facet $F:=\left(*,\ldots,*,\left[s_{n-1}\cdot
  \frac{2\pi}{m_{n-1}},(s_{n-1}+1)\cdot
  \frac{2\pi}{m_{n-1}}\right]\right)$. As  
noted in Remark~\ref{R_hsl}, the homeomorphism 
$$
\phi^{n-1}_{s_{n-1}\cdot \frac{2\pi}{m_{n-1}},(s_{n-1}+1)\cdot
  \frac{2\pi}{m_{n-1}}}: 
(\theta_1,\ldots,\theta_{n-1})\mapsto \left(\theta_1,\ldots,\theta_{n-2},
\left(\theta_{n-1}-s_{n-1}\cdot \frac{2\pi}{m_{n-1}}\right)
\cdot \frac{m_{n-1}}{2}\right)
$$
sends the facet $F$ into $(*,\ldots,*,[0,\pi])$, and its boundary into 
$(*,\ldots,*,0)\cup (*,\ldots,*,\pi)$. The boundary $(*,\ldots,*,0)\cup
(*,\ldots,*,\pi)$ of the hemisphere $(*,\ldots,*,[0,\pi])$ is an
$(n-2)$-dimensional sphere, let us use the homeomorphism $\Pi_n$ defined
in Proposition~\ref{P_proj} to send this into the standard $(n-2)$-sphere.
It is easy to verify that $\Pi_n\circ \phi^{n-1}_{s_{n-1}\cdot
  \frac{2\pi}{m_{n-1}},(s_{n-1}+1)\cdot \frac{2\pi}{m_{n-1}}}$
establishes a 
bijection between the faces contained in $F$ and the faces of
$L(m_1,\ldots,m_{n-3},2\cdot m_{n-2})$. By induction we may thus state
that the faces properly contained in $F$ form a $CW$-complex covering
the boundary of $F$.
\end{proof}

As a consequence of our proof we see that the lune complexes
$L(m_1,\ldots,m_{n-1})$ have the following recursive property:
\begin{corollary}
\label{C_nest}
The poset of all faces contained in an arbitrary facet of
$L(m_1,\ldots,m_{n-1})$ is isomorphic to the face poset of
$L(m_1,\ldots,m_{n-3},2\cdot m_{n-2})$. 
\end{corollary}

\begin{example}
{\em Figure~\ref{F_L33} represents the lune complex $L(3,3)$. It has 
$8$ vertices (of which $6$ are visible on the picture, the invisible
  ones are marked with an empty circle), $9$ edges (the $3$ invisible
  ones are marked with dashed lines), and $3$ facets (of which only
  $(*,[2\pi/3,4\pi/3])$ is entirely visible.) The boundary of each facet
  is a hexagon, isomorphic to $L(6)$.  
}
\end{example}
\begin{figure}[h]
\begin{center}
\input{op-01.pstex_t}
\end{center}
\caption{The complex $L(3,3)$}
\label{F_L33}
\end{figure}
We conclude this section with another embedding result that
sometimes complements the role of Corollary~\ref{C_nest}.
\begin{proposition}
\label{P_nest2}
The partially ordered set of faces of length at most $n-2$ of\\
$L(m_1,\ldots,m_{n-1})$ is isomorphic to the face poset of
$L(m_1,\ldots,m_{n-4}, 2\cdot m_{n-3})$.
\end{proposition}
\begin{proof}
If a spherical vector has length at most $n-2$ then the $(n-2)$-nd
coordinate in its simplified code is $0$, $\pi$, or $*$, and the last
coordinate is always $*$. Removing the last coordinate from all
such spherical vectors establishes a bijection with the hemisphere 
$\{(x_1,\ldots,x_{n-1})\::\: x_1^2+\cdots +x_{n-1}^2=1, x_{n-1}=0\}$. 
Consider the projection $\Pi_{n-1}$ described in
Proposition~\ref{P_hsb}, taking this set into the standard
$(n-3)$-sphere. This projection, combined with removing the last star,
takes a face with code 
$$\left(*,\ldots,*,\left[s_r\cdot \frac{\pi}{m_r},(s_r+1)\cdot
  \frac{\pi}{m_r}\right], s_{r+1}\cdot \frac{\pi}{m_{r+1}},
  \ldots,s_{n-3}\cdot \frac{\pi}{m_{n-3}},0,*\right)$$  
into 
$$\left(*,\ldots,*,\left[s_r\cdot \frac{\pi}{m_r},(s_r+1)\cdot \frac{\pi}{
    m_r}\right], s_{r+1}\cdot \frac{\pi}{m_{r+1}}, \ldots,s_{n-3}\cdot \frac{\pi}{m_{n-3}}\right),$$ 
and a face with code 
$$\left(*,\ldots,*,\left[s_r\cdot \frac{\pi}{m_r},(s_r+1)\cdot
    \frac{\pi}{m_r}\right], s_{r+1}\cdot \frac{\pi}{m_{r+1}},
    \ldots,s_{n-3}\cdot 
    \frac{\pi}{m_{n-3}},\pi,*\right)$$  
into 
$$\left(*,\ldots,*,\left[s_r\cdot \frac{\pi}{m_r},(s_r+1)\cdot
    \frac{\pi}{m_r}\right], s_{r+1}\cdot \frac{\pi}{m_{r+1}}, 
    \ldots,(2m_{n-3}-s_{n-3})\cdot \frac{\pi}{m_{n-3}}\right).$$ 
Considering also the other possible face codes, the statement becomes a
    trivial verification of the definitions.
\end{proof}


\section{The flag $f$-vector of the lune complex}
\label{S_ce}

In this section we present a fundamental recursion formula for the 
flag $f$-vectors of Eulerian posets of the form 
$P_1\left(L(m_1,\ldots,m_{n-1})\right)$. (Since the lune complexes are
$CW$-spheres, the partially ordered sets considered are in fact
Eulerian.) To simplify our notation, for every 
$CW$-sphere $\Omega$ we will use $f_S(\Omega)$ as a shorthand for 
$f_S(P_1(\Omega))$. This can not lead to confusion, since every
saturated chain enumerated in $f_S(P_1(\Omega))$ contains $\1$, so this
element may be removed from all chains at once and we are left with an
equivalent enumeration question.

\begin{proposition} 
\label{P_frec}
For $n\geq 4$ and $S\neq\emptyset$ the flag number
    $f_S(L(m_1,\ldots,m_{n-1}))$ is equal to
$$
\frac{m_{n-1}}{2}\cdot
f_S(L(m_1,\ldots,m_{n-3},2m_{n-2}))
-\frac{m_{n-1}-2}{2}\cdot f_S(L(m_1,\ldots,m_{n-4},2m_{n-3}))$$
if $S\cap \{n,n-1\}=\emptyset$, and 
$$
2^{|S\cap \{n\}|}\cdot \frac{m_{n-1}}{2} \cdot f_{S\setminus
  \{n\}}(L(m_1,\ldots,m_{n-3},2m_{n-2}))$$ 
if $S\cap \{n,n-1\}\neq\emptyset$.
\end{proposition}
\begin{proof}
Consider the case $S\cap \{n,n-1\}\neq\emptyset$ first. Every sub-coatom
in an Eulerian poset is covered by exactly two atoms. Hence,
a set $S$ not containing $n$ (but containing $n-1$) satisfies 
$$
f_{S\cup\{n\}}(L(m_1,\ldots,m_{n-1}))=
2\cdot f_S(L(m_1,\ldots,m_{n-1})). 
$$
By this observation, if our formula is correct when $n\in S$ then it is
also correct in the case when $n\not\in S$ but $n-1\in S$. 
Therefore, w.l.o.g.\ we may assume $n\in S$. First we choose
the facet $F$ in the $S$-chain and then the rest below it. There are 
$m_{n-1}$ ways to choose $F$.
By Corollary~\ref{C_nest}, the interval $[\0,F]$ is
isomorphic to the face poset of $L(m_1,\ldots,m_{n-3},2m_{n-2})$.  
Hence there are $f_{S\setminus \{n\}}(L(m_1,\ldots,m_{n-3},2m_{n-2})$
options to choose the rest of the $S$-chain.

