# Nevanlinna extremal measures for polynomials related to $q^{-1}$-Fibonacci
polynomials

F. Štampach1

###### Abstract

The aim of this paper is the study of $q^{-1}$-Fibonacci polynomials with
$0<q<1$. First, the $q^{-1}$-Fibonacci polynomials are related to a
$q$-exponential function which allows an asymptotic analysis to be worked out.
Second, related basic orthogonal polynomials are investigated with the
emphasis on their orthogonality properties. In particular, a compact formula
for the reproducing kernel is obtained that allows to describe all the
N-extremal measures of orthogonality in terms of basic hypergeometric
functions and their zeros. Two special cases involving $q$-sine and $q$-cosine
are discussed in more detail.

1Department of Applied Mathematics, Faculty of Information Technology, Czech
Technical University in Prague, Thákurova 9, 160 00 Praha, Czech Republic

_Keywords_ : $q$-Fibonacci polynomials, orthogonal polynomials, measure of
orthogonality, N-extremal measures, $q$-exponential, $q$-sine, $q$-cosine

_2010 Mathematical Subject Classification_ : 11B39, 33D45, 30E05

##  1 Introduction

In [13], Carlitz introduced the $q$-Fibonacci polynomials $\varphi_{n}(x;q)$
by

$\varphi_{n}(x;q)=\sum_{2k<n}{n-k-1\brack k}_{q}q^{k^{2}}x^{n-2k-1},$ (1)

for $n\in\mathbb{Z}_{+}$ (nonnegative integers). The $q$-binomial coefficient
is defined by

${n\brack k}_{q}=\frac{(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}},$

for $k=0,1,\dots,n$ and $n\in\mathbb{Z}_{+}$, where

$(a;q)_{n}=\prod_{k=0}^{n-1}\left(1-aq^{k}\right)$

is the standard notation for the $q$-Pochhammer symbol. In this paper, we
intensively use the theory of basic hypergeometric series and, in the
notation, we will strictly follow Gasper and Rahman’s book [18]. In addition,
if it is not stated otherwise, we always assume $0<q<1$.

The polynomials $\varphi_{n}(x;q)$ satisfy the second-order recurrence

$\varphi_{n+1}(x;q)=x\varphi_{n}(x;q)+q^{n-1}\varphi_{n-1}(x;q),\quad
n\in\mathbb{N},$ (2)

with the initial conditions $\varphi_{0}(x;q)=0$ and $\varphi_{1}(x;q)=1$. Let
us mention that $\varphi_{n}(1;q)$ are polynomials in $q$ first considered by
I. Schur in conjunction with his proof of the Rogers-Ramanujan identities
[31]. It can be referred as the $q$-analogue to the Fibonacci numbers $F_{n}$
since clearly $\varphi_{n}(1;1)=F_{n}$, see [12]. For a different definition
of $q$-Fibonacci numbers, see also [21].

One of our primary intention is the study of orthogonality of related
polynomials

$T_{n}(x;q)=(-{\rm i})^{n}q^{-n/2}\varphi_{n+1}({\rm i}q^{1/2}x;q),\quad
n=-1,0,1,2,\dots.$ (3)

They fulfill the second-order difference equation

$T_{n+1}(x;q)=xT_{n}(x;q)-q^{n-1}T_{n-1}(x;q),\quad n\in\mathbb{Z}_{+},$ (4)

with the initial conditions $T_{-1}(x;q)=0$ and $T_{0}(x;q)=1$. Identity (1)
together with (3) yield the explicit expression

$T_{n}(x;q)=\sum_{2k\leq n}{n-k\brack k}_{q}(-1)^{k}q^{k(k-1)}x^{n-2k},\quad
n\in\mathbb{Z}_{+}.$

The polynomials $\varphi_{n}(x;q)$ as well as $T_{n}(x;q)$ attract attention
of mathematicians for their beautiful properties. Let us mention at least
their intriguing relation to the Rogers-Ramanujan function, associated
continued fraction, and the celebrated Rogers-Ramanujan identities [2, 3, 4,
17, 19]. Some algebraic and combinatorial properties of these polynomials have
also been studied in [13, 16].

By the spectral theorem for orthogonal polynomials, the sequence
$\\{T_{n}(x;q)\\}_{n=0}^{\infty}$ obeying the recurrence rule (4), constitutes
a system of polynomials orthogonal with respect to a positive Borel measure on
$\mathbb{R}$ for any $q>0$. In the case $0<q<1$, polynomials $T_{n}(x;q)$ are
a particular case of orthogonal polynomials studied by Al-Salam and Ismail in
[2], consult also [19, Sec. 13.6]. The respective measure of orthogonality
$\mu_{q}$ is unique and its Stieltjes transform can be expressed as a
generalized Rogers-Ramanujan continued fraction.

Concerning the asymptotic behavior of $T_{n}(x;q)$, for $n\rightarrow\infty$
and $0<q<1$, one has

$\lim_{n\rightarrow\infty}x^{-n}T_{n}(x;q)=A_{q}\left(q^{-1}x^{2}\right)\quad\forall
x\neq 0,$

where

$A_{q}(z)=\,_{0}\phi_{1}\\!\left(\,;0;q,-qz\right)$

is the Ramanujan’s entire function. With the aid of Markov’s theorem, one can
express the Stieltjes transform of $\mu_{q}$ in the form

$\int_{\mathbb{R}}\frac{\mbox{d}\mu_{q}(x)}{1-xz}=\frac{A_{q}\left(z^{2}\right)}{A_{q}\left(q^{-1}z^{2}\right)},$
(5)

see [2, Thm. 3.2, 3.4]. Equality (5) holds for all $z\in\mathbb{C}$ for which
$A_{q}\left(q^{-1}z^{2}\right)\neq 0$. Recall that the set of zeros of the
Ramanujan’s entire function $A_{q}$ is composed of denumerable number of
positive isolated points accumulating at $\infty$, [2, 20]. Let us denote
these zeros as $z_{j}=z_{j}(q)$, for $j\in\mathbb{N}$. The orthogonality
relation for $\\{T_{n}(x;q)\\}_{n=0}^{\infty}$ reads

$\sum_{j=1}^{\infty}\frac{A_{q}(qz_{j})}{z_{j}A_{q}^{\prime}(z_{j})}T_{n}\left(\pm
q^{-1/2}z_{j}^{-1/2}\right)T_{m}\left(\pm
q^{-1/2}z_{j}^{-1/2}\right)=-2q^{n(n-1)/2}\delta_{mn},$

where $T_{n}(\pm y)T_{m}(\pm y)$ is a shorthand for the expression

$T_{n}\left(y;q\right)T_{m}\left(y;q\right)+T_{n}\left(-y;q\right)T_{m}\left(-y;q\right)\\!,$

see [2, Thm. 3.3, Cor. 4.5].

In the case $q=1$, the polynomials $T_{n}(x;1)$ are essentially the Chebyshev
polynomials of the second kind $U_{n}$. More precisely, one has
$T_{n}(x;1)=U_{n}\left(x/2\right)$. This family of polynomials is deeply
investigated and very well known [15].

In this paper we focus on the case $q>1$. This case is particularly
interesting for there are infinitely many measures of orthogonality. In other
words, the corresponding Hamburger moment problem is in the indeterminate case
if $q>1$. Solutions of the indeterminate moment problem are characterized with
the aid of four entire functions used in the Nevanlinna parametrization. In
the case under investigation, these functions are expressed as certain
$q$-extensions of sine $\mathcal{S}_{q}$ and cosine $\mathcal{C}_{q}$, for the
functions $\mathcal{S}_{q}$ and $\mathcal{C}_{q}$ naturally arise in the
asymptotic formulas for polynomials $T_{n}(x;q)$, as $n\rightarrow\infty$.

This paper is organized as follows. In Section 2, we recall certain version of
$q$-exponential function $\mathcal{E}_{q}$, associated $q$-trigonometric
functions $\mathcal{S}_{q}$ and $\mathcal{C}_{q}$, and their hyperbolic
counterparts $\mathcal{S}h_{q}$ and $\mathcal{C}h_{q}$. These functions
appeared at many various places and have been studied extensively in past,
though not in the context of $q$-Fibonacci polynomials. Section 2 recalls
their basic properties and above all an addition formula for the product of
two $q$-exponential functions which is an essential identity needed for
derivation of main results of this paper.

Section 3 is devoted to the study of relations between the $q$-exponential
$\mathcal{E}_{q}$ and the $q^{-1}$-Fibonacci polynomials
$\varphi_{n}(x;q^{-1})$. We consider $q^{-1}$-Fibonacci polynomials with
$0<q<1$ which is the same, though in some cases more convenient, as
considering $q$-Fibonacci polynomials with $q>1$. First of all, we derive a
formula for $q^{-1}$-Fibonacci polynomials in terms of the $q$-exponential
$\mathcal{E}_{q}$. As a direct consequence, we obtain asymptotic formulas for
$\varphi_{n}(x;q^{-1})$, as $n\rightarrow\infty$. Depending on the parity of
the index $n$, the functions $\mathcal{S}h_{q}$ and $\mathcal{C}h_{q}$ arise
as the leading term in the asymptotic formulas. Further, we prove a
$q$-version of the Euler–Cassini formula for $q^{-1}$-Fibonacci polynomials.
We use a method based on a very general identity which could be used
analogously in numerous similar situations. As an application, we obtain
several more subtle identities between $q^{-1}$-Fibonacci polynomials and the
functions $\mathcal{E}_{q}$, $\mathcal{S}h_{q}$, and $\mathcal{C}h_{q}$.

