An Explicit Example of Polynomials Orthogonal on the Unit Circle

An Explicit Example of Polynomials Orthogonal
on the Unit Circle with a Dense Point Spectrum
Generated by a Geometric Distribution

Alexei ZHEDANOV

A. Zhedanov

School of Mathematics, Renmin University of China, Beijing 100872, China
<EMAIL_ADDRESS>
Received November 02, 2020, in final form December 19, 2020; Published online
December 21, 2020

We present a new explicit family of polynomials orthogonal on the unit circle
with a dense point spectrum. This family is expressed in terms of
$q$-hypergeometric function of type ${{}_{2}}\phi_{1}$. The orthogonality
measure is the wrapped geometric distribution. Some “classical” properties of
the above polynomials are presented.

polynomials orthogonal on the unit circle; wrapped geometric dustribution;
dense point spectrum

33D45; 42C05

##  1 Introduction

Let $\Phi_{n}(z)$ be monic polynomials
$\Phi_{n}(z)=z^{n}+O\big{(}z^{n-1}\big{)}$ defined through the recurrence
relation [17]

$\displaystyle\Phi_{n+1}(z)=z\Phi_{n}(z)-\bar{a}_{n}\Phi^{*}_{n}(z),\qquad\Phi_{0}(z)=1,$

where

$\Phi_{n}^{*}(z)=z^{n}\bar{\Phi}_{n}(1/z)$

and where $\bar{\Phi}_{n}(z)$ means complex conjugation of expansion
coefficients of the polynomial $\Phi_{n}(z)$. The recursion parameters

$a_{n}=-\bar{\Phi}_{n+1}(0)$

are called the Verblunsky (sometimes also reflection, Schur etc.) parameters
[17].

Under the condition

$\displaystyle|a_{n}|<1,\qquad n=0,1,2,\dots$ (1.1)

the polynomials $\Phi_{n}(z)$ are orthogonal on the unit circle with respect
to a positive measure ${\rm d}\sigma(\theta)$

$\displaystyle\int_{0}^{2\pi}\Phi_{n}\big{(}{\rm e}^{{\rm
i}\theta}\big{)}\bar{\Phi}_{m}\big{(}{\rm e}^{-{\rm i}\theta}\big{)}{\rm
d}\sigma(\theta)=h_{n}\delta_{nm},$ (1.2)

where

$\displaystyle
h_{n}=\big{(}1-|a_{0}|^{2}\big{)}\big{(}1-|a_{1}|^{2}\big{)}\cdots\big{(}1-|a_{n-1}|^{2}\big{)}$
(1.3)

are normalization constants (which are nonzero due to condition (1.1)). In
this case $\Phi_{n}(z)$ are called the orthogonal polynomials on the unit
circle (OPUC).

Note that orthogonality relation (1.2) is equivalent to conditions [17]

$\displaystyle I_{nj}\equiv\int_{0}^{2\pi}\Phi_{n}\big{(}{\rm e}^{{\rm
i}\theta}\big{)}{\rm e}^{-{\rm i}j\theta}\,{\rm
d}\sigma(\theta)=h_{n}\delta_{nj},\qquad j=0,1,2,\dots,n.$ (1.4)

Equivalently, OPUC $\Phi_{n}(z)$ can be constructed in terms of trigonometric
moments $\sigma_{n}$. The latter are defined as

$\displaystyle\sigma_{n}=\int_{0}^{2\pi}{\rm e}^{{\rm i}n\theta}\,{\rm
d}\sigma(\theta),\qquad n=0,\pm 1,\pm 2,\dots.$