Assume from now on $S\cap \{n,n-1\}=\emptyset$. We distinguish two
sub-cases depending on whether the top element of the $S$-chain has
length at least $n-1$ or less. If the length of the top element is at least
$n-1$ then its simplified code has last coordinate $t_{n-1}\cdot
\frac{2\pi}{m_{n-1}}$ for some integer $t_{n-1}\in [0,m_{n-1}]$, and it
is contained in precisely two facets of $L(m_1,\ldots,m_{n-1})$:  
$$\begin{array}{rl}
\mbox{in}&
\left(*,\ldots,*,\left[t_{n-1}\cdot \frac{2\pi}{
    m_{n-1}},(t_{n-1}+1)\cdot
    \frac{2\pi}{m_{n-1}}\right]\right),\\
\mbox{and in}&
\left(*,\ldots,*,\left[(t_{n-1}-1)\cdot
  \frac{2\pi}{m_{n-1}},t_{n-1}\cdot
  \frac{2\pi}{m_{n-1}}\right]\right).
\end{array}
$$ 
(If $t_{n-1}$ is $0$ resp.\ $m_{n-1}$ then $t_{n-1}-1$ resp.\
  $t_{n-1}+1$ must be understood ``modulo $m_{n-1}$''.) If we count each
  such chain below each facet, then we count each such chain exactly twice. 
By Corollary~\ref{C_nest}, below each
facet we have a copy of $L(m_1,\ldots,m_{n-3},2m_{n-2})$, hence each such
chain is counted in $\frac{m_{n-1}}{2}\cdot
f_S(L(m_1,\ldots,m_{n-3},2m_{n-2}))$ exactly once. 
If the length of the top element is less
than $n-1$, then it (and the rest of the chain) is contained in all
facets of $L(m_1,\ldots,m_{n-1})$. Such chains are thus overcounted in 
$\frac{m_{n-1}}{2}\cdot f_S(L(m_1,\ldots,m_{n-3},2m_{n-2}))$ precisely
$\frac{m_{n-1}}{2}-1$ times. By Proposition~\ref{P_nest2} the effect of
this overcounting may be offset by subtracting $(m_{n-1}-2)/2\cdot
f_S(L(m_1,\ldots,m_{n-4},2m_{n-3}))$. 
\end{proof}
We may transform Proposition~\ref{P_frec}
into the following recursion formula for the $ce$-index:
\begin{proposition}
\label{P_cerec}
The $ce$-index of $L(m_1,\ldots,m_{n-1})$ satisfies
\begin{eqnarray*}
\Phi_{ce}(L(m_1,\ldots,m_{n-1}))&=&
\frac{m_{n-1}}{2}\cdot \Phi_{ce}(L(m_1,\ldots,m_{n-3},2m_{n-2}))\cdot c\\
&&-\frac{m_{n-1}-2}{2}\cdot \Phi_{ce}(L(m_1,\ldots,m_{n-4},2m_{n-3}))\cdot e^2
\end{eqnarray*}
for $n \geq 4$.
\end{proposition}
\begin{proof}
Considering~(\ref{E_ce}) it is sufficient to prove the appropriate
formula for each entry $L_S$ in the flag $L$-vector of the posets
involved. We may restrict our attention to even sets $S$ (since the $ce$-index
is a polynomial of $c$ and $e^2$). 

Assume first $n\not\in S$. Then every set $T$ containing $[1,n]\setminus
S$ contains $\{n\}$, and so applying Proposition~\ref{P_frec} to the
right hand side of
\begin{equation}
\label{E_cesp}
L_S(L(m_1,\ldots,m_{n-1}))=
(-1)^{n-|S|}\sum_{T\supseteq [1,n]\setminus S} \left(\frac{-1}{
  2}\right)^{|T|} f_T(L(m_1,\ldots,m_{n-1}))
\end{equation}
yields only terms of the form $2\cdot \frac{m_{n-1}}{2}\cdot
f_{T\setminus\{n\}} (L(m_1,\ldots,m_{n-3},2 m_{n-2}))$. Replacing $T$ with $T\setminus
\{n\}$ in each summand on the right hand side yields  
$$
(-1)^{n-|S|}\sum_{[1,n-1]\setminus S\subseteq T\subseteq [1,n-1]} 
\left(\frac{-1}{2}\right)^{|T|} 
2\cdot \frac{m_{n-1}}{2}\cdot
f_T(L(m_1,\ldots,\ldots,m_{n-3},2 m_{n-2})) 
$$
which is exactly $\frac{m_{n-1}}{2}\cdot L_S(L(m_1,\ldots, m_{n-3},2
m_{n-2}))$. Therefore our recursion formula is true for the coefficients
of $ce$ words ending with $c$. 

We are left with the case $n\in S$, i.e., the proof of our statement for
the coefficients of $ce$-words ending with $e$ . Since $S$ must be even,
in this case $S$ must 
also contain $n-1$, and $[1,n]\setminus S$ is a subset of $[1,n-2]$. 
The right hand side of~(\ref{E_cesp}) may be rewritten as
\begin{eqnarray*}
(-1)^{n-|S|}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{[1,n\setminus S\subseteq T\subseteq [1,n-2]}
\!\!\!\left(\frac{-1}{2}\right)^{|T|}
\!\!\!\!\!\!\!\!\!\!  &&
\left(
f_T(L(m_1,\ldots,m_{n-1}))
-\frac{1}{2} f_{T\cup\{n-1\}}(L(m_1,\ldots,m_{n-1}))
\right.\\
&&\left.
-\frac{1}{2} f_{T\cup\{n\}}(L(m_1,\ldots,m_{n-1}))
-\!\frac{1}{4} f_{T\cup\{n-1,n\}}(L(m_1,\ldots,m_{n-1}))
\right)
\end{eqnarray*}
Applying Proposition~\ref{P_frec} to each term in this sum, the
multiples of the terms\\
$f_T(L(m_1,\ldots,m_{n-3},2 m_{n-2}))$
cancel, and the multiples of the terms\\
$f_T(L(m_1,\ldots,m_{n-3},2 m_{n-2}))$ add up to
$$L_S(L(m_1,\ldots,m_{n-1}))=-\frac{m_{n-1}-2}{2}L_{S\setminus
  \{n-1,n\}}(L(m_1,\ldots,m_{n-3},2 m_{n-2})).$$
\end{proof}
Introducing the symbol $L()$ for the zero-dimensional
sphere (with two vertices), the recursion formulas of
Propositions~\ref{P_frec} and~\ref{P_cerec} can be easily shown to
extend to the case $n=3$. The $ce$-index of the (face poset of)
$L(m_1,\ldots,m_{n-1})$ for $n\leq 4$ is shown in Table~\ref{t_T}.
(Note that $L(m_1)$ is a cycle with $m_1$ vertices and $m_1$ edges.) 