In Section 4, we introduce a family of orthogonal polynomials $P_{n}(x;q)$
related to $q^{-1}$-Fibonacci polynomials. We derive limit relations for
$P_{n}(x;q)$, as $n\rightarrow\infty$, as a straightforward consequence of the
knowledge of the asymptotic behavior of $q^{-1}$-Fibonacci polynomials. This
provides an alternative way of asymptotic analysis of $P_{n}(x;q)$ to the
usual method based on an appropriate generating function formula for
$P_{n}(x;q)$ and Darboux’s theorem as it was originally done by Chen and
Ismail in [14, Sec. 3].

The second part of Section 4 contains the most important results concerning
orthogonality of the polynomials $P_{n}(x;q)$. First, we recall that functions
from the Nevanlinna parametrization can be expressed in terms of the functions
$\mathcal{S}_{q}$ and $\mathcal{C}_{q}$. Next, with the aid of the addition
formula from Section 2, we find a compact expression for the reproducing
kernel for the polynomials $P_{n}(x;q)$. These two ingredients allow us to
describe all N-extremal measures in a closed form and write down the
corresponding orthogonality relations for $P_{n}(x;q)$. In addition, we point
out two special cases of orthogonality relations written in terms of the
$q$-sine $\mathcal{S}_{q}$ and its zeros only, and similar relation with the
$q$-cosine $\mathcal{C}_{q}$ and its zeros.

##  2 Functions $\mathcal{E}_{q}$, $\mathcal{S}_{q}$ and $\mathcal{C}_{q}$

There are several known $q$-deformations of the exponential function. Two
commonly known are due to Jackson:

$E_{q}(z)=\sum_{n=0}^{\infty}\frac{q^{n(n-1)/2}}{(q;q)_{n}}z^{n}\quad(z\in\mathbb{C})\quad\mbox{
and }\quad e_{q}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{(q;q)_{n}}\quad(|z|<1),$

see [18, §1.3].

The one-parameter generalization

$E_{q}^{(\alpha)}(z)=\sum_{n=0}^{\infty}\frac{q^{\alpha
n^{2}/2}}{(q;q)_{n}}z^{n}$

where $\alpha\geq 0$, was studied by Atakishieyev in [6], see also references
therein. Obviously, one has $E_{q}^{(0)}(z)=e_{q}(z)$ and
$E_{q}^{(1)}(z)=E_{q}(q^{1/2}z)$. Ismail shows in [19, Sec. 14.1] that
$E_{q}^{(\alpha)}$ are entire functions of order zero, hence they have
infinitely many zeros. The particular case corresponding to the value
$\alpha=1/2$ plays a crucial role in asymptotic formulas for
$q^{-1}$-Fibonacci polynomials, as $n\rightarrow\infty$. An asymptotic and
numerical study of zeros of $E_{q}^{(1/2)}$ has been made in [28]. For the
sake of simplicity, we use the notation $\mathcal{E}_{q}(z)$ for
$E_{q}^{(1/2)}(z)$. Thus, we consider the function

$\mathcal{E}_{q}(z)=E_{q}^{(1/2)}(z)={}_{1}\phi_{1}\left(0;-q^{1/2};q^{1/2},-q^{1/4}z\right)=\sum_{n=0}^{\infty}\frac{q^{n^{2}/4}}{(q;q)_{n}}z^{n},\quad
z\in\mathbb{C}.$

Let us also remark that this function can be obtained as a rescaled limit of
yet another more general family of $q$-exponential functions introduced by
Ismail and Zhang in [24] and later extensively studied mainly by Suslov, see
[36]. Particularly, the function $\mathcal{E}_{q}$ as well as the
corresponding trigonometric functions appears in Suslov’s book in [36, Sec.
2.5]. The function $\mathcal{E}_{q}$ fulfills the second order $q$-difference
equation

$\mathcal{E}_{q}(z)-\mathcal{E}_{q}(qz)=zq^{1/4}\mathcal{E}_{q}(q^{1/2}z).$
(6)

Corresponding $q$-sine and $q$-cosine functions can be introduced by formulas

$\mathcal{S}_{q}(z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n(n+1)}}{(q;q)_{2n+1}}z^{2n+1}\quad\mbox{
and
}\quad\mathcal{C}_{q}(z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n^{2}}}{(q;q)_{2n}}z^{2n},$
(7)

for $z\in\mathbb{C}$. With this choice, the $q$-version of Euler’s identity

$\mathcal{E}_{q}(\mathrm{i}z)=\mathcal{C}_{q}(z)+\mathrm{i}q^{1/4}\mathcal{S}_{q}(z)$
(8)

holds. Clearly,

$\lim_{q\rightarrow 1-}\mathcal{S}_{q}((1-q)z)=\sin z\quad\mbox{ and
}\quad\lim_{q\rightarrow 1-}\mathcal{C}_{q}((1-q)z)=\cos z.$

Alternatively, functions $\mathcal{S}_{q}$ and $\mathcal{C}_{q}$ can be
written as the ${}_{1}\phi_{1}$ function with the base $q^{2}$,

$\mathcal{S}_{q}(z)=\frac{z}{1-q}\,_{1}\phi_{1}(0;q^{3};q^{2},q^{2}z^{2})\quad\mbox{and}\quad\mathcal{C}_{q}(z)=\,_{1}\phi_{1}(0;q;q^{2},qz^{2})$
(9)

and in this form, they were introduced by Koornwinder and Swarttouw in [26,
Sec. 5]. The formulas in (9) allow to express the functions $\mathcal{S}_{q}$
and $\mathcal{C}_{q}$ in terms of the third Jackson (or Hahn-Exton or
${}_{1}\phi_{1}$) $q$-Bessel function

$J_{\nu}^{(3)}(z;q)=\frac{(q^{\nu+1};q){}_{\infty}}{(q;q)_{\infty}}\,z^{\nu}\,_{1}\phi_{1}(0;q^{\nu+1};q,qz^{2}),$

see, for example, [25, 26]. Namely, one has

$\mathcal{S}_{q}(z)=\frac{(q^{2};q^{2})_{\infty}}{(q;q^{2})_{\infty}}z^{1/2}J_{1/2}^{(3)}(z;q^{2})\quad\mbox{
and
}\quad\mathcal{C}_{q}(z)=\frac{(q^{2};q^{2})_{\infty}}{(q;q^{2})_{\infty}}z^{1/2}J_{-1/2}^{(3)}(q^{-1/2}z;q^{2})$
(10)

which, if compared with the classical formulas

$\sin z=\sqrt{\frac{\pi z}{2}}J_{1/2}(z)\quad\mbox{ and }\quad\cos
z=\sqrt{\frac{\pi z}{2}}J_{-1/2}(z),$

gives another reason to refer to $\mathcal{S}_{q}$ and $\mathcal{C}_{q}$ as
$q$-sine and $q$-cosine, respectively. For some additional properties
concerning $\mathcal{S}_{q}$ and $\mathcal{C}_{q}$, consult also paper of
Bustoz and Cardoso [11].

Recall that the functions $\mathcal{S}_{q}$ and $\mathcal{C}_{q}$ have
infinite number of real interlacing zeros which are all simple. This may be
deduced from results of [25, Sec. 3] for $J_{\nu}^{(3)}$ applying the
relations (10). Further, it is straightforward to verify that

$\mathcal{S}_{q}(z)-\mathcal{S}_{q}(qz)=z\mathcal{C}_{q}(q^{1/2}z)\quad\mbox{
and
}\quad\mathcal{C}_{q}(z)-\mathcal{C}_{q}(qz)=-zq^{1/2}\mathcal{S}_{q}(q^{1/2}z).$
(11)

Consequently, functions $\mathcal{S}_{q}$ and $\mathcal{C}_{q}$ are linearly
independent solutions to the second-order $q$-difference equation

$u(q^{2}z)+\left(q^{2}z^{2}-(1+q)\right)u(qz)+qu(z)=0.$

It is also remarkable that there is a symmetry while interchanging $q$ and
$q^{-1}$, namely

$\mathcal{E}_{q^{-1}}(q^{-1/4}z)=\mathcal{E}_{q}(-q^{1/4}z)$

and

$q^{-1/4}\mathcal{S}_{q^{-1}}(q^{-1/4}z)=-q^{1/4}\mathcal{S}_{q}(q^{1/4}z)\quad\mbox{
and }\quad\mathcal{C}_{q^{-1}}(q^{-1/4}z)=\mathcal{C}_{q}(q^{1/4}z).$

At last, let us define the corresponding $q$-analogue to the hyperbolic sine
and cosine by formulas

$\mathcal{S}h_{q}(z)=-\mathrm{i}\mathcal{S}_{q}(\mathrm{i}z)\quad\mbox{ and
}\quad\mathcal{C}h_{q}(z)=\mathcal{C}_{q}(\mathrm{i}z),$ (12)

for all $z\in\mathbb{C}$.

Finally, we recall an addition formula for $\mathcal{E}_{q}$. It has been
originally derived by Rahman [30] and reproved by Suslov, see [36, Thm. 3.6].
Since several important results stated below are a consequence of this formula
we provide an independent simple proof.