Then polynomials $\Phi_{n}(z)$ have the explicit expression

$\displaystyle\Phi_{n}(z)=(\Delta_{n})^{-1}\left|\begin{matrix}\sigma_{0}&\sigma_{1}&\dots&\sigma_{n}\\\
\sigma_{-1}&\sigma_{0}&\dots&\sigma_{n-1}\\\ \dots&\dots&\dots&\dots\\\
\sigma_{1-n}&\sigma_{2-n}&\dots&\sigma_{1}\\\
1&z&\dots&z^{n}\end{matrix}\right|,$

where

$\displaystyle\Delta_{n}=\left|\begin{matrix}\sigma_{0}&\sigma_{1}&\dots&\sigma_{n-1}\\\
\sigma_{-1}&\sigma_{0}&\dots&\sigma_{n-2}\\\ \dots&\dots&\dots&\dots\\\
\sigma_{1-n}&\sigma_{2-n}&\dots&\sigma_{0}\end{matrix}\right|$

are Toeplitz determinants which are all positive $\Delta_{n}>0$,
$n=0,1,2,\dots$. Note the symmetry property of the trigonometric moments

$\displaystyle\sigma_{-n}=\bar{\sigma}_{n}.$ (1.5)

Explicit examples of polynomials orthogonal on unit circle are very
interesting from different point view. By “explicit examples” we mean that all
main objects: the parameters $a_{n}$, the moments $\sigma_{n}$, the measure
$\sigma(\theta)$ and the polynomials themselves $\Phi_{n}(z)$ have explicit
expressions in terms of special functions. Usually, in most known explicit
examples the parameters $a_{n}$ are given by elementary functions of $n$ while
the OPUC $\Phi_{n}(z)$ are expressed in terms of hypergeometric functions
(either ordinary or basic). A list of known explicit examples can be found,
e.g., in Simon’s monograph [17].

In [21, 27] new explicit examples of OPUC were presented. In these examples
polynomials $\Phi_{n}(z)$ are expressed in terms of elliptic hypergeometric
function ${{}_{3}}E_{2}(z)$ while the moments $\sigma_{n}$ and the recurrence
parameters $a_{n}$ have simple expressions in terms of elliptic functions. The
most interesting property of the OPUC of these examples is that they are
orthogonal on the unit circle with respect to a dense point measure. This
means that the function $\sigma(\theta)$ is a step function with infinitely
many points $\theta_{s}$ of jumps, and these points are dense on the interval
$[0,2\pi]$. In terms of the distribution function this can be presented as

$\displaystyle\rho(\theta)=\sum_{s=-\infty}^{\infty}M_{s}\delta(\theta-\theta_{s}),$

where $\rho(\theta)$ is a distribution defined as ${\rm
d}\sigma(\theta)=\rho(\theta){\rm d}\theta$, $\delta(\theta)$ is the Dirac
delta function and $M_{s}$ are concentrated masses located at points of jumps
$\theta_{s}$. The spectral points $z_{s}=\exp({\rm i}\theta_{s})$ are dense on
the unit circle.

Then orthogonality relation (1.2) can then be presented as

$\sum_{s=-\infty}^{\infty}M_{s}\Phi_{n}\big{(}{\rm e}^{{\rm
i}\theta_{s}}\big{)}\bar{\Phi}_{m}\big{(}{\rm e}^{-{\rm
i}\theta_{s}}\big{)}=h_{n}\delta_{nm}.$

From general considerations (see, e.g., [17]) it follows that polynomials
orthogonal with respect to such dense point measures are rather generic if one
assumes some natural restrictions upon behavior of the recurrence parameters
$a_{n}$. On the other hand, such measures are very important from physical
point of view, because they correspond to the phenomenon of the Anderson
localization [10, 17].

Usually examples of OPUC with dense point spectrum are related to sequences of
the parameters $a_{n}$ which behave (quasi) stochastically inside the interval
$|a_{n}|<1$ [17]. OPUC in [21, 27] provide perhaps the first known examples of
a pure point dense measure on the unit circle where both the coefficients
$a_{n}$ and the moments $\sigma_{n}$ are given explicitly by analytic
functions in $n$.

In this paper we propose a much simpler explicit example of polynomials
orthogonal on the unit with respect to a (wrapped) geometric distribution
which is dense on the unit circle. Polynomials themselves are expressed in
terms of basic hypergeometric function ${{}_{2}}\phi_{1}(q;z)$ with $|q|=1$.