\begin{table}[h]
\begin{eqnarray*}
\Psi_{ce}(\emptyset)&=&1\\
\Psi_{ce}(L())&=&c\\
\Psi_{ce}(L(m_1))&=&\frac{m_1}{2}c^2-\left(\frac{m_1}{2}-1\right) e^2\\
\Psi_{ce}(L(m_1,m_2))&=& 
m_1\frac{m_2}{2}c^3-(m_1-1)\frac{m_2}{2}
e^2c-\left(\frac{m_2}{2}-1\right)ce^2\\  
\Psi_{ce}(L(m_1,m_2,m_3))&=& m_1m_2\frac{m_3}{2}c^4-(m_1-1)m_2\frac{m_3}{2}
e^2c^2-\left(m_2-1\right)\frac{m_3}{2}ce^2c\\ 
&&-m_1\left(\frac{m_3}{2}-1\right)c^2e^2
+\left(m_1-1\right) \left(\frac{m_3}{2}-1\right)e^4. 
\\
\end{eqnarray*}
\caption{The $ce$-index of the face poset of $L(m_1,\ldots,m_{n-1})$ for
$n\leq 4$.}
\label{t_T}
\end{table}

\section{Spherical shellability of the lune complex}
\label{S_sshell}

The main result of this section is the following.
\begin{theorem}
\label{T_sshell}
For $n\geq 2$ and $m_1,\ldots,m_{n-2}\geq 0$, the lune complex
$L(m_1,\ldots,m_{n-1})$ is spherically shellable.
\end{theorem}
\begin{proof}
We proceed by induction on $n$ and the $m_i$'s. For $n=2$ 
the lune complex $L(m_1)$ is a cycle with $m_1$ vertices and $m_1$
edges, easily shown to be spherically shellable. Consider next $n\geq 3$
and $m_{n-1}=2$. The lune complex $L(m_1,\ldots,m_{n-2},2)$ has two
(closed) facets: the hemisphere $\overline{F_1}=(*,\ldots,[0,\pi])$ and
the hemisphere $\overline{F_2}(*,\ldots,[\pi,2\pi])$. The boundary of
both facets is the same, and it 
is isomorphic to the $CW$-complex $L(m_1,\ldots,m_{n-3},2m_{n-2})$, as
noted in the proof of Theorem~\ref{T_Lsphere}. Hence axiom {\rm ($S$-a)}
is satisfied by the induction hypothesis, while {\rm ($S$-b)} is never
applicable when we have only two facets.

Assume finally $n\geq 3$ and $m_{n-1}\geq 3$. 
By our induction hypothesis we may assume that the complex
$L(m_1,\ldots,m_{n-1}-1)$ has an $S$-shelling
$F_1,\ldots,F_{m_{n-1}}$. Due to the ``rotational symmetry'' (the map
$(\theta_1,\ldots,\theta_{n-1})\mapsto  
(\theta_1,\ldots,\theta_{n-2},\theta_{n-1}+2\pi/(m_{n-1}-1))$ induces an 
automorphism of $L(m_1,\ldots,m_{n-1}-1)$) we may assume that 
the closure of the last facet is 
$$\overline{F_{m_{n-1}-1}}
=\left(*,\ldots,*,\left[\frac{m_{n-1}-2}{m_{n-1}-1}\cdot
 2\pi,2\pi\right]\right).$$
A homeomorphic copy of
$L(m_1,\ldots,m_{n-1})$ may be obtained from $L(m_1,\ldots,m_{n-1}-1)$
by subdividing $\overline{F_{m_{n-1}-1}}$
into two (closed) facets
$$
\overline{F_{m_{n-1}-1}'}
=\left(*,\ldots,*,\left[\frac{m_{n-1}-2}{m_{n-1}-1}\cdot
  2\pi,\frac{m_{n-1}-1.5}{m_{n-1}-1}\cdot 2\pi\right]\right)\quad\mbox{and}
$$
$$
\overline{F_{m_{n-1}-1}''}
=\left(*,\ldots,*,\left[\frac{m_{n-1}-1.5}{m_{n-1}-1}\cdot
  2\pi,2\pi\right]\right)  
$$
and replicating the appropriate face structure at the intersection of
the subdividing closed facets. The homeomorphism from the subdivided complex to
$L(m_1,\ldots,m_{n-1})$ may be given by 
$$
(\theta_1,\ldots,\theta_{n-1})\mapsto
(\theta_1,\ldots,\theta_{n-2},\theta_{n-1}') 
$$
where 
$$
\theta_{n-1}'=
\left\{
\begin{array}{ll}
\frac{m_{n-1}-1}{m_{n-1}}\cdot \theta_{n-1}
&\mbox{when $0\leq\theta_{n-1}\leq \frac{m_{n-1}-2}{m_{n-1}-1}\cdot
  2\pi$,}\\
&\\
2\cdot \frac{m_{n-1}-1}{m_{n-1}}\cdot
\theta_{n-1}+\frac{2-m_{n-1}}{m_{n-1}}\cdot 2\pi
&\mbox{when $\frac{m_{n-1}-2}{m_{n-1}-1}\cdot
  2\pi\leq\theta_{n-1}\leq 2\pi$.}\\
\end{array}
\right.
$$
All we need to show then is that
$F_1,\ldots,F_{m_{n-2}},F_{m_{n-1}}',F_{m_{n-1}}''$ is an $S$-shelling 
of the subdivided complex. In other words, we only need to verify that 
{\rm ($S$-b)} is satisfied by the facet $F_{m_{n-1}}'$. The intersection
of the boundary of $F_{m_{n-1}}'$ with the closure of the previously
added facets is isomorphic to the positive hemisphere $0\leq
\theta_{n-2}\leq \pi$ in $L(m_1,\ldots,m_{n-3},2m_{n-2})$.
Its semisuspension may be geometrically realized by adding the lune
$(*,\ldots,*,[\pi,2\pi])$ to those  facets of
$L(m_1,\ldots,m_{n-3},2m_{n-2})$ which are contained in the positive
hemisphere. Thus the semisuspension is  isomorphic to
$L(m_1,\ldots,m_{n-3},m_{n-2}+1)$. This isomorphism is induced by the
homeomorphism $(\theta_1,\ldots,\theta_{n-2})\mapsto
(\theta_1,\ldots,\theta_{n-3},\theta_{n-2}'')$ where  
$$
\theta_{n-2}''=
\left\{
\begin{array}{ll}
\frac{2m_{n-2}}{m_{n-2}+1}\cdot \theta_{n-2}
&\mbox{when $0\leq\theta_{n-2}\leq\pi$,}\\
&\\
\frac{2\theta_{n-2}}{m_{n-2}+1}\
+\frac{2(m_{n-2}-1)\pi}{m_{n-2}+1}
&\mbox{when $\pi\leq\theta_{n-2}\leq 2\pi$.}\\
\end{array}
\right.
$$
Again, by our induction hypothesis, $L(m_1,\ldots,m_{n-3},m_{n-2}+1)$ has
an $S$-shelling and, because of the ``rotational symmetry'' mentioned
above, we may choose any of its facets to be the first one.
\end{proof}
The proof of Theorem~\ref{T_sshell}, together
with~(\ref{E_Slemma}) provides the following recursion formula for the
$cd$-index of $L(m_1,\ldots,m_{n-1})$:
\begin{equation}
\label{E_Lrec}
\begin{array}{rcl}
\Phi_{cd}(L(m_1,\ldots,m_{n-1}))&=&
\Phi_{cd}(L(m_1,\ldots,m_{n-3},2\cdot m_{n-2}))c\\
&&+(m_{n-1}-2)\Phi_{cd}(L(m_1,\ldots,m_{n-3},m_{n-2}+1))c\\
&&-(m_{n-1}-2)\Phi_{cd}(L(m_1,\ldots,m_{n-4},2m_{n-3}))(c^2-d).\\
\end{array}
\end{equation}
In fact, every time we increase $m_{n-1}$ by $1$, we subdivide a closed facet
$\overline{\sigma}$ into two facets $\overline{\sigma_1}$ and
$\overline{\sigma_2}$ in such a way that the semisuspension of
$\Gamma=\overline{\sigma_1}\cap\overline{\sigma_2}$ is isomorphic to
$L(m_1,\ldots,m_{n-3},m_{n-2}+1)$, while the boundary of $\Gamma$ is
clearly isomorphic to $L(m_1,\ldots,m_{n-4},2m_{n-3})$. To obtain
$L(m_1,\ldots,m_{n-1})$ from $L(m_1,\ldots,m_{n-2},2)$ we need to
perform such a subdivision $(m_{n-1}-2)$ times. Finally, 
$$
\Phi_{cd}(L(m_1,\ldots,m_{n-2},2))=\Phi_{cd}(L(m_1,\ldots,m_{n-3},2m_{n-2}))c
$$
is obvious. 