######  Proposition 1.

For $u,v\in\mathbb{C}$, it holds

$\displaystyle\mathcal{E}_{q}(u)\mathcal{E}_{q}(-v)={}_{3}\phi_{3}\left[\begin{matrix}0,&u^{-1}vq^{1/2},&uv^{-1}q^{1/2}\\\
q^{1/2},&-q^{1/2},&-q\end{matrix}\ ;q,uvq^{1/2}\right]$ $\displaystyle\hskip
66.0pt+q^{1/4}\frac{u-v}{1-q}\,_{3}\phi_{3}\left[\begin{matrix}0,&u^{-1}vq,&uv^{-1}q\\\
q^{3/2},&-q^{3/2},&-q\end{matrix}\ ;q,uvq\right]\\!.$ (13)

###### Proof.

By multiplication, we obtain

$\mathcal{E}_{q}(u)\mathcal{E}_{q}(-v)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n^{2}/4}}{(q;q)_{n}}v^{n}\sum_{k=0}^{n}{n\brack
k}_{q}q^{k(k-1)/2}\left(-q^{\frac{1-n}{2}}\frac{u}{v}\right)^{k}.$

The inner sum equals $(q^{(1-n)/2}uv^{-1};q)_{n},$ as it follows from the
formula [18, Exer. 1.2]

$\sum_{k=0}^{n}{n\brack k}_{q}q^{k(k-1)/2}(-z)^{k}=(z;q)_{n}\,.$

Now, it suffices to write separately the series with odd and even summation
index $n$ and take into account that

$(q^{(1-n)/2}uv^{-1};q)_{n}=\begin{cases}(-1)^{k}u^{k}v^{-k}q^{-k^{2}/2}(u^{-1}vq^{1/2};q)_{k}(uv^{-1}q^{1/2};q)_{k},&\mbox{if
}n=2k,\\\
(-1)^{k}u^{k}v^{-k}q^{-k(k+1)/2}(u^{-1}vq;q)_{k}(uv^{-1};q)_{k+1},&\mbox{if
}n=2k+1.\end{cases}$

In this way, one arrives at the expression

$\displaystyle\mathcal{E}_{q}(u)\mathcal{E}_{q}(-v)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n^{2}/2}}{(q;q)_{2n}}(u^{-1}vq^{1/2};q)_{n}(uv^{-1}q^{1/2};q)_{n}\
u^{n}v^{n}$ $\displaystyle\hskip
66.0pt+q^{1/4}(u-v)\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n(n+1)/2}}{(q;q)_{2n+1}}(u^{-1}vq;q)_{n}(uv^{-1}q;q)_{n}\
u^{n}v^{n}.$

∎

As an immediate consequence of (13) and (8), one has the following statement,
cf. also [36, Thm. 3.8].

######  Corollary 2.

For $u,v\in\mathbb{C}$, one has

$\mathcal{C}_{q}(u)\mathcal{C}_{q}(v)+q^{1/2}\mathcal{S}_{q}(u)\mathcal{S}_{q}(v)={}_{3}\phi_{3}\left[\begin{matrix}0,&u^{-1}vq^{1/2},&uv^{-1}q^{1/2}\\\
q^{1/2},&-q^{1/2},&-q\end{matrix}\ ;q,-uvq^{1/2}\right]$ (14)

and

$\mathcal{S}_{q}(u)\mathcal{C}_{q}(v)-\mathcal{C}_{q}(u)\mathcal{S}_{q}(v)=\frac{u-v}{1-q}\,_{3}\phi_{3}\left[\begin{matrix}0,&u^{-1}vq,&uv^{-1}q\\\
q^{3/2},&-q^{3/2},&-q\end{matrix}\ ;q,-uvq\right]\\!.$ (15)

Note that by setting $u=q^{1/2}v$ in (14) one gets

$\mathcal{C}_{q}(q^{1/2}v)\mathcal{C}_{q}(v)+q^{1/2}\mathcal{S}_{q}(q^{1/2}v)\mathcal{S}_{q}(v)=1.$
(16)

In addition, from (8) and (14), it readily follows

$|\mathcal{E}_{q}(ix)|^{2}=\mathcal{C}_{q}^{2}(x)+q^{1/2}\mathcal{S}_{q}^{2}(x)={}_{2}\phi_{2}\left[\begin{matrix}0,&q^{1/2}\\\
-q^{1/2},&-q\end{matrix}\ ;q,-x^{2}q^{1/2}\right],\quad\mbox{for
}x\in\mathbb{R}.$ (17)

##  3 Relations between $\mathcal{E}_{q}$ and $q^{-1}$-Fibonacci polynomials

Recall that rather than assuming $q>1$, it is more convenient to write
$q^{-1}$ instead of $q$ in (3) and still suppose $0<q<1$. Then one arrives at
the $q^{-1}$-Fibonacci polynomials with the explicit expression:

$\varphi_{n}(x;q^{-1})=\sum_{2k<n}{n-k-1\brack
k}_{q}q^{k(k+1-n)}x^{n-2k-1},\quad\mbox{ for }n\in\mathbb{Z}_{+}.$ (18)

Note the sequence

$\left\\{q^{(n-1)^{2}/2}\varphi_{n}\left(q^{-(n-1)/2}x;q^{-1}\right)\right\\}_{n=0}^{\infty}$

satisfies the same recurrence (2) and initial conditions as
$\\{\varphi_{n}(x;q)\\}_{n=0}^{\infty}$ Hence, we have the following relation
between $q$-Fibonacci and $q^{-1}$-Fibonacci polynomials,

$\varphi_{n}(x;q)=q^{(n-1)^{2}/2}\varphi_{n}\left(q^{-(n-1)/2}x;q^{-1}\right),\quad\forall
n\in\mathbb{Z}_{+},$

cf. [13, Eq. (2.6)].

First, we derive an expression for the $q^{-1}$-Fibonacci polynomials
$\varphi_{n}(x;q^{-1})$ in terms of the $q$-exponential function
$\mathcal{E}_{q}$. A limit relation for $\varphi_{n}(x;q^{-1})$, as
$n\rightarrow\infty$, then readily follows.

######  Proposition 3.

For all $n\in\mathbb{Z}_{+}$, one has

$\varphi_{n}(x;q^{-1})=\frac{1}{2}q^{-(n-1)^{2}/4}\left(\mathcal{E}_{q}(x)\mathcal{E}_{q}(-q^{n/2}x)-(-1)^{n}\mathcal{E}_{q}(-x)\mathcal{E}_{q}(q^{n/2}x)\right).$

###### Proof.

For $n\in\mathbb{Z}_{+}$ and $x\in\mathbb{C}$, put

$\chi_{n}^{\pm}=(\mp 1)^{n}q^{-n(n-2)/4}\mathcal{E}_{q}(\pm q^{n/2}x).$

By using (6), one readily verifies that the sequences $\\{\chi_{n}^{+}\\}$ and
$\\{\chi_{n}^{-}\\}$ fulfill the recurrence (2) with $q$ replaced by $q^{-1}$.
Next, with the aid of the identity (13), one obtains

$\chi_{0}^{+}\chi_{1}^{-}-\chi_{0}^{-}\chi_{1}^{+}=q^{1/4}\left(\mathcal{E}_{q}(x)\mathcal{E}_{q}(-q^{1/2}x)+\mathcal{E}_{q}(-x)\mathcal{E}_{q}(q^{1/2}x)\right)=2q^{1/4}.$

Finally, the sequence

$\left\\{\frac{\chi_{0}^{+}\chi_{n}^{-}-\chi_{0}^{-}\chi_{n}^{+}}{\chi_{0}^{+}\chi_{1}^{-}-\chi_{0}^{-}\chi_{1}^{+}}\right\\}_{n=0}^{\infty}$

coincides with $\\{\varphi_{n}(x;q^{-1})\\}_{n=0}^{\infty}$ since it satisfies
the same recurrence with the same initial conditions. The statement now
follows. ∎

The following corollary readily follows from (8), (12), and Proposition 3.

######  Corollary 4.

For $n\in\mathbb{Z}_{+}$, one has

$\varphi_{2n+1}(x;q^{-1})=q^{-n^{2}}\left(\mathcal{C}h_{q}(x)\mathcal{C}h_{q}(q^{n+1/2}x)-q^{1/2}\mathcal{S}h_{q}(x)\mathcal{S}h_{q}(q^{n+1/2}x)\right)$
(19)

and

$\varphi_{2n}(x;q^{-1})=q^{-n(n-1)}\left(\mathcal{S}h_{q}(x)\mathcal{C}h_{q}(q^{n}x)-\mathcal{C}h_{q}(x)\mathcal{S}h_{q}(q^{n}x)\right).$
(20)

In addition, the identities (19), (20) together with the formulas (14), (15),
and (12) allow to express $q^{-1}$-Fibonacci polynomials as terminating
${}_{3}\phi_{3}$ series. These identities can be also readily verified by
using (1).

######  Corollary 5.

For $n\in\mathbb{Z}_{+}$, one has

$\varphi_{2n+1}(x;q^{-1})=q^{-n^{2}}\,_{3}\phi_{3}\left[\begin{matrix}0,&q^{-n},&q^{n+1}\\\
q^{1/2},&-q^{1/2},&-q\end{matrix}\ ;q,q^{n+1}x^{2}\right]$

and

$\varphi_{2n}(x;q^{-1})=q^{-n(n-1)}\frac{1-q^{n}}{1-q}x\,_{3}\phi_{3}\left[\begin{matrix}0,&q^{-n+1},&q^{n+1}\\\
q^{3/2},&-q^{3/2},&-q\end{matrix}\ ;q,q^{n+1}x^{2}\right]\\!.$

######  Proposition 6.

For all $x\in\mathbb{C}$, the limit relations

$\lim_{n\rightarrow\infty}q^{n(n-1)}\varphi_{2n}(x;q^{-1})=\mathcal{S}h_{q}(x)\quad\mbox{
and
}\quad\lim_{n\rightarrow\infty}q^{n^{2}}\varphi_{2n+1}(x;q^{-1})=\mathcal{C}h_{q}(x)$

hold.