##  2 Wrapped geometric distribution and corresponding OPUC

Let $q$ be a fixed point belonging to the unit circle $|q|=1$ and not a root
of unity, i.e., we demand that $q^{n}\neq 1$ for all natural integer
$n=0,1,\dots$. Choose a real parameter $p$ within the unit interval $0<p<1$.
Define the measure on the unit circle as

$\displaystyle\rho(\theta)=(1-p)\sum_{s=0}^{\infty}p^{s}\delta(\theta-s\chi),$
(2.1)

where $\chi$ is a fixed irrational parameter $0<\chi<1$ such that

$\displaystyle q=\exp(2\pi{\rm i}\chi).$ (2.2)

Irrationality of $\chi$ means that the set of points $z_{s}=q^{s}$,
$s=0,1,\dots$ (i.e., the location of jumps of the measure) is dense on the
unit circle. The weights (i.e., the concentrated masses) $w_{s}$ at the points
$z_{s}$ form the geometric sequence: $w_{s}=p^{s}$, $s=0,1,2,\dots$.

Corresponding trigonometric moments are

$\displaystyle\sigma_{n}=\sum_{s=0}^{\infty}z_{s}^{n}w_{s}=(1-p)\sum_{s=0}^{\infty}q^{sn}p^{s}=\frac{1-p}{1-pq^{n}},\qquad
n=0,\pm 1,\pm 2,\dots.$ (2.3)

Note that the measure (2.1) can be interpreted as the wrapped geometric
distribution on the unit circle (see, e.g., [8, 15] for definition and
discussion of wrapped distributions on the unit circle).

Relation $\sigma_{0}=1$ means that the measure (2.1) has the standard
normalization condition.

One can present expression (2.3) as

$\displaystyle\sigma_{n}=\frac{(p;q)_{n}}{(pq;q)_{n}},$ (2.4)

where the $q$-shifted factorial ($q$-Pochhammer symbol) is defined as [4, 9]
$(a;q)_{0}=1$ and

$\displaystyle(a;q)_{n}=(1-a)(1-aq)\cdots\big{(}1-aq^{n-1}\big{)}$

for positive $n=1,2,\dots$ and

$\displaystyle(a;q)_{n}=\frac{1}{\big{(}aq^{n};q\big{)}_{-n}}$

for negative $n=-1,-2,\dots$.

It is known that the Laurent biorthogonal Pastro polynomials $P(z;a,b)$ [16]
depending on two arbitrary parameters $a$, $b$ can be uniquely defined through
their moments [22]

$\displaystyle\sigma_{n}=\frac{(a;q)_{n}}{(b;q)_{n}},\qquad n=0,\pm 1,\pm
2,\dots.$ (2.5)

Explicitly, these polynomials are given by [16, 22]

$\displaystyle P(z;a,b)=\mu_{n}\,{{}_{2}}\phi_{1}\left({q^{-n},b\atop
aq^{1-n}};qz\right),$ (2.6)

where $\mu_{n}$ is an appropriate normalization factor to fulfill the
condition $P_{n}(z)=z^{n}+O\big{(}z^{n-1}\big{)}$. The definition and notation
of the basic hypergeometric function ${{}_{m}}\phi_{n}(z)$ is standard (see,
e.g., [4, 9]). For example, in the special case $m=n+1$ we have the
expressions

${{}_{m+1}}\phi_{m}\left({a_{1},a_{2}\dots,a_{m+1}\atop
b_{1},b_{2}\dots,b_{m}};z\right)=\sum_{s=0}^{\infty}\frac{(a_{1};q)_{s}(a_{2};q)_{s}\cdots(a_{m+1};q)_{s}}{(q;q)_{s}(b_{1};q)_{s}(b_{2};q)_{s}\cdots(b_{m};q)_{s}}z^{s}.$