\section{A sequence of orthogonal polynomials represented by lune complexes}
\label{S_ops}

In the following we assume that $\nu_1,\nu_2,\ldots$ is an infinite
sequence of positive numbers satisfying $\nu_1=1$ and $\nu_n\geq 2$ for
$n\geq 2$. We define the polynomials $Q_{-1}(x),Q_0(x), Q_1(x),\ldots$ by 
setting $Q_{-1}(x)=0$, $Q_0(x)=1$, and the recursion formula  
\begin{equation}
\label{E_Qrec}
Q_n(x)=\nu_n\cdot x\cdot Q_{n-1}(x)-(\nu_n-1)\cdot
Q_{n-2}(x)\quad\mbox{for $n\geq 1$}.
\end{equation}
(As a consequence, $Q_1(x)=x$.) 
This sequence is not monic, the leading coefficient of $Q_n(x)$ is 
$\nu_1\cdots\nu_n$. Introducing $P_{-1}(x)=0$ and 
$P_n(x)=Q_n(x)/(\nu_1\cdots\nu_n)$ for $n\geq 1$ we obtain the monic and
symmetric OPS
$\{P_n (x)\}_{n=0}^{\infty}$  satisfying the recurrence
$$
P_n(x)=x\cdot P_{n-1}(x)-
\frac{\nu_n-1}{\nu_{n-1}\nu_n}P_{n-2}(x)\quad\mbox{for $n\geq 1$}. 
$$
Since, for each $n$, $Q_n(x)$ differs from $P_n(x)$ only by a nonzero
constant factor, the polynomials $\{Q_n(x)\}_{n=0}^{\infty}$ form a
(non-monic) symmetric OPS. As an illustration of the power of spherical
shellings we provide a new proof of the following theorem. 
\begin{theorem}
\label{T_nonneg}
Each polynomial $Q_n(x)$ may be written as a non-negative combination of
products of powers of $x$ and $(x^2-1)$.
\end{theorem}
Before explaining how spherical shellability of the lune complexes may be
used to prove this theorem, let us review how Theorem~\ref{T_nonneg}
follows from the classical theory. Introducing $\lambda_1=1$ and 
$\lambda_n=\frac{\nu_n-1}{\nu_{n-1}\nu_n}$ for $n\geq 2$, 
the sequence $\{\lambda_{n+1}\}_{n=1}^{\infty}$ is a chain sequence with
parameter sequence $g_n=(1-1/\nu_n)$. Hence, as noted at the end of the
preliminary Section~\ref{ss_op}, the true interval of orthogonality of
the OPS $\{P_n (x)\}_{n=0}^{\infty}$ is a subset of $[-1,1]$. In other
words, every zero of each $P_n(x)$ is contained in $[-1,1]$, and the
same holds for the zeros of the $Q_n(x)$, since for
every $n$, the polynomials $Q_n(x)$ and $P_n(x)$ differ at most by a
nonzero constant factor. A ``classical'' proof of
Theorem~\ref{T_nonneg} may be then concluded using the following
observation, due to Ismail and Stanton~\cite{Ismail-Stanton}.

\begin{lemma}
\label{L_SI}
Assume that all zeros of the polynomial $q(x)$ are simple and real, and
that $q(x)$ is a linear combination of only even or only odd powers of
$x$. Then the polynomial $q(x)$ is a non-negative linear combination of
polynomials of the form $(x^2-1)^n$ or $x(x^2-1)^n$ if and only if all
zeros of $q(x)$ lie in the interval $[-1,1]$.
\end{lemma}
\begin{proof}
Assume first $q(x)$ is a non-negative linear combination of
polynomials of the form $(x^2-1)^n$ or $x(x^2-1)^n$. Assume also that
$q(x)$ is a combination of even powers of $x$ (the odd case is
similar). Then only terms of the form $(x^2-1)^n$ occur in $q(x)$ and
$q(x)>0$ when $|x|>1$. Thus all zeros lie in $[-1,1]$.
Assume, conversely, that all zeros of $q(x)$ lie in $[-1,1]$. We restrict
our attention again to the case of even powers, the odd case is similar.
Then $q(x)=(x^2-r_1^2)\cdots(x^2-r_n^2)$
for some $r_1,..,r_n\in [0,1]$. Since $x^2-r_i^2=(x^2-1)+(1-r_i^2)$, each 
factor has a non-negative expansion so the product will too.
\end{proof}

For sequences of integer $\nu_i$'s,
Theorem~\ref{T_nonneg} is a relatively easy consequence of
Theorem~\ref{T_sshell}. To see this, let us introduce the auxiliary
polynomials $R_n(y_1,y_2,\ldots,y_{n-1};x)$ given by $R_0=1$,
$R_1(x)=x$, $\di R_2(y_1;x)=x+\frac{y_1-2}{2}(x^2-1)$, and the recursion
formula 
\begin{equation}
\label{E_Rrec}
\begin{array}{rcl}
R_n(y_1,\ldots,y_{n-1};x)&=&
R_{n-1}(y_1,\ldots,y_{n-3},2\cdot y_{n-2};x)x\\
&&+(y_{n-1}-2)R_{n-1}(y_1,\ldots,y_{n-3},y_{n-2}+1;x)x\\
&&-(y_{n-1}-2)R_{n-2}(y_1,\ldots,y_{n-4},2y_{n-3};x)
\left(x^2-\frac{x^2-1}{2}\right).\\  
\end{array}
\end{equation}
As an immediate consequence of equation~(\ref{E_Lrec}) we may observe
that for any sequence of integers $m_1,m_2,\ldots,m_n,\ldots$, satisfying
$m_i\geq 2$ for all $i$, the value of $R(y_1,\ldots,y_{n-1};x)$ is the  
image of\\
$\Phi_{ce}(L(m_1,\ldots,m_{n-1}))$ under the homomorphism
induced by $c\mapsto x$ and $e\mapsto 1$. (Under this homomorphism,
$d=(c^2-e^2)/2$ goes into $(x^2-1)/2$.) Applying the same homomorphism
to Proposition~\ref{P_cerec} we obtain the recursion formula 
\begin{equation}
\begin{array}{rcl}
R(m_1,\ldots,m_{n-1};x)&=&
\frac{m_{n-1}}{2}\cdot R(m_1,\ldots,m_{n-3},2m_{n-2};x)\cdot x\\
&&-\frac{m_{n-1}-2}{2}\cdot R(m_1,\ldots,m_{n-4},2m_{n-3}; x)
\end{array}
\end{equation}
Comparing this recursion formula to~(\ref{E_Qrec}) it follows by trivial 
induction on $n$ that 
\begin{equation}
\label{E_QR}
Q_n(x)=R_n(\nu_2,\nu_3,\ldots,\nu_{n-1},2\cdot\nu_{n};x)
\end{equation}
for any $n\geq 1$, if all the $\nu_i$'s are integers and at least $2$. 
As an immediate consequence of Theorem~\ref{T_sshell}, the polynomials 
$R(m_1,\ldots,m_{n-1};x)$ are non-negative combinations of terms of the
form $x^i(x^2-1)^j$. (By Stanley's result~\cite[Theorem
  2.2]{Stanley-flag}, the $cd$-index associated to a spherically shellable  
$CW$-sphere has nonnegative coefficients.) This concludes the proof for
sequences of integer $\nu_i$'s.  