###### Proof.

Send $n\to\infty$ in identities (19) and (20). ∎

The last formulas relating $q^{-1}$-Fibonacci polynomials with the functions
$\mathcal{E}_{q}$, $\mathcal{S}h_{q}$, and $\mathcal{C}h_{q}$ follows from an
identity which may be viewed as a $q$-analogue of the famous Euler-Cassini’s
identity for Fibonacci numbers. We present a general way of proving such
identities by using a function called $\mathfrak{F}$ that was originally
introduced in [33]. This function takes as its argument a complex sequence
satisfying certain convergence condition. Here, it suffices to recall the
definition of $\mathfrak{F}$ applied to any $n$-tuple of complex variables
$(x_{1},x_{2},\dots,x_{n})$. This definition can be formulated as follows:

$\mathfrak{F}(x_{1},x_{2},\dots,x_{n})=\det X^{(n)},\mbox{ for
}n\in\mathbb{N},\mbox{ and }\mathfrak{F}(\emptyset)=1,$

where $X^{(n)}$ is the $n\times n$ matrix with entries

$X_{ij}^{(n)}=\begin{cases}1,&\mbox{ if }i=j,\\\ x_{i},&\mbox{ if }|i-j|=1,\\\
0,&\mbox{ otherwise}.\end{cases}$

The function $\mathfrak{F}$ satisfies the relation

$\mathfrak{F}(x_{1},x_{2},\dots,x_{k+1})=\mathfrak{F}(x_{1},x_{2},\dots,x_{k})-x_{k}x_{k+1}\,\mathfrak{F}(x_{1},x_{2},\dots,x_{k-1}),\quad
k\in\mathbb{N}$ (21)

and the identity

$\displaystyle\mathfrak{F}(x_{1},x_{2},\dots,x_{n})\mathfrak{F}(x_{2},x_{3},\dots,x_{m})-\mathfrak{F}(x_{1},x_{2},\dots,x_{m})\mathfrak{F}(x_{2},x_{3},\dots,x_{n})$
$\displaystyle\hskip
160.0pt=\left(\prod_{j=1}^{n}x_{j}x_{j+1}\right)\mathfrak{F}(x_{n+2},x_{n+3},\dots,x_{m})$
(22)

which holds true for $m,n\in\mathbb{Z}_{+}$, $n<m$, see [34, Lemma 1].

By using (21), one easily verifies that the sequence
$\\{p_{n}\\}_{n=0}^{\infty}$ determined by the second-order recurrence
$p_{n+1}=\alpha_{n}p_{n}+\beta_{n}p_{n-1}$, for $n\in\mathbb{Z}_{+}$, and
initial conditions $p_{-1}=0$ and $p_{0}=1$, can be expressed with the aid of
the function $\mathfrak{F}$, see, for example, the general formula [35, Eq.
(19)]. For $q^{-1}$-Fibonacci polynomials the formula reads

$\varphi_{n+1}(x;q^{-1})=x^{n}\,\mathfrak{F}\left(\left\\{{\rm
i}x^{-1}q^{-(2k-1)/4}\right\\}_{k=1}^{n}\right),\quad n\in\mathbb{Z}_{+}.$
(23)

Further we derive the well known $q$-Euler-Cassini formula for
$\varphi_{n}(x;q^{-1})$ first proved by MacMahon [27, Chp. III, Sec. VII] and
reproved several times. See, for example, [9, 17] for combinatorial proofs.
Another proof for the more general polynomials introduced by Al-Salam and
Ismail in [2] was given in [23] (the presented method can be applied to this
more general case as well). For related investigations consult also [5] and
references therein.

######  Proposition 7.

For $m,n\in\mathbb{Z}_{+}$, $n\leq m$, it holds

$\displaystyle
q^{\frac{n}{2}}\varphi_{m}(x;q^{-1})\varphi_{n+1}(q^{-\frac{1}{2}}x;q^{-1})-q^{\frac{m}{2}}\varphi_{n}(x;q^{-1})\varphi_{m+1}(q^{-\frac{1}{2}}x;q^{-1})$
$\displaystyle\hskip
200.0pt=(-1)^{n}q^{n-\frac{mn}{2}}\varphi_{m-n}(q^{\frac{n}{2}}x;q^{-1}).$
(24)

###### Proof.

By substituting

$x_{k}={\rm i}x^{-1}q^{-(2k-1)/4}$

in (22) and using identity (23) one arrives at the relation (24) for $n<m$.
Clearly, in the case $n=m$, both sides of (24) vanish. ∎

######  Proposition 8.

For all $n\in\mathbb{Z}_{+}$ and $x\in\mathbb{C}$, the following relations
hold true:

$\displaystyle i)$
$\displaystyle\varphi_{2n+1}(q^{-1/2}x;q^{-1})\mathcal{S}h_{q}(x)-q^{-n}\varphi_{2n}(x;q^{-1})\mathcal{C}h_{q}(q^{-1/2}x)=q^{-n^{2}}\mathcal{S}h_{q}(q^{n}x),$
$\displaystyle ii)$
$\displaystyle\varphi_{2n+1}(q^{-1/2}x;q^{-1})\mathcal{C}h_{q}(x)-q^{-n+1/2}\varphi_{2n}(x;q^{-1})\mathcal{S}h_{q}(q^{-1/2}x)=q^{-n^{2}}\mathcal{C}h_{q}(q^{n}x),$
$\displaystyle iii)$
$\displaystyle\varphi_{2n+1}(x;q^{-1})\mathcal{S}h_{q}(q^{-1/2}x)-q^{n}\varphi_{2n+2}(q^{-1/2}x;q^{-1})\mathcal{C}h_{q}(x)=q^{-n^{2}}\mathcal{S}h_{q}(q^{n+1/2}x),$
$\displaystyle iv)$
$\displaystyle\varphi_{2n+1}(x;q^{-1})\mathcal{C}h_{q}(q^{-1/2}x)-q^{n+1/2}\varphi_{2n+2}(q^{-1/2}x;q^{-1})\mathcal{S}h_{q}(x)=q^{-n^{2}}\mathcal{C}h_{q}(q^{n+1/2}x),$
$\displaystyle v)$
$\displaystyle\varphi_{n+1}(q^{-1/2}x;q^{-1})\mathcal{E}_{q}(x)-q^{-(2n-1)/4}\varphi_{n}(x;q^{-1})\mathcal{E}_{q}(q^{-1/2}x)=(-1)^{n}q^{-n^{2}/4}\mathcal{E}_{q}(q^{n/2}x).$

###### Proof.

This statement follows from Propositions 6 and 7. First, we indicate the
derivation of identity $i)$. Relations $ii)$, $iii)$, and $iv)$ can be derived
in a similar way. In the second part of the proof, we verify equality $v)$.

By writing $2m$ instead of $m$ and $2n$ instead of $n$ in (24), and
multiplying both sides by $q^{m(m-1)-n}$, one finds

$\displaystyle\varphi_{2n+1}(q^{-1/2}x;q^{-1})\left[q^{m(m-1)}\varphi_{2m}(x;q^{-1})\right]-q^{-n}\varphi_{2n}(x;q)\left[q^{m^{2}}\varphi_{2m+1}(q^{-1/2}x;q^{-1})\right]$
$\displaystyle\hskip
200.0pt=q^{-n^{2}}\left[q^{(m-n)(m-n-1)}\varphi_{2(m-n)}(q^{n}x;q^{-1})\right]\\!.$

To obtain identity $i)$, it is sufficient to send $m\rightarrow\infty$ and
apply Proposition 6 to the expressions in square brackets.

Next, recall that by (8) and (12), one has

$\mathcal{E}_{q}(z)=\mathcal{C}h_{q}(z)+q^{1/4}\mathcal{S}h_{q}(z),$ (25)

for all $z\in\mathbb{C}$. By multiplying the equation $i)$ by $q^{1/4}$ and
adding to the equation $ii)$, one obtains

$\varphi_{2n+1}(q^{-1/2}x;q^{-1})\mathcal{E}_{q}(x)-q^{-n+1/4}\varphi_{2n}(x;q^{-1})\mathcal{E}_{q}(q^{-1/2}x)=q^{-n^{2}}\mathcal{E}_{q}(q^{n}x)$

where the formula (25) has been used. This verifies $v)$ for an even index.
Similarly, by multiplying the equation $iii)$ by $q^{1/4}$ and adding to the
equation $iv)$, one obtains

$\varphi_{2n+1}(x;q^{-1})\mathcal{E}_{q}(q^{-1/2}x)-q^{n+1/4}\varphi_{2n+2}(q^{-1/2}x;q^{-1})\mathcal{E}_{q}(x)=q^{-n^{2}}\mathcal{E}_{q}(q^{n+1/2}x).$

which is the identity $v)$ with an odd index. ∎

##  4 Related orthogonal polynomials, reproducing kernel, and N-extremal
measures of orthogonality

In this section, we draw our attention to a family of orthogonal polynomials
associated to $q^{-1}$-Fibonacci polynomials. The main goal is the description
of all their measures of orthogonality which are N-extremal solutions of the
corresponding Hamburger moment problem. Main references concerning the
Hamburger moment problem are classical treatises [1] and [32], for a brief
summary, see also [19, Sec. 21.1].