Note the Laurent biorthogonal polynomials (LBP) can be considered as a
generalization of the OPUC. Their main distinction from OPUC is that the
moments $\sigma_{n}$ do not satisfy, in general, the symmetry condition (1.5).
The LBP can also be characterized by the three-term recurrence relation of
$R_{I}$ type [7, 24]

$\displaystyle P_{n+1}(z)+g_{n}P_{n}(z)=z(P_{n}(z)+d_{n}P_{n-1}(z)),\qquad
P_{0}=1,\qquad P_{-1}=0$ (2.7)

with some recurrence coefficients $g_{n}$, $d_{n}$.

In contrast to the case of orthogonal polynomials, the recurrence relation
(2.7) can be presented in the form of the generalized eigenvalue problem [25]

$\displaystyle J_{1}{\mathbf{P}}(z)=zJ_{2}{\mathbf{P}}(z),$ (2.8)

where $J_{1}$, $J_{2}$ are upper and lower bidiagonal matrices acting on the
vector

$\mathbf{P}(z)=(P_{0}(z),P_{1}(z),\dots).$

Comparing expressions (2.5) and (2.4) we can conclude that the OPUC
corresponding to the wrappedd geometric distribution are special case of the
Pastro polynomials with $a=p$, $b=qp$.

This allows us to present the main result

######  Theorem 2.1.

The polynomials $\Phi_{n}(z)$ orthogonal on the unit circle with respect to
the wrapped geometric distribution (2.1) have the explicit expression

$\displaystyle\Phi_{n}(z)=\mu_{n}\>{{}_{2}}{\phi}_{1}\left({q^{-n},pq\atop
pq^{1-n}};zq\right),$ (2.9)

where

$\displaystyle\mu_{n}=q^{-n}\frac{(q;q)_{n}\big{(}pq^{1-n};q\big{)}_{n}}{\big{(}q^{-n};q\big{)}_{n}(pq;q)_{n}}=p^{n}\frac{\big{(}p^{-1};q\big{)}_{n}}{(qp;q)_{n}}.$
(2.10)

One can directly check that the polynomials (2.9) satisfy orthogonality
relations (1.4). Indeed, one has

$\displaystyle
I_{nj}=(1-p)\sum_{s=0}^{\infty}\Phi_{n}\big{(}q^{s}\big{)}q^{-sj}p^{s}=(1-p)\mu_{n}\sum_{s=0}^{\infty}\sum_{k=0}^{n}\frac{\big{(}q^{-n};q\big{)}_{k}(pq;q)_{k}}{(q;q)_{k}\big{(}pq^{1-n};q\big{)}_{k}}q^{(1+s)k}q^{-js}p^{s}.$

Performing summation over $s$ we get

$\displaystyle
I_{nj}=\frac{(1-p)\mu_{n}}{1-pq^{-j}}\sum_{k=0}^{n}\frac{\big{(}q^{-n};q\big{)}_{k}(pq;q)_{k}\big{(}pq^{-j};q\big{)}_{k}}{(q;q)_{k}\big{(}pq^{1-n}\big{)}_{k}\big{(}pq^{1-j};q\big{)}_{k}}q^{k}=\frac{(1-p)\mu_{n}}{1-pq^{-j}}\,{{}_{3}}{\phi}_{2}\left({q^{-n},pq,pq^{-j}\atop
pq^{1-n},pq^{1-j}};q\right).$

The above expression can be simplified by the $q$-Saalschütz formula [4, 9]:

$\displaystyle{{}_{3}}{\phi}_{2}\left({q^{-n},a,b\atop
c,abc^{-1}q^{1-n}};q\right)=\frac{(c/a;q)_{n}(c/b;q)_{n}}{(c;q)_{n}(c/(ab);q)_{n}}.$