However, only two small observations are necessary to extend the validity of
our new argument to sequences of arbitrary real $\nu_i$'s. 
First, we may observe that equation (\ref{E_QR}) is valid for any
sequence of real numbers $\nu_2,\nu_3,\ldots$. In fact, keeping the
$\nu_i$'s as variables it is obvious from~(\ref{E_Qrec}) that $Q_n(x)$
is a polynomial expression of $\nu_2,\ldots,\nu_n$ and $x$, and so is 
$R_n(\nu_2,\nu_3,\ldots,\nu_{n-1},2\cdot\nu_{n};x)$ in light
of~(\ref{E_Rrec}). Both polynomials agree for infinitely many
independent (integer) values of $\nu_2,\ldots,\nu_n$ and $x$, so
they are equal as polynomial expressions. But then they are also equal
when we substitute non-integer values as $\nu_i$'s. 
Hence it is sufficient to show that the polynomials 
$R(r_1,\ldots,r_{n-1};x)$ are non-negative combinations of terms of the
form $x^i(x^2-1)^j$ whenever all $r_i$'s are at least $2$, even if they
are not integers. A slightly stronger statement is easily proven by
induction:

\begin{proposition}
Given $n\geq 1$, and any sequence of real numbers $r_1,\ldots,r_{n-1}$
satisfying $r_i\geq 2$ for all $i$, the polynomial
$R_n(r_1,\ldots,r_{n-1};x)$ is a non-negative combinations of terms of the
form $x^i(x^2-1)^j$. Moreover, increasing $r_n$ while leaving all other
$r_i$'s unchanged cannot decrease any coefficient in such a combination.    
\end{proposition}
\begin{proof}
As an immediate consequence of~(\ref{E_Rrec}) we may write 
\begin{equation}
\label{E_R2rec}
R_n(y_1,\ldots,y_{n-2},2;x)=
R_{n-1}(y_1,\ldots,y_{n-3},2\cdot y_{n-2};x)x.
\end{equation}
Using this observation, we may rewrite~(\ref{E_Rrec}) as 
\begin{equation}
\label{E_Rsrec}
\begin{array}{rcl}
R_n(y_1,\ldots,y_{n-1};x)&=&
R_{n-1}(y_1,\ldots,y_{n-3},2\cdot y_{n-2};x)x\\
&&+(y_{n-1}-2)\left(R_{n-1}(y_1,\ldots,y_{n-3},y_{n-2}+1;x)\right.\\
&& \ \ \left.-R_{n-1}(y_1,\ldots,y_{n-4},y_{n-3},2;x)\right)x\\
&&+(y_{n-1}-2)R_{n-2}(y_1,\ldots,y_{n-4},2y_{n-3};x)\frac{x^2-1}{2}.  
\end{array}
\end{equation}
By our induction hypothesis, $R_{n-1}(r_1,\ldots,r_{n-3},2 r_{n-2};x)$
and $R_{n-2}(r_1,\ldots,r_{n-4},2 r_{n-3};x)$ are non-negative 
combinations of terms of the form $x^i(x^2-1)^j$ whenever the $r_i$'s
are at least $2$. The same also holds for the difference 
$$R_{n-1}(r_1,\ldots,r_{n-3},r_{n-2}+1;x) -
R_{n-1}(r_1,\ldots,r_{n-4},r_{n-3},2;x)$$   
since, by our induction
hypothesis, the coefficients cannot decrease when we increase $y_{n-2}$
from $2$ to $r_{n-2}+1$. Finally the last variable $y_{n-1}$ occurs only
as factor $(y_{n-2}-2)$ in two of the terms, so the coefficients of the
terms of the form $x^i(x^2-1)^j$ in $R_n(r_1,\ldots,r_{n-1};x)$ can not
decrease when we increase $r_{n-1}$.
\end{proof}

\begin{remark}
{\em 
Although the chain-sequence approach, as well as the spherical shelling
argument, prove ``essentially'' the same result, the actual representation
that could be obtained following either argument is different. Using
Lemma~\ref{L_SI} we obtain a positive combination where each power of
$(x^2-1)$ is multiplied by at most the first power of $x$, while the
spherical shelling approach yields a positive combination of terms of
the form $x^{n-2r}(x^2-1)^r$. 
}
\end{remark}


\begin{remark}
{\em
The careful reader will notice that in the ``classical'' approach the
condition {$\nu_i\geq 2$} may be relaxed to $\nu_i>1$, since this
condition already guarantees $0<1-1/\nu_i<1$, and so $\lambda_n$ is a
chain sequence. The same is probably also true about the spherical
shelling approach, 
since it is possible to construct lune complexes $L(m_1,\ldots,m_{n-1})$
even when some $m_i$'s satisfying $i<n-1$ are equal to $1$. For example,
removing all vertices except $(0,*)$ and $(\pi,*)$ from
Figure~\ref{F_L33} (and merging the edges meeting at the removed
vertices) yields the lune complex $L(1,3)$, with $2$ vertices: $(0,*)$
and $(\pi,*)$; $3$ edges: $([0,\pi],0)$, $([0,\pi],2\pi/3)$, and
$([0,\pi],2\pi/3)$; and $3$ faces (having the same codes as in
$L(3,3)$. As indicated by the notational ambivalence $([0,\pi],0)=(*,0)$,
allowing $m_i=1$ would induce some confusion that would need extra
consideration at every step. We prefer to avoid this complication in
this first presentation of lune complexes.   
Theorem~\ref{T_nonneg} easily follows from the classical
theory anyway, and Theorem~\ref{T_c},
inspired by a more direct approach to the $cd$-index calculation,
gives a much more explicit statement under the more general conditions.
This section is only an illustration of the possible
usefulness of spherical shellings in the theory of orthogonal
polynomials.}
\end{remark}

\section{Explicit $cd$-index formula}
\label{S_expl}
The proof of Theorem~\ref{T_nonneg} motivates to introduce the following
sequence of partially ordered sets.