Recall that polynomials $P_{n}$ and $Q_{n}$, $n\in\mathbb{Z}_{+}$, are
orthogonal polynomials of the first and second kind, respectively, if they are
solutions of the recurrence relation

$\beta_{n-1}u_{n-1}+(\alpha_{n}-z)u_{n}+\beta_{n}u_{n+1}=0,\quad
n\in\mathbb{N},$

satisfying initial conditions $P_{0}(z)=1$,
$P_{1}(z)=(z-\alpha_{0})/\beta_{0}$, $Q_{0}(z)=0$ and $Q_{1}(z)=1/\beta_{0}$,
see [1, Chp. 1]. It is to be assumed that $\alpha_{n}\in\mathbb{R}$ and
$\beta_{n}\in\mathbb{R}\setminus\\{0\\}$, for all $n\in\mathbb{Z}_{+}$.

For $n=-1,0,1,2,\dots$, define

$P_{n}(x;q)=(-{\rm i})^{n}q^{n(n+1)/4}\varphi_{n+1}({\rm i}q^{-1/2}x;q^{-1}).$
(26)

The polynomials $P_{n}(x;q)$ form the family of orthogonal polynomials of the
first kind with $\alpha_{n}=0$ and $\beta_{n}=q^{-n/2}$, $n\in\mathbb{Z}_{+}$.
In addition, they are the symmetric counterpart to the polynomials
$T_{n}(x;q^{-1})$ since their monic version coincides with $T_{n}(x;q^{-1})$,
namely

$q^{-n(n-1)/4}P_{n}(x;q)=T_{n}(x;q^{-1}).$

For the corresponding orthogonal polynomials of the second kind $Q_{n}(x;q)$,
one has

$Q_{n}(x;q)=P_{n-1}(q^{1/2}x;q)=(-{\rm i})^{n-1}q^{n(n-1)/4}\varphi_{n}({\rm
i}x;q^{-1}),\quad n\in\mathbb{Z}_{+}.$ (27)

Taking into account the equation (18), one easily verifies that

$P_{2n-1}(0;q)=Q_{2n}(0;q)=0\;\mbox{ and
}\;P_{2n}(0;q)=Q_{2n+1}(0;q)=(-1)^{n}q^{n/2}$ (28)

where $n\in\mathbb{Z}_{+}$. Recall that the Hamburger moment problem
associated with a family of orthogonal polynomials of the first kind $P_{n}$
(and of the second kind $Q_{n}$) is in the indeterminate case if and only if

$\sum_{n=0}^{\infty}\left(P_{n}^{2}(0)+Q_{n}^{2}(0)\right)<\infty,$

see, for example, [1, Chp. 2]. For polynomials $P_{n}(0;q)$ and $Q_{n}(0;q)$,
this condition is clearly true for all $q$ such that $0<q<1$, as it readily
follows from (28).

Further recall that, by the Nevanlinna theorem, all measures of orthogonality
$\mu_{\varphi}$ of polynomials $P_{n}$ are in one-to-one correspondence with
functions $\varphi$ belonging to the one-point compactification of the space
of Pick functions. Recall that Pick functions are defined and holomorphic on
the open complex halfplane $\mathop{\mathrm{missing}}{Im}\nolimits z>0$, with
values in the closed halfplane $\mathop{\mathrm{missing}}{Im}\nolimits z\geq
0$. The correspondence is established by identifying the Stieltjes transform
of the measure $\mu_{\varphi}$,

$\int_{\mathbb{R}}\frac{\mbox{d}\mu_{\varphi}(x)}{z-x}=\frac{A(z)\varphi(z)-C(z)}{B(z)\varphi(z)-D(z)}\,,\
\ z\in\mathbb{C}\setminus\mathbb{R}.$

Entire functions $A$, $B$, $C$, $D$ are local uniform limits of polynomials
$A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$, respectively, where

$\displaystyle
A_{n+1}(z)=\beta_{n}\left(Q_{n+1}(z)Q_{n}(0)-Q_{n+1}(0)Q_{n}(z)\right),$ (29)
$\displaystyle
B_{n+1}(z)=\beta_{n}\left(P_{n+1}(z)Q_{n}(0)-Q_{n+1}(0)P_{n}(z)\right),$ (30)
$\displaystyle
C_{n+1}(z)=\beta_{n}\left(Q_{n+1}(z)P_{n}(0)-P_{n+1}(0)Q_{n}(z)\right),$ (31)
$\displaystyle
D_{n+1}(z)=\beta_{n}\left(P_{n+1}(z)P_{n}(0)-P_{n+1}(0)P_{n}(z)\right),$ (32)

see [19, Sec. 21.1].

The particular class of measures of orthogonality is composed by measures
$\mu_{t}$ associated with the Pick function
$\varphi(z)=t\in\mathbb{R}\cup\\{\infty\\}$. Measures $\mu_{t}$ are called
N-extremal and are purely discrete. The support of $\mu_{t}$ is an unbounded
set of isolated points which is known to be equal to the zero set

$\mathfrak{Z}_{t}=\\{x\in\mathbb{R}\mid B(x)t-D(x)=0\\}.$ (33)

Hence

$\mu_{t}=\sum_{x\in\mathfrak{Z}_{t}}\rho(x)\delta_{x}$ (34)

where

$\rho(x)=\mathop{\mathrm{missing}}{Res}\nolimits_{z=x}\frac{A(z)t-C(z)}{B(z)t-D(z)}=\frac{A(x)t-C(x)}{B^{\prime}(x)t-D^{\prime}(x)}=\frac{1}{B^{\prime}(x)D(x)-B(x)D^{\prime}(x)}$
(35)

since, for $x\in\mathfrak{Z}_{t}$, $t=D(x)/B(x)$ and the general identity

$A(z)D(z)-B(z)C(z)=1,$ (36)

holds true for all $z\in\mathbb{C}$. The orthogonality relation for
polynomials of the first kind then reads

$\sum_{x\in\mathfrak{Z}_{t}}\rho(x)P_{n}(x)P_{m}(x)=\delta_{mn},\quad\forall
m,n\in\mathbb{Z}_{+}.$ (37)

Thus, the determination of the Nevanlinna functions, $B$ and $D$ in
particular, is essential for the description of N-extremal measures of
orthogonality. Formulas for $A$, $B$, $C$ and $D$ can be, in turn, established
if the asymptotic behavior of $P_{n}$ and $Q_{n}$ is known for
$n\rightarrow\infty$, as one observes from formulas (29)-(32). In the case of
$P_{n}(x;q)$ and $Q_{n}(x;q)$, the asymptotic formulas follow from Proposition
6 and the relations (26) and (27). We arrive at the following limits:

$\lim_{n\rightarrow\infty}(-1)^{n}q^{-n/2}P_{2n}(x;q)=\mathcal{C}_{q}(q^{-1/2}x)$
(38)

and

$\lim_{n\rightarrow\infty}(-1)^{n}q^{-n/2}P_{2n+1}(x;q)=q^{1/2}\mathcal{S}_{q}(q^{-1/2}x).$
(39)

An alternative way of derivation of the last two limit relations was presented
by Chen and Ismail [14]. It is based on Darboux’s method applied to a suitable
generating function formula for $P_{n}(x;q)$. According to this method, the
leading asymptotic term of $P_{n}(x;q)$ is determined by the singularity of
the generating function which is located most closely to the origin, cf. [29,
Section 8.9]. The explicit form of the generating function for $P_{n}(x;q)$ is
as follows [14, Thm. 3.1]

$\sum_{n=0}^{\infty}P_{n}(x;q)t^{n}=\sum_{k=0}^{\infty}\frac{q^{k(k-1)/4}x^{k}t^{k}}{(-q^{1/2}t^{2};q)_{k+1}},\quad|t|<q^{-1/4}.$

Let us note there is another (entire) generating function formula which is
formulated in the following proposition in terms of $q^{-1}$-Fibonacci
polynomials.

######  Proposition 9.

For all $x,s\in\mathbb{C}$, it holds

$\sum_{n=0}^{\infty}q^{n(n-1)/2}\varphi_{n+1}(x;q^{-1})(-s)^{n}={}_{2}\phi_{2}\left[\begin{matrix}q,&x^{-1}s\\\
0,&0\end{matrix}\ ;q,sx\right]\\!.$ (40)

###### Proof.

Suppose $x\in\mathbb{C}$ is fixed and denote by $V(s)$ the function on the
right side of (40). $V(s)$ is a well defined entire function which is readily
seen to satisfy the $q$-difference equation

$V(s)=1-s(x-s)V(qs).$ (41)

Writing the power series expansion of $V(s)$ at $s=0$ in the form

$V(s)=\sum_{n=0}^{\infty}v_{n}q^{n(n-1)/2}(-s)^{n}$

and inserting the series into (41) one finds that the coefficients $v_{n}$
obey the recurrence $v_{n+1}=xv_{n}+q^{-n}v_{n-1}$, for $n\in\mathbb{N}$, with
the initial conditions $v_{0}=1$ and $v_{1}=x$. Necessarily,
$v_{n}=\varphi_{n+1}(x;q^{-1})$ for all $n\in\mathbb{Z}_{+}$. ∎

By applying the limit $n\to\infty$ in the equations (29)-(32), using (38),
(39), and (27), one obtains formulas for the Nevanlinna functions

$A(z;q)=q^{-1/2}D(q^{1/2}z;q)=\mathcal{S}_{q}(z)\quad\mbox{and}\quad
C(z;q)=-B(q^{1/2}z;q)=\mathcal{C}_{q}(z).$ (42)

######  Remark 10.

Note that in the particular case of the functions from (42) the general
identity (36) becomes (16).