We thus have

$\displaystyle
I_{nj}=\frac{(1-p)\mu_{n}}{1-pq^{-j}}\frac{\big{(}q^{-n};q\big{)}_{n}\big{(}q^{j+1-n};q\big{)}_{n}}{\big{(}pq^{1-n};q\big{)}_{n}\big{(}p^{-1}q^{j-n};q\big{)}_{n}}.$
(2.11)

The factor $\big{(}q^{j+1-n};q\big{)}_{n}$ in (2.11) becomes zero when
$j=0,1,\dots,n-1$ and hence

$I_{nj}=0,\qquad j=0,1,\dots,n-1,$

which is equivalent to orthogonality relation (1.4). It remains to show that
$I_{nn}>0$. After simple calculations one can arrive at the expression

$\displaystyle I_{nn}=h_{n}=\frac{|(q;q)_{n}|^{2}}{|(pq;q)_{n}|^{2}}p^{n}$
(2.12)

from which it is clear that $h_{n}>0$ for all $n=0,1,2,\dots$ due to condition
$q^{n}\neq 1$.

Explicit expression for the the recurrence parameters $a_{n}$ follows from
(2.9) and (2.10):

$\displaystyle{\bar{a}}_{n-1}=-\Phi_{n}(0)=-\mu_{n}=-p^{n}\frac{\big{(}p^{-1};q\big{)}_{n}}{(qp;q)_{n}}.$

For the square of absolute values we have rather simple expression

$\displaystyle|a_{n-1}|^{2}={\bar{a}}_{n-1}a_{n-1}=\frac{(1-p)^{2}}{1+p^{2}-p\big{(}q^{n}+q^{-n}\big{)}}=\frac{1}{1+\beta\sin^{2}(\chi\pi
n)},$ (2.13)

where

$\displaystyle\beta=\frac{4p}{(1-p)^{2}}$

and where the parameter $\chi$ is the same as in (2.2).

It is seen from (2.13) that the values $|a_{n}|$ oscillate inside the interval

$\displaystyle\frac{1-p}{1+p}<|a_{n}|<1,\qquad n=0,1,\dots.$

Because of irrationality of $\chi$ the absolute value $|a_{n}|$ achieves the
boundaries of this interval with any prescribed accuracy (never achieving
exact boundary values). Note that $a_{-1}=-1$ which corresponds to the
standard initial conditions for OPUC [5, 17]. It is easily verified that
expression (2.12) for $h_{n}$ agrees with relation (1.3).

The OPUC (2.9) can be considered as $|q|=1$ analogs of the OPUC introduced by
Askey in [1] (see also [3] for more general OPUC of Askey’s type).

##  3 “Classicality” of the polynomials $\boldsymbol{\Phi_{n}(z)}$

The OPUC (2.9) possess “classical” properties which make them similar to
classical orthogonal polynomials.

First of all, they satisfy the three-term recurrence relation (2.7) where the
recurrence coefficients are

$\displaystyle g_{n}=\frac{q^{n}-p}{1-pq^{n+1}},\qquad
d_{n}=-\frac{p(1-q^{n})^{2}}{(1-pq^{n})(1-pq^{n+1})}.$

Moreover, the polynomial $\Phi_{n}(z)$ possess a remarkable duality property.
Indeed, one can rewrite polynomials $\Phi_{n}(z)$ in a different form

$\displaystyle\Phi_{n}(z)=p^{n}\frac{(q;q)_{n}}{(pq;q)_{n}}z^{n}\,{{}_{3}}{\phi}_{2}\left({q^{-n},p^{-1},z^{-1}\atop
q,0};q\right),$ (3.1)

which can be obtained from (2.9) by standard transformation formulas [4, 9].

From this formula the duality property

$\displaystyle A_{s}\Phi_{s}(q^{n})=A_{n}\Phi_{n}(q^{s})$ (3.2)

follows, where

$\displaystyle A_{n}=\frac{(pq;q)_{n}}{(q;q)_{n}}p^{-n}.$

This property resembles corresponding duality properties for the classical
orthogonal polynomials from the Askey scheme [2, 11, 19]. The main difference
is that the polynomials $\Phi_{n}(z)$ satisfy the generalized eigenvalue
problem (2.8) instead of the ordinary eigenvalue problem for orthogonal
polynomials.