\begin{definition}
Given a sequence of positive integers $\nu_1,\nu_2,\ldots$ satisfying
$\nu_1=1$ and\\ 
$\nu_n\geq 2$ for $n\geq 2$ let $Q_n=Q_n(\nu_2,\ldots,\nu_n)$
be the face poset of the lune complex\\
$L(\nu_2,\ldots,\nu_{n-1},2\nu_n)$. 
\end{definition}
As we have seen in the previous section, the sequence $\Phi_{ce}(Q_n)$
is a non-commutative generalization of the OPS $\{Q_n(x)\}_{n=0}^{\infty}$
defined by~(\ref{E_Qrec}). As noted in the proof of Theorem~\ref{T_Lsphere},
the faces contained in any facet of $L(\nu_2,\ldots,\nu_{n-1},2\nu_n)$
replicate the face structure of $L(\nu_2,\ldots,\nu_{n-2},2\nu_{n-1})$. 
\begin{corollary}
\label{C_Qc}
For every coatom $c$ of $Q_n$, the interval $[\0,c]\subset Q_n$ is
isomorphic to $Q_{n-1}$.    
\end{corollary}
As an immediate consequence of Proposition~\ref{P_cerec} we have
\begin{equation}
\Phi_{ce}(Q_n)=\nu_{n}\Phi_{ce}(Q_{n-1})c+(1-\nu_n)\Phi_{ce}(Q_{n-2})e^2.
\end{equation}
Similarly, (\ref{E_Lrec}) implies
\begin{equation}
\label{E_Qcdrec}
\Phi_{cd}(Q_n)=\nu_{n}\Phi_{cd}(Q_{n-1})c+(1-\nu_n)\Phi_{cd}(Q_{n-2})(c^2-2d).
\end{equation}
Introducing the sequence of polynomials $\imath_0, \imath_1(t_1),\ldots,
\imath_n(t_1,\ldots,t_n),\ldots$ given by  
\begin{equation}
\label{E_idef}
\begin{array}{rcl}
\imath_0&=&1\quad\mbox{and}\\
\imath_n(t_1,\ldots,t_n)&=&1+(t_1-1)+(t_1-1)(t_2-1)+\cdots+(t_1-1)\cdots
(t_n-1)\quad\mbox{for $n\geq 1$,}\\
\end{array}
\end{equation} 
the coefficient of a $cd$-word in $\Phi_{cd}(Q_n)$ may be
described as follows:
\begin{proposition} 
\label{P_Qcdc}
The coefficient of $c^{k_1}dc^{k_2}d\cdots dc^{k_{r-1}}dc^{k_r}$ in
$\Phi_{cd}(Q_n)$ is
$$
2^r\prod_{i=1}^{r-1} \left(\nu_{k_1+\cdots+k_i+2i}-1\right)\cdot
\prod_{j=1}^r \imath_{k_j}\left(\nu_{k_1+\cdots+k_{j-1}+2j-1},\ldots,\nu_{k_1+\cdots+k_{j-1}+k_j+2j-2}\right).
$$
\end{proposition}
\begin{proof}
Recalling the fact that $c$ has degree $1$ and $d$ has degree $2$, the
$i$-th $d$ from the left arises from the letters at position
$k_1+\cdots+k_i+2i-1$ and $k_1+\cdots+k_i+2i$ in the $ab$-index. By
inspection of~(\ref{E_Qcdrec}) it is clear that a letter $d$ is
introduced only when the contribution of the second term is considered
at the appropriate place,
and so the $i$-th $d$ contributes precisely a factor of
$2\left(\nu_{k_1+\cdots+k_i+2i}-1\right)$ to the coefficient of the
$cd$-word. 

The contribution of the term $c^{k_j}$ is obtained by substituting the
$\nu_i$'s corresponding to the positions covered into a function 
$I_{k_j}(t_1,\cdots,t_{k_j})$, described recursively by
$$I_k(t_1,\ldots,t_k)=t_k I_{k-1}(t_1,\ldots,t_{k-1})+(1-t_k)\cdot
I_{k-2}(t_1,\ldots,t_{k-2})$$
where $I_0=1$ and $I_1(t_1)=t_1=t_1-1+1$ are the appropriate initial
conditions. Straightforward induction shows
$I_n(t_1,\ldots,t_n)=\imath_n(t_1,\ldots,t_n)$.   
\end{proof}
\begin{corollary}
\label{C_Qcdc}
Every $cd$-word in $\Phi_{cd}(Q_n)$ has a strictly positive
coefficient. 
\end{corollary}

Let us observe now, that in the proof Proposition~\ref{P_Qcdc} we used
no other properties of of the posets 
$Q_n$ than the recursion~(\ref{E_Qcdrec}) for their $cd$-indices. Given
any sequence $\{\widetilde{Q}_n\}_{n=0}^{\infty}$ of $cd$-polynomials 
satisfying $\widetilde{Q}_0=1$, $\widetilde{Q}_1=c$, and a recursion
formula 
$\widetilde{Q}_n
=\nu_{n}\widetilde{Q}_{n-1}c
+(1-\nu_n)\widetilde{Q}_{n-2}(c^2-2d) 
$
for all $n\geq 2$, the coefficients of the $cd$-words in
$\widetilde{Q}_n$ are given by Proposition~\ref{P_Qcdc}. As a
consequence, all $cd$-words have non-negative coefficients if all
$\nu_i$'s satisfy $\nu_i\geq 1$. Consider now the linear transformation
from $cd$-polynomials to polynomials in one variable induced
by sending $c$ into $x$ and $d$ into $(x^2-1)/2$. (Equivalently, we send
$e^2$ into $1$.) Then the sequence of polynomials
$\{\widetilde{Q}_n\}_{n=0}^{\infty}$ goes into a sequence 
$\{Q_n(x)\}_{n=0}^{\infty}$ that satisfies $Q_0(x)=1$, $Q_1(x)=x$ and
the recursion formula $Q_n(x)
=\nu_{n} x Q_{n-1}(x)
+(1-\nu_n)Q_{n-2}(x)$. If $\nu_i>1$ for $i\geq 2$ then we obtain an OPS, and 
now it is an immediate consequence of Proposition~\ref{P_Qcdc} that
every $Q_n(x)$ in this sequence is a positive combination of terms of
the form $x^i(x^2-1)^j$. Moreover, we obtain the following explicit
formula for the coefficients. 

\begin{theorem}
\label{T_c}
Assume that the OPS $\{Q_n(x)\}_{n=0}^{\infty}$ is given by $Q_0(x)=1$,
$Q_1(x)=x$, and
$$Q_n(x)=\nu_{n} x Q_{n-1}(x)+(1-\nu_n)Q_{n-2}(x)\quad\mbox{for $n\geq
  2$,}$$
where $\nu_n>1$ for $n\geq 2$. Then each $Q_n(x)$ may be written as a
  positive combination of terms of the form $x^{n-2r}(x^2-1)^r$. The
  coefficient of $x^{n-2r}(x^2-1)^r$ is 
$$
\sum_{k_1+\cdots+k_r=n-2r}\prod_{i=1}^{r-1} \left(\nu_{k_1+\cdots+k_i+2i}-1\right)\cdot
\prod_{j=1}^r \imath_{k_j}\left(\nu_{k_1+\cdots+k_{j-1}+2j-1},\ldots,\nu_{k_1+\cdots+k_{j-1}+k_j+2j-2}\right).
$$
Here the functions $\imath_k(t_1,\ldots,t_k)$ are the ones given
in~(\ref{E_idef}). 
\end{theorem}
In fact, the image of $c^{k_1}dc^{k_2}d\cdots
dc^{k_{r-1}}dc^{k_r}$ is
$x^{k_1+\cdots+k_r}\left(\frac{x^2-1}{2}\right)^r$ (here
$k_1+\cdots+k_r=n-2r$), so Theorem~\ref{T_c} follows from the proof of
Proposition~\ref{P_Qcdc} and the observation that the factors $2^r$ and
$\left(\frac{1}{2}\right)^r$ cancel. 