From the general theory, it is well known that for any $t\in\mathbb{R}$, the
functions $A(z)t+C(z)$ and $B(z)t+D(z)$ have infinite number of zeros, all of
them being real and simple. Moreover, zeros of $A(z)t+C(z)$ and $B(z)t+D(z)$
interlace. The same holds true for functions $A(z)$ and $B(z)$ which
correspond to the case $t=\infty$, see [1, Chp. 2]. From this and (42) one
immediately obtains the following statement.

######  Proposition 11.

All zeros of the functions $\mathcal{S}_{q}$ and $\mathcal{C}_{q}$ are real
and simple. In addition, zeros of the functions $\mathcal{S}_{q}(z)$ and
$\mathcal{C}_{q}(q^{-1/2}z)$ interlace, as well as zeros of the functions
$\mathcal{S}_{q}(q^{-1/2}z)$ and $\mathcal{C}_{q}(z)$ (as functions in $z$).

As shown by Buchwalter and Cassier in [10], see also [7, Sec. 1], the
reproducing kernel for orthogonal polynomials of the first kind can be
expressed in terms of functions $B(z)$ and $D(z)$, namely

$K(u,v)=\sum_{n=0}^{\infty}P_{n}(u)P_{n}(v)=\frac{B(u)D(v)-D(u)B(v)}{u-v}\,.$
(43)

In the case of polynomials $P_{n}(x;q)$, it is possible to express the
reproducing kernel in a closed form.

######  Theorem 12.

The formula for the reproducing kernel for orthogonal polynomials $P_{n}(x;q)$
reads

$K(u,v)=\frac{1}{1-q}\,_{3}\phi_{3}\left[\begin{matrix}0,&u^{-1}vq,&uv^{-1}q\\\
q^{3/2},&-q^{3/2},&-q\end{matrix}\ ;q,-uv\right]\\!.$ (44)

###### Proof.

The statement follows from the formulas (43), (42), and (15). ∎

######  Corollary 13.

It holds

$B^{\prime}(u;q)D(u;q)-B(u;q)D^{\prime}(u;q)=\frac{1}{1-q}\,_{3}\phi_{3}\left[\begin{matrix}0,&q,&q\\\
q^{3/2},&-q^{3/2},&-q\end{matrix}\ ;q,-u^{2}\right]\\!,$

for all $u\in\mathbb{C}$.

###### Proof.

It suffices to send $v\rightarrow u$ in the formula (44). ∎

Further, we focus on the description of all N-extremal measures of
orthogonality for polynomials $P_{n}(x;q)$. We use a different parametrization
of N-extremal measures $\mu_{t}$ substituting for $t$ the quotient of
$q$-cosine and $q$-sine functions of a new parameter $u$. This
reparametrization together with the formula (13) allow to express all
N-extremal measures in a compact form. A similar idea was used by Ismail and
Masson in the case of $q$-Hermite polynomials [22].

######  Theorem 14.

If $0<q<1$ and $u\in\mathbb{R}$, then the orthogonality relation for
$P_{n}(x;q)$ reads

$\sum_{k=1}^{\infty}\left(\,{}_{3}\phi_{3}\left[\begin{matrix}0,&q,&q\\\
q^{3/2},&-q^{3/2},&-q\end{matrix}\
;q,-\lambda_{k}^{2}(u)\right]\right)^{\\!-1}\\!P_{n}(\lambda_{k}(u);q)P_{m}(\lambda_{k}(u);q)=\frac{1}{1-q}\delta_{mn}$

where $\lambda_{1}(u),\lambda_{2}(u),\lambda_{3}(u),\dots$ stand for zeros of
the function (46). Moreover, the map

$u\mapsto\mu_{u}(q)=(1-q)\sum_{k=1}^{\infty}\left(\,{}_{3}\phi_{3}\left[\begin{matrix}0,&q,&q\\\
q^{3/2},&-q^{3/2},&-q\end{matrix}\
;q,-\lambda_{k}^{2}(u)\right]\right)^{\\!-1}\delta_{\lambda_{k}(u)}$

represents one-to-one mapping from $[0,s_{1}(q))$ onto the set of all
N-extremal measures associated with polynomials $P_{n}(x;q)$ where $s_{1}(q)$
stands for the lowest positive zero of $\mathcal{S}_{q}$.

###### Proof.

Function

$u\mapsto\frac{\mathcal{C}_{q}(u)}{\mathcal{S}_{q}(u)}$

is bijection of the interval $[0,s_{1}(q))$ onto $\mathbb{R}\cup\\{\infty\\}$.
Hence, by identifying

$t=\frac{\mathcal{C}_{q}(u)}{\mathcal{S}_{q}(u)}$ (45)

in (34), the one-to-one mapping $u\mapsto\mu_{u}(q)$ from the interval
$[0,s_{1}(q))$ onto the set of all N-extremal measures of orthogonality for
$P_{n}(x;q)$ is established. The support of $\mu_{u}(q)$ coincides with the
set of zeros of the function $z\mapsto B(z;q)t-D(z;q)$ where $t$ is related to
$u$ by (45). By using (42), one finds $\mu_{u}(q)$ is supported by set of
zeros of the function

$z\mapsto\mathcal{C}_{q}(q^{-1/2}z)\mathcal{C}_{q}(u)+q^{1/2}\mathcal{S}_{q}(q^{-1/2}z)\mathcal{S}_{q}(u)$

which, according to (14), equals

$z\mapsto{}_{3}\phi_{3}\left[\begin{matrix}0,&u^{-1}z,&uz^{-1}q\\\
q^{1/2},&-q^{1/2},&-q\end{matrix}\ ;q,-uz\right]\\!.$ (46)

In addition, the formula given in Corollary 13 coincides with the reciprocal
of the function $\rho$ from (35). Thus, we get the description of all the
N-extremal measures $\mu_{u}(q)$ in terms of ${}_{3}\phi_{3}$ in the form
stated as the second part of the statement. The orthogonality relation then
follows from (37). ∎

######  Remark 15.

The general theory on the Nevanlinna parametrization provides us with the
following generalization of Proposition 11: _For $u\in\mathbb{R}$, all the
zeros $\lambda_{1}(u),\lambda_{2}(u),\dots$ of function (46) are real and
simple. In addition, if we denote the function from (46) by
$f_{u}(\,\cdot\,;q)$, then, for any $u_{1},u_{2}\in[0,s_{1}(q))$ such that
$u_{1}\neq u_{2}$, zeros of functions $f_{u_{1}}(\,\cdot\,;q)$ and
$f_{u_{2}}(\,\cdot\,;q)$ interlace._

Let us draw our attention to two special cases. Let the sequences

$0<s_{1}(q)<s_{2}(q)<s_{3}(q)<\dots\;\mbox{ and }\quad
0<c_{1}(q)<c_{2}(q)<c_{3}(q)<\dots$

denote all positive zeros of $\mathcal{S}_{q}$ and $\mathcal{C}_{q}$,
respectively. Then there is an orthogonality relation for $P_{n}(x;q)$ in
terms of the function $\mathcal{S}_{q}$ and its zeros only.

######  Proposition 16.

The orthogonality relation

$(1-q)P_{n}(0;q)P_{m}(0;q)-\\!\\!\sum_{k\in\mathbb{Z}\setminus\\{0\\}}\frac{\mathcal{S}_{q}(qs_{k}(q))}{s_{k}(q)\mathcal{S}_{q}^{\prime}(s_{k}(q))}P_{n}\left(q^{1/2}s_{k}(q);q\right)P_{m}\left(q^{1/2}s_{k}(q);q\right)=\delta_{mn}$

holds for all $m,n\in\mathbb{Z}_{+}$, where $s_{-k}(q)=-s_{k}(q)$.

###### Proof.

Consider the case $t=0$ in the original parametrization of N-extremal measures
(33), (34), and (35). The corresponding measure is supported by the set of
zeros of $D(z;q)=q^{1/2}\mathcal{S}_{q}(q^{-1/2}z)$, i.e., on the set

$\\{0\\}\cup\left\\{\pm q^{1/2}s_{k}(q)\mid k\in\mathbb{N}\right\\}.$

Further, for the function $\rho$, one has

$\rho\left(q^{1/2}s_{k}(q)\right)=\frac{C(q^{1/2}s_{k}(q);q)}{D^{\prime}(q^{1/2}s_{k}(q);q)}=\frac{\mathcal{C}_{q}(q^{1/2}s_{k}(q))}{\mathcal{S}_{q}^{\prime}(s_{k}(q))}=-\frac{\mathcal{S}_{q}(qs_{k}(q))}{s_{k}(q)\mathcal{S}_{q}^{\prime}(s_{k}(q))}$

where the last equality holds true due to the first relation of (11). Clearly,
the expression for $\rho\left(q^{1/2}s_{k}(q)\right)$ is valid for all nonzero
$k\in\mathbb{Z}$, with the convention $s_{-k}(q)=-s_{k}(q)$. Further, one has

$\rho(0)=\frac{\mathcal{C}_{q}(0)}{\mathcal{S}_{q}^{\prime}(0)}=1-q,$

as it follows from the very definition of $\mathcal{S}_{q}$ and
$\mathcal{C}_{q}$ (7). The rest then follows from the general relation (37). ∎

One more orthogonality relation for $P_{n}(x;q)$, this time in terms of the
function $\mathcal{C}_{q}$ and its zeros only, is as follows.