From the duality property one can derive the second-order $q$-difference
equation

$\displaystyle
B_{s+1}\Phi_{n}\big{(}q^{s+1}\big{)}+g_{s}\Phi_{n}\big{(}q^{s}\big{)}=q^{n}\big{(}\Phi_{n}\big{(}q^{s}\big{)}+B_{s}^{-1}d_{s}\Phi_{n}\big{(}q^{s-1}\big{)}\big{)},$
(3.3)

where

$B_{s}=\frac{A_{s-1}}{A_{s}}=\frac{p\big{(}1-q^{s}\big{)}}{1-pq^{s}}.$

Equation (3.3) can also be presented in the form of the generalized eigenvalue
problem

$\displaystyle L\Phi_{n}(z)=q^{n}M\Phi_{n}(z),$ (3.4)

where the first-order $q$-difference operators $L$, $M$ act on the argument
$z$ of the polynomials.

Relations (2.8) and (3.4) mean that the polynomials possess the bispectrality
property: they satisfy simultaneously two GEVP. Concerning definition and
general theory of bispectrality see, e.g., [6]. For orthogonal polynomials
from the Askey scheme this property is well known [9]. For biorthogonal
polynomials and rational functions the bispectrality is known for some special
families. The most general from them are elliptic biorthogonal functions [18].
However the general theory of bispectrality for systems satisfying GEVP is not
yet developed (see, e.g., [20, 23] for algebraic description of bispectrality
on the “lowest” level of hypergeometric functions ${{}_{3}}F_{2}(1)$).

The duality property implies that for $z=q^{s}$, $s=0,1,2,\dots$ the
hypergeometric function in (3.1) reduces to a polynomial of degree $s$ of the
argument $q^{-n}$.

It is well known (see, e.g., [17]) that if $z_{0}$ is a point on the unit
circle corresponding to a concentrated mass $M_{0}$ then the relation

$\displaystyle\sum_{n=0}^{\infty}\frac{|\Phi_{n}(z_{0})|^{2}}{h_{n}}=1/M_{0},$

holds, where the normalization coefficient $h_{n}$ is defined in (1.3).

In our case this means that for every spectral point $z_{s}=q^{s}$,
$s=0,1,\dots$ there exists the identity

$\displaystyle\sum_{n=0}^{\infty}\frac{\big{|}\Phi_{n}\big{(}q^{s}\big{)}\big{|}^{2}}{h_{n}}={M_{s}}^{-1}=\frac{p^{-s}}{1-p}.$
(3.5)

Identity (3.5) follows easily from the duality property (3.2) and from
orthogonality relation.

So far, we have considered the case when $q$ is not a root of unity. If,
otherwise, $q$ is a primitive root of unity

$\displaystyle q=\exp\left(\frac{2\pi{\rm i}M}{N}\right)$

with coprime integers $M$, $N$, then there are only $N$ distinct mass points
on the unit circle located at $z_{s}=q^{s}$, $s=0,1,2,\dots,N-1$. In this case
the polynomials $\Phi_{n}(z)$ are orthogonal on vertices of a regular $N$-gon
with respect to the finite wrapped geometric distribution:

$\displaystyle\sum_{s=0}^{N-1}\Phi_{n}\big{(}q^{s}\big{)}\bar{\Phi}_{m}\big{(}q^{-s}\big{)}\big{(}1-p^{N}\big{)}p^{s}=h_{n}\delta_{nm},\qquad
n,m=0,1,\dots,N-1.$

See [26] for other explicit examples of polynomials orthogonal on the vertices
of regular polygons.