Admittedly, the proof of Theorem~\ref{T_c} may be presented without any
reference to the face posets of lune complexes, but it seems to be more
difficult to come up with the idea of the ``underlying'' 
non-commutative polynomials without the the inspiration from
the theory of $cd$-indices of Eulerian posets. 

\section{Connection to the Tchebyshev posets}
\label{S_Tch}

The self-similarity property of Corollary~\ref{C_Qc}
also holds for the duals of the {\em Tchebyshev posets}
introduced by the present author in~\cite{Hetyei-Tch}. Moreover, the
setting $\nu_n=2$ for $n\geq 2$ in~(\ref{E_Qrec}) yields precisely the
Tchebyshev polynomials of the first kind. Hence it is worth observing
the following connection:
\begin{theorem}
The Eulerian poset $Q_n(2,\ldots,2)$ is isomorphic to the dual of the
Tchebyshev poset $T_n$ introduced in~\cite{Hetyei-Tch}.  
\end{theorem}
Before outlining the proof, let us recall the (momentarily) most convenient 
definition of $T_n$. The Tchebyshev poset
$T_n$ has a unique minimum element $\0=(-1,1)$ and a unique
maximum element $\1=(-(n+1),-(n+2))$. All other elements of $T_n$
are pairs of nonzero integers $(x,y)$ such that $|x|<|y|$ and 
$|x|,|y|\in\{1,2,\ldots,n+1\}$, with the restriction that for $|y|=n+1$ we
must have $y=-(n+1)$. For $(x_1,y_1),(x_2,y_2)\in T_n$ the partial order
is defined by 
\begin{equation}
\label{E_Tdef}
(x_1,y_1)<(x_2,y_2) \Leftrightarrow
\left((|y_1|<|y_2|) 
\wedge \left((|y_1|<|x_2|) \vee (y_1=x_2) \vee (x_1=x_2)\right)\right). 
\end{equation}
This definition is easily dualized as follows. Let us introduce 
$$x'=\sign(y)(n+2-|y|)\quad \mbox{and}\quad  y'=\sign(x)(n+2-|x|)\quad  
\mbox{for every $(x,y)\in T_n\setminus \{\0,\1\}$}.$$  
Then $|x|<|y|$ is equivalent to $|x'|<|y'|$ and
$|x|,|y|\in\{1,2,\ldots,n+1\}$ is equivalent to
$|x'|,|y'|\in\{1,2,\ldots,n+1\}$. The restriction on the sign of $y$ when 
$|y|=n+1$ is equivalent to setting $x=-1$ whenever $|x|=1$.
Given $(x_1,y_1)$ and $(x_2,y_2)$  from
$T_n\setminus \{\0,\1\}$, let us describe the condition for
$(x_1,y_1)<^*(x_2,y_2)$ in the dual order, in terms of $(x_1',y_1')$ and
$(x_2',y_2')$. In other words, we are describing the condition for 
$(x_1,y_1)>(x_2,y_2)$ in the original order. Easy substitution into
(\ref{E_Tdef}) yields:
\begin{equation}
\label{E_T*def}
(x_1',y_1')<^*(x_2',y_2') \Leftrightarrow
\left((|x_1'|<|x_2'|) 
\wedge \left((|x_2'|>|y_1'|) \vee (x_2'=y_1') \vee (y_2'=y_1')\right)\right). 
\end{equation}
It is easy to verify that $|x'|$ is the rank of $(x',y')\in
T_n^*\setminus\{\0,\1\}$. We will use this labeling of the elements of 
$T_n^*\setminus\{\0,\1\}$ to encode the elements 
of $Q_n(2,\ldots,2)\setminus\{\0,\1\}$, i.e., the proper faces of the
$(n-1)$-dimensional lune complex $L(2,\ldots,2,4)$. Using
Corollary~\ref{C_Lequiv}, consider the simplified code of each face in
this complex, that is, let us replace every
coordinate after the first $0$ or $\pi$ with $*$. Thus the code of every
vertex will be of one of the following forms:
\begin{flushleft}
$(0,*,\ldots,*)$, $(\pi,*,\ldots,*)$,\\ 
$(\pi/2,\ldots,\pi/2,0,*,\ldots,*)$, 
$(\pi/2,\ldots,\pi/2,\pi,*,\ldots,*)$,\\
$(\pi/2,\ldots,\pi/2,0)$, $(\pi/2,\ldots,\pi/2)$,
$(\pi/2,\ldots,\pi/2,\pi)$, or $(\pi/2,\ldots,\pi/2,3\pi/2)$.
\end{flushleft}
All the vertices have rank $1$ in the face poset, hence the first
coordinate of their code $(x',y')$ should satisfy $x'=1$. Let us set
$|y'|=\ell+1$ where $\ell$ is the length of code the vertex. Choose $y'$
to be negative if $\theta_{\ell}=0$ and let $y'$ have positive sign if 
$\theta_{\ell}=\pi$. This rule uniquely determines the code $(x',y')$ of
a vertex, except for the vertices $(\pi/2,\ldots,\pi/2)$ and
$(\pi/2,\ldots,3\pi/2)$ which are the only vertices of length $n$. 
Let us associate $(-1,-(n+1))$ to $(\pi/2,\ldots,\pi/2)$
and $(-1,(n+1))$ to $(\pi/2,\ldots,3\pi/2)$. Obviously we defined a
bijection between the vertices of the lune complex and the rank $1$
elements of $T_n^*$.

A lune of dimension more than zero but less than $(n-1)$ will have a
code of the form
$$(*,\ldots,*,[\alpha,\beta],\pi/2,\ldots,\pi/2,\theta,*,\ldots,*)$$ 
where 
$[\alpha,\beta]$ is either $[0,\pi/2]$ or $[\pi/2,\pi]$ and $\theta\in
\{0,\pi\}$ unless $\theta$ is the last coordinate. 
(The number of entries between $[\alpha,\beta]$ and $\theta$ may
be zero.) Again we must choose $|x'|$ to be the rank of our lune, which
is the dimension of the lune plus one. Consistently with the sign choice
for the vertices let us set $x'$ to be negative if $[\alpha,\beta]$
contains $0$ (i.e., $[\alpha,\beta]=[0,\pi/2]$), and let us set $x'$
to be positive of $[\alpha,\beta]$ contains $\pi$ (i.e.,
$[\alpha,\beta]=[\pi/2,\pi]$). Set again $|y'|=\ell+1$, where $\ell$ is
the length of the lune. Consistently with the sign choice for the
vertices, set the sign of $y'$ to be negative if $\theta\in\{0,\pi/2\}$
and positive if $\theta\in\{\pi,3\pi/2\}$. (The possibilities
$\theta=\pi/2$ or $\theta=3\pi/2$ occur only if the lune has length
$n$.)

It is easy to verify that this correspondence between the faces of the lune
complex and elements of $T_n^*$ up to rank at most $(n-2)$ is order
preserving. In fact, if the lune (or vertex) $\lambda_1$ associated
$(x_1',y_1')$ to is contained in the lune $\lambda_2$ associated to
$(x_2',y_2')$, then the $\lambda_2$ must have larger dimension, and so
larger rank (this is condition $|x_1'|<|x_2'|$). If this is
satisfied, then $\lambda_1\subset\lambda_2 $ holds when either
the first non-star coordinate of $\lambda_2$ has higher index 
than the length $\lambda_1$ (equivalent to $|y_1|<|x_2|$) or 
the interval $[\alpha,\beta]$ in the code $\lambda_2$ 
exactly at index $\ell(\lambda_1)$ (thus containment at this
coordinate is the only issue, and it is equivalent to $x_2'=y_1'$), or
the interval $[\alpha,\beta]$ in the code $\lambda_2$ occurs at a
coordinate where the coordinate of $\lambda_1$ is $\pi/2$. In this last
case the subsequent coordinates of $\lambda_1$ and $\lambda_2$ must
agree, which is equivalent to $x_2'=y_1'$. 