######  Proposition 17.

The orthogonality relation

$-\\!\\!\sum_{k\in\mathbb{Z}\setminus\\{0\\}}\frac{\mathcal{C}_{q}(qc_{k}(q))}{c_{k}(q)\mathcal{C}_{q}^{\prime}(c_{k}(q))}P_{n}\left(q^{1/2}c_{k}(q);q\right)P_{m}\left(q^{1/2}c_{k}(q);q\right)=\delta_{mn}$

holds for all $m,n\in\mathbb{Z}_{+}$, where $c_{-k}(q)=-c_{k}(q)$.

###### Proof.

The proof proceeds in the same way as the proof of Proposition 16.
Nevertheless, we provide the details for the sake of completeness.

Consider the case $t=\infty$ in the original parametrization of N-extremal
measures (33), (34), and (35). The corresponding measure is supported by the
set of zeros of $B(z;q)=-\mathcal{C}_{q}(q^{-1/2}z)$, i.e., on the set

$\left\\{\pm q^{1/2}c_{k}(q)\mid k\in\mathbb{N}\right\\}.$

For the function $\rho$, one has

$\rho\left(q^{1/2}c_{k}(q)\right)=\frac{A(q^{1/2}c_{k}(q);q)}{B^{\prime}(q^{1/2}c_{k}(q);q)}=\frac{-q^{1/2}\mathcal{S}_{q}(q^{1/2}c_{k}(q))}{\mathcal{C}_{q}^{\prime}(c_{k}(q))}=-\frac{\mathcal{C}_{q}(qc_{k}(q))}{c_{k}(q)\mathcal{C}_{q}^{\prime}(c_{k}(q))}$

where the last equality holds true due to the second relation of (11).
Clearly, the expression for $\rho\left(q^{1/2}c_{k}(q)\right)$ is valid for
all nonzero $k\in\mathbb{Z}$, with the convention $c_{-k}(q)=-c_{k}(q)$. The
orthogonality relation is then a consequence of (37). ∎

######  Remark 18.

Apart from $N$-extremal measures, there is another exceptional family of
measures of orthogonality $\mu_{\varphi}$ corresponding to constant Pick
functions $\varphi(z)=\beta+i\gamma$, with $\beta\in\mathbb{R}$ and
$\gamma>0$. Let us denote these measures by $\mu_{\beta,\gamma}$. It turns out
that $\mu_{\beta,\gamma}$ is an absolutely continuous measure supported on
$\mathbb{R}$ with the density

$\frac{\mbox{d}\mu_{\beta,\gamma}}{\mbox{d}x}=\frac{\gamma}{\pi}\big{(}\big{(}\beta
B(x)-D(x)\big{)}^{2}+\gamma^{2}B(x)^{2}\big{)}^{-1},$

see [8]. In particular, the measure $\mu_{0,q^{1/4}}$ has the density

$\frac{\mbox{d}\mu_{0,q^{1/4}}}{\mbox{d}x}=\frac{1}{q^{1/4}\pi}\left|\mathcal{E}_{q}(iq^{-1/2}x)\right|^{-2},$

which one verifies by using formulas (42) and (8). This was already observed
by Chen and Ismail in [14, Thm. 3.4]. Let us point out that, with the aid of
(17), the density of $\mu_{0,q^{1/4}}$ can be written in a more explicit form
in terms of ${}_{2}\phi_{2}$. Namely,

$\frac{\mbox{d}\mu_{0,q^{1/4}}}{\mbox{d}x}=\left(\pi
q^{1/4}{}_{2}\phi_{2}\left[\begin{matrix}0,&q^{1/2}\\\
-q^{1/2},&-q\end{matrix}\ ;q,-q^{-1/2}x^{2}\right]\right)^{-1}\\!.$

## Acknowledgements

The author wishes to acknowledge gratefully the partial support from grant No.
GA13-11058S of the Czech Science Foundation.

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# Dual-probe 1D hybrid fs/ps rotational CARS for simultaneous single-shot
temperature, pressure, and O2/N2 measurements

David Escofet-Martin  Corresponding author<EMAIL_ADDRESS>Anthony O.
Ojo  Institute for Multiscale Thermofluids, University of Edinburgh,
Edinburgh, UK.  Joshua Collins  Institute for Multiscale Thermofluids,
University of Edinburgh, Edinburgh, UK.  Nils Torge Mecker  Institute for
Multiscale Thermofluids, University of Edinburgh, Edinburgh, UK.  Mark Linne
Institute for Multiscale Thermofluids, University of Edinburgh, Edinburgh, UK.
Brian Peterson  Institute for Multiscale Thermofluids, University of
Edinburgh, Edinburgh, UK.

###### Abstract

We employ dual-probe one-dimensional (1D) femtosecond (fs)/picosecond (ps)
hybrid rotational coherent anti-Stokes Raman spectroscopy (HRCARS) to
investigate simultaneous temperature, pressure, and O2/N2 measurements for
gas-phase diagnostics. The dual-probe HRCARS technique allows for simultaneous
measurements from the time and frequency-domain. A novel approach for
measuring pressure, which offers high accuracy (<1%) and precision (0.42%), is
presented. The technique is first demonstrated in a chamber for a range of
pressures (1-1.5 bar). This technique shows an impressive capability of
resolving 1D pressure gradients arising from a N2 jet impinging on a surface,
both in laminar and turbulent conditions. The technique is shown to be capable
of resolving single-shot pressure gradients (0.04 bar/mm) originating from
kinetic energy conversion to pressure and resolves characteristic O2/N2
structures from laminar and turbulent mixing.

††journal: ol

Temperature is a key thermodynamic variable in reactive flows, because
chemical reaction rates are exponentially temperature dependent. Spatially
resolved pressure relates to molecular motion and forces, and it is critical
for aerodynamic studies. Relative species concentrations are essential when
evaluating turbulent mixing, which is prevalent in many practical applications
where two gases have to be combined. There is thus a need for spatially
resolved, one-dimensional (1D) gas-phase measurements of pressure, temperature
and species, which would support flow simulations and optimize operation of
system components in practical devices such as gas turbines.

Laser techniques for pressure and temperature are non-intrusive [1]. In gas-
phase thermometry, coherent anti-Stokes Raman spectroscopy (CARS) is esteemed
for measurement precision and accuracy [2]. Historically, nanosecond (ns) CARS
provided zero-dimensional (0D) temperature measurements through highly
resolved spectra [3], and simultaneous temperature and pressure measurements
were performed by utilizing different spectral features in the frequency-
domain [4, 5].

New technological developments such as high energy picosecond (ps) and
femtosecond (fs) lasers have granted the extension of CARS measurements from
0D to 1D [6] and to two-dimensional (2D) [7]. These developments permitted
time-domain experiments to resolve collisional behavior. Time-resolved CARS
has been used to infer Raman coherence decay information [8, 9] and
temperature [10, 11].

Encoding time-domain information into the frequency-domain has been done by
chirped-probe-pulse (CPP) CARS [12] but remains complex and limited to 0D.
Recently, Prince et al [13] introduced hybrid fs/ps CARS (HCARS). This
technique employs fs beams for the pump/Stokes and a ps beam for the probe.
With fs/ps HCARS, frequency-domain information is time-resolved due to the
short ps probe, while time-domain information can be obtained by scanning the
probe delay [14, 15]. The significance of this new configuration can be
observed in measurements applied in a wide range of pressures, temperatures,
and species [16, 17, 18, 14, 15, 19, 20]. Multiple-probe HCARS experiments
have been demonstrated in 0D to correct temperature for time-varying
collisional environments [21]. The frequency-domain pressure dependence of
HCARS has been exploited to measure pressure [22]. Using this pressure
dependence and multiple probes, simultaneous pressure and temperature
measurements were performed in 0D [23] and 1D [24]. However, a trade-off
between pressure sensitivity and signal level exists when evaluating pressure
[24]. This limits the measurements accuracy and precision. Single-shot
temporally and spectrally resolved coherences have been demonstrated in 0D
[25], limitations in detector dimensionality complicate 1D measurements.

This Letter introduces a new dual-probe 1D hybrid fs/ps rotational CARS
(HRCARS) approach for simultaneous single-shot temperature, pressure, and
O2/N2 ratio measurements. This analysis exploits the strong time-domain
pressure dependence of the CARS signal for accurate and precise pressure
measurements and overcomes challenges within the frequency-domain.

The dual-probe 1D fs/ps HRCARS technique is demonstrated here in two
environments. The first is an optically accessible chamber [26] to evaluate
the accuracy of the pressure measurements. The second is a high speed N2 jet
impinging on a surface, to demonstrate the utility of the technique in
capturing 1D pressure gradients.