##  4 Concluding remarks

In contrast to examples of OPUC obtained in [27], the polynomials (2.6) have
non-real moments $\sigma_{n}$ and hence the coefficients $a_{n}$ are non-real
as well. This means that it is impossible to associate with OPUC (2.6)
polynomials orthogonal on an interval of the real line. In [27] explicit
examples of polynomials orthogonal with dense point spectrum on an interval
were presented using standard Szegő mapping from OPUC to an interval of the
real line. We mention also examples of OPUC and ordinary orthogonal
polynomials with dense point spectrum presented in [13, 14].

The OPUC (2.6) allow a trivial modification which shifts all spectral points
on the unit circle on the same constant angle $\varphi$, i.e., we can consider
the same weights $w_{s}=p^{s}(1-p)$ located at the points

$\theta_{s}=2\pi\chi s+\varphi,\qquad s=0,1,2,\dots.$

Equivalently, this means that the new spectral points will be
$\tilde{z}_{s}={\rm e}^{{\rm i}\varphi}q^{s}$, $s=0,1,2,\dots$.

Such transformation is equivalent to a simple rotation of the argument of OPUC
[5, 17]:

$\tilde{\Phi}_{n}(z)={\rm e}^{-{\rm i}\varphi n}\Phi_{n}\big{(}{\rm e}^{{\rm
i}\varphi}z\big{)}.$

Another modification of the OPUC (2.6) is more substantional. It leads to
Laurent biorthogonal polynomials orthogonal on the unit circle with dense
point measure.

Indeed, assume that the spectral points on the unit circle are the same:
$z_{s}=q^{s},\,s=0,1,2,\dots$. Take the weights:

$\displaystyle w_{s}=p^{s}\frac{\big{(}q^{k};q\big{)}_{s}}{(q;q)_{s}},\qquad
0<p<1,\qquad k=1,2,3,\dots.$ (4.1)

For $k=1$ we return to the case of the wrapped geometric distribution. For
$k>1$ the moments are

$\displaystyle\sigma_{n}=\sum_{s=0}^{\infty}\frac{\big{(}q^{k};q\big{)}_{s}}{(q;q)_{s}}p^{s}q^{sn}.$
(4.2)

By $q$-binomial theorem [4, 9] the above sum is simplified to

$\displaystyle\sigma_{n}=\frac{\big{(}q^{k}pq^{n};q\big{)}_{\infty}}{\big{(}pq^{n};q\big{)}_{\infty}}=\frac{1}{\big{(}pq^{n};q\big{)}_{k}}=\frac{(p;q)_{n}}{(p;q)_{k}\big{(}pq^{k};q\big{)}_{n}}.$

Usually, the convergence problem for $q$-series like (4.2) with $|q|=1$ is
highly nontrivial (see, e.g., [12]). In our case however this problem does not
appear because for integer $k$ there is cancellation of almost all terms
(apart of a finite number of initial ones) in denominators of the coefficients
in (4.2). Hence the convergence for $0<p<1$ still takes place.

For fixed $k$ the moments $\sigma_{n}$ coincide (up to a constant factor) with
the moments (2.5) for the Pastro polynomials with $a=p$, $b=pq^{k}$. Hence
from (2.6) we have explicit expression for them

$\displaystyle P(z)=\mu_{n}{{}_{2}}\phi_{1}\left({q^{-n},pq^{k}\atop
pq^{1-n}};qz\right),$

where

$\mu_{n}=\frac{p^{n}\big{(}p^{-1};q\big{)}_{n}}{\big{(}pq^{k};q\big{)}_{n}}.$

These polynomials are NOT OPUC (apart from the already considered case $k=1$)
because the weights (4.1) are not positive and hence the moments $\sigma_{n}$
do not satisfy symmetry property (1.5).

Existence of other explicit examples of OPUC with dense point spectrum is an
interesting open problem.

### Acknowledgements

The author is indebted for F.A. Grünbaum, A. Magnus, V. Spiridonov, S.
Tsujimoto and L. Vinet for discussions and for anonymous referees for valuable
remarks. The author is gratefully holding Simons CRM Professorship and is
funded by the National Foundation of China (Grant No. 11771015).

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