Finally, we may extend our order preserving bijection to the
elements of rank $n$ by associating 
$(-n,-(n+1))$ to $(*,\ldots,*,[0,\pi/2])$,
$(n,-(n+1))$ to $(*,\ldots,*,[\pi/2,\pi])$,
$(n,(n+1))$ to\\ $(*,\ldots,*,[\pi,3\pi/2])$, and
$(-n,(n+1))$ to $(*,\ldots,*,[3\pi/2,2\pi])$. Since we are constructing
a bijection between graded partially ordered sets, it is sufficient to
verify that the corresponding coatoms cover the corresponding
sub-coatoms. This is most easily shown ``pictorially'':
%\begin{center}
\par\noindent
\input{op-02.pstex_t}
%\end{center}
\par\noindent
corresponds to
%\begin{center}
\par\noindent
\input{op-03.pstex_t}
%\end{center}
\par\noindent
Note that on the first picture the labels correspond to $n=3$, but for
larger $n$ the only difference is that each label needs to be
prepended with the appropriate number of stars.

\section{Concluding remarks}
\label{S_cr}

As mentioned in the Introduction, an interesting
continuation of the research presented in this paper could be exploring
the potential connections between the theory of chain sequences and
flag-enumeration in Eulerian posets, deciding along the way 
whether there is a closer relation to 
spherical shellings or to the approach taken in
Section~\ref{S_expl}. If the theory of chain sequences turns
out to be unrelated to either of these methods, then its non-commutative
generalization (if it exists) could provide a new approach to proving
non-negativity results for Eulerian posets. 

It is also worth exploring what the study of lune complexes may tell us
about systems of symmetric orthogonal polynomials.
The first question is whether the polynomials $R(y_1,\ldots,y_n;x)$
given by~(\ref{E_Rrec}) (introduced in connection with spherical shellings)
have any further significance in the theory of
orthogonal polynomials. Second, the issue of ``translating''
invariants of Eulerian posets could be raised. If we do not insist on
finding non-negativity results, ``almost every'' symmetric OPS is
equivalent (up to replacing each polynomial with a nonzero constant
multiple) to an OPS defined by a recursion formula of the form
$Q_n(x)=\nu_{n} xQ_{n-1}(x)+(1-\nu_n)Q_{n-2}(x)$. Whenever the
$\nu_i$'s are positive integers such that an associated poset
$Q_n(\nu_2,\ldots,\nu_n)$ exists, invariants like the ``toric
$h$-vector'' (for the definition see \cite[Section 3.14]{Stanley-EC1})
are polynomial expressions of the $\nu_i$'s. Hence the definition of
such invariants may be extended to almost all symmetric OPS (and perhaps
even to the ``singular ones'', if ``taking limits'' is possible). It is
very natural to ask, what is the meaning of such invariants in the
theory of orthogonal polynomials. Finally, a new connection between
certain orthogonal polynomials and statistics on words may be established by
passing through the lune complex representation. In fact,  
the non-negativity of the $cd$-index of the Tchebyshev
posets was shown in~\cite{Hetyei-Tch} using a shelling of the order
complex, and not spherical shelling. That approach not only provided
explicit formulas for the coefficients of the $cd$-index, but also
established a connection to a certain statistics on words. It is
worth trying to generalize that method to the order complexes of the
face posets of lune complexes with the same aim.

To conclude, let us remind the reader how having a sequence of face posets of
$CW$-spheres closed under taking boundary complexes of facets may bring
us closer to proving Stanley's conjecture~\cite[Conjecture 2.1]{Stanley-flag}. 
Given a sequence $\Omega_1,\Omega_2,\ldots$ of $CW$-spheres (one for each
dimension), such that
each face of each $\Omega_i$ has the same  face structure as some $\Omega_j$,
we may restrict our attention to those Eulerian posets $P$ for which
every interval $[\0,x]$ satisfying $x\neq \1$ is isomorphic to some
$P_1(\Omega_i)$. In the case when each $\Omega_i$ is the boundary complex of
a simplex, one obtains the class of Gorenstein$^*$ simplicial posets,
that was treated by Stanley in~\cite{Stanley-flag}. A first step towards 
handling the cubical case (where each $\Omega_i$ is the boundary complex
of a cube) was proposed by Ehrenborg and
Hetyei in~\cite{Ehrenborg-Hetyei}. Consideration of the special case
when each $\Omega_i$ (for $i\geq 2$) is of the form $L(2,2,\ldots,4)$
was proposed in~\cite{Hetyei-Tch} and, before
considering $\Omega_i=L(\nu_2,\ldots,\nu_i,2\nu_i)$ with other
$\nu_i$'s, it is still advisable to  
investigate that special case first. As noted in~\cite{Hetyei-Tch}, it is easy
to derive the analogues of the Dehn-Sommerville equations for ``spheres
of dual Tchebyshev cells'', but it is unknown whether a sufficient number
of linearly independent examples exists (in analogy to the simplicial
and the cubical case). It is probably harder to construct such examples
(because of the lack of polytopality), but if they exist, one 
obtains a special case of Stanley's conjecture that is very different
from the simplicial and cubical cases yet ``not more difficult'' (as
far as the number of unknowns per rank is concerned). 
It is to be expected that, for any such restricted case, the
study of $cd$-indices (or, equivalently, flag $f$-vectors) may be
reduced to the study of (non-flag) $f$-vectors of certain
$CW$-spheres. In particular, Stanley reduces to the nonnegativity of the
$cd$-index of a simplicial Gorenstein$^*$ poset to the nonnegativity of its
simplicial $h$-vector in~\cite{Stanley-flag}, and Ehrenborg and Hetyei
show an analogous reduction for cubical posets
in~\cite{Ehrenborg-Hetyei}. The nonnegativity of the simplicial
$h$-vector of simplicial Cohen-Macaulay posets was shown in Stanley's
earlier paper~\cite{Stanley-fvect}. In the cubical case, the
nonnegativity of the analogous $h$-vector (originally defined by
Adin~\cite{Adin}) is still a conjecture, i.e., the cubical analogue
of~\cite{Stanley-fvect} is still missing. It is not
implausible to suspect that, if finding the analogous results for  
for $\Omega_i=L(2,2,\ldots,2,4)$ is feasible, then the corresponding argument
for $\Omega_i=L(\nu_2,\ldots,\nu_i,2\nu_i)$ with arbitrary $\nu_i$'s
would involve finding a generalized $h$-vector that is a linear
expression of the face numbers, with coefficients from ${\mathbb
  Q}[\nu_2,\nu_3,\ldots]$. Once a sufficient number of such simplified
questions is answered, the proof of Stanley's general conjecture may perhaps be
found more easily.  


\section*{Acknowledgments}
I wish to thank to Tam\'as Erd\'elyi and Paul N\'evai for pointing in
the right direction and to Dennis Stanton for fruitful interaction,
and useful advice. I am also grateful to an anonymous referee for
substantial corrections.

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\end{document}








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