Figure 1 illustrates the experimental setup for the N2 impinging jet. The
setup is modified from previous HRCARS experimental layouts [7] to include two
probe beams. The seed (Vitara) of a Ti:sapphire amplifier delivering 35 fs
pulses with 7.5 mJ at 1 kHz is locked to the ninth harmonic of the oscillator
frequency (80 MHz) of a diode pumped Nd:YAG providing 20 ps pulses with 20 mJ
at 50 Hz.

Figure 1: Schematic of the experimental setup for the N2 impinging jet
measurements.

The pump/Stokes energy is adjusted using a variable attenuator [waveplate plus
polarizing beam splitter (BS)]. We use dispersion compensation (DC) mirrors to
precompensate for dispersion effects caused by the cylindrical lens (CL)1 and
spherical lens (SL)2. A compensation of $-$220 fs2 provides a near Fourier-
transformed pulse after going through the lenses. A combination of a CL1
(f1=300 mm) and a SL2 (f2=300mm) create a laser sheet that intersects the
probe beam sheet. The probe beam is split (with a waveplate plus polarizing
BS) into 2 probes: 5% to probe 1 ($\tau_{1}\approx\ 40ps$) and 95% to probe 2
($\tau_{2}\approx 270\ ps$). Two translation stages are used to control the
relative delays between the three beams. Two pairs of CLs3,4 (f3,4=150 mm) and
CLs7,8 (f7,8=150 mm), with the optical power aligned with the y axis and
separated by 2f, are placed in the optical path. The CLs4,8 image a knife-edge
into the probe volume with a magnification slightly $<$ 1\. The knife-edge
prevents scattering of the beam from the impinging surface. CLs5,9 (f5,9=250
mm) with the optical power along the x axis create two laser sheets. The laser
sheets cross at the probe volume with an angle of 5∘ with the pump/Stokes. The
beams are then collimated with CLs6,10 (f6,10=400 mm). The mirrors placed
between the probe volume and CL11 are positioned such that both path lengths
are the same. The probe volume (0.055 x 1.2 x 4.5 mm3, $\Delta x$ x $\Delta z$
x $\Delta y$) is relay imaged into the entrance of the spectrometer using CL11
and CL12 (f11=500 mm, f12=250 mm). The anti-Stokes and Stokes signals are
separated using two angle-tunable short-pass (SP) filters. The CL13 (f13=100
mm) focuses the light into the spectrometer. The 0.75 m spectrometer is
equipped with a 2400 gr/mm grating, which disperses the HRCARS signal onto an
electron multiplying charge coupled device (EMCCD). The dispersion is 0.18
cm-1/pixel with a spectral resolution of 0.75 cm-1 (FWHM). The magnification
of the CLs7,8 and CL11 is set to $\approx$0.8. The EMCCD is binned 2x1 in the
y direction and is operated at 25 Hz with 1 ms exposure for a spatial to pixel
relationship of 30 $\mu$m/pixel. Figure 2(a) shows a representative single-
shot spectrogram of the CARS intensity, where the two probes are imaged one on
top of the other. The resolution of the imaging system is 60 $\mu$m, found by
evaluating the sharpness of the imaged knife-edge. A total length of
$\approx$4.5 mm is evaluated from the surface. 1D HRCARS spectrograms are
corrected for the finite bandwith of the pump/Stokes pulses by scanning the
probe beam delay with respect to the pump/Stokes in an argon environment. A
weighted average from these delays is used to correct the experimental
spectrograms.

Figure 2: (a) Single-shot spatially and temporally resolved spectrogram of the
dual-probe fs/ps HRCARS near the symmetry axis for the laminar N2 impinging
jet (374x1024 pixels). (b) Single-shot spatially and temporally resolved
projected intensity of the dual-probe fs/ps HRCARS.

For the chamber experiments, the aforementioned knife-edges (Fig. 1) together
with CLs1,3,4,7,8 are removed, and SL2 is substituted for a CL with f=300 mm.
Due to the increased dispersion from the chamber windows (12 mm fused silica),
the number of bounces between the DC mirrors is increased to provide -440 fs2.
The measured pulse after passing through the window is 37 fs (FWHM). The
energy employed for the pump/Stokes beam is significantly reduced (<3 mJ) due
to supercontinuum light generation in the chamber windows.

The CARS model used here is detailed in [27, 28]. Pressure dependent
linewidths are included with self-broadened linewidths for N2 [20] and O2-N2
for O2 [29]. For the frequency-domain fitting, a three-dimensional library
with varying temperature, pressure, and O2/N2 ratio was employed.

The spatially resolved intensities for different probe delays shown in Fig.
2(b) are key to studying the time-domain pressure fitting. The intensity from
integrating the CARS signal is

$I_{CARS}(T,P,\tau)\propto\int\mathcal{F}\left[P^{(3)}(t,\tau,T,P)\right]d\omega$
(1)

where $P^{(3)}$ is the third order polarization

$P^{(3)}(t,\tau,T,P)\propto\left(\frac{i}{\hbar}\right)^{3}E_{pr}(t-\tau)R_{CARS}(t,T,P)e^{i\omega_{pr}(t-\tau)}.$
(2)

$E_{pr}$ is the normalized electric field envelope of the probe, $\tau$ is the
relative delay between the probe and the pump/Stokes, and $R_{CARS}$ is the
molecular response function

$R_{CARS}(t,T,P)=\frac{P}{kT}\sum_{J}I_{J;J+2}(T)\times
e^{\frac{t}{\hbar}\left(i\Delta
E_{J;J+2}-\frac{1}{2}\Gamma_{J;J+2}(T,P)\right)},$ (3)

where ${P}/{kT}$ represents the number density effect, $I_{J;J+2}$ is the
Raman transition strength, $\Delta E_{J;J+2}$ is the frequency of the
transitions, and $\Gamma_{J;J+2}$ is the linewidths.

Given electric fields for pump/Stokes and probe, the N2 HRCARS intensity is a
function of probe delay, temperature, and pressure [Eq. (1)-(3)]. Figure 3(a)
shows the simulated ICARS for two probe delays ($\tau_{1}=40$ ps,
$\tau_{2}=270$ ps) and the Iratio=ICARSτ1/ICARSτ2 for a range of pressures of
0.9-1.1 bar and temperatures of 280-310 K. The Raman response
($R_{CARS}(t,T,P)$) contains two pressure dependent terms: number density
(${P}/{kT}$) and linewidths ($\Gamma_{J;J+2}(T,P)$). For probe 1 (ICARSτ1,
$\tau_{1}=40$ ps), the number density contribution dominates the signal
intensity for both decreasing temperature and increasing pressure. For probe 2
(ICARSτ2, $\tau_{2}=270$ ps), the signal decreases for increasing pressure due
to the exponential dependence on linewidth, which overcomes the quadratic
signal increase due to number density.

Figure 3: (a) Simulated intensity as a function of pressure (0.9-1.1 bar) and
temperature (280-310 K) for early probe ($\tau_{1}=40$ ps), late probe
($\tau_{2}=270$ ps) and $I_{ratio}$(b) Simulated relation for Iratio, pressure
and temperature.

The Iratio for spatially correlated measurements is not a function of number
density or pump/Stokes laser energy, and the pressure sensitivity is increased
when compared to any of the individual probes. The relation between the
simulated $I_{ratio}$ and the measured $I_{ratio}$ is not complete until a
calibration measurement is performed. This accounts for the energy profiles of
both individual probe beams, among other things, and is performed separately
for chamber and jet experiments, both at ambient pressure (1.01 bar) and
temperature. To better utilize the detector dynamic range, the energy split of
probes 1 and 2 is adjusted so that $I_{ratio}\approx 1$ under the calibration
conditions.

Figure 3(b) shows the simulated normalized Iratio for a range of temperatures
of 280-310 K and pressures of 0.9-1.6 bar. The surface plot illustrates the
strong sensitivity of the Iratio with respect to pressure; a variation from 1
to 1.6 bar produces an Iratio change greater than five for this temperature
range. The Iratio sensitivity with respect to pressure is predominantly a
function of $\Delta\tau=\tau_{2}-\tau_{1}$ and can be adjusted to match the
expected measured pressure to the dynamic range of the detector.

The fitting procedure starts by assuming ambient pressure in a frequency-
domain fit for probe 1, so we obtain temperature ($T_{1}$). This temperature
is then used to obtain pressure using the Iratio-pressure relation. The
calculated pressure is then used to refit probe 1, resulting in the final
temperature (${T}$). With the final temperature and the Iratio-pressure
relation, we can obtain the final pressure (${P}$) that has already converged.
The statistics displayed in Table 1 are for spatially and time-resolved
measurements in the chamber, from evaluating the central 20 spectral rows in
50 shots. The measurements precision for the chamber is hindered compared to
the N2 impinging jet experiments due to the reduced pump/Stokes energy ($<$3
mJ).

Table 1: Pressure and temperature measurements in the chamber $\overline{T_{1}}$(K) |  $\overline{T}$(K) |  $\overline{P}$(bar) |  $\frac{\sigma_{T}}{\overline{T}}$ (%) |  $\frac{\sigma_{P}}{\overline{P}}$ (%) |  $\overline{P_{trans.}}$(bar)
---|---|---|---|---|---
296.43 | 296.43 | 1.01 | 1.36 | 0.83 | 1.01
298.59 | 296.59 | 1.22 | 1.14 | 1.29 | 1.21
299.73 | 296.87 | 1.31 | 1.08 | 1.72 | 1.30
300.17 | 296.64 | 1.38 | 1.04 | 1.91 | 1.37
301.30 | 296.83 | 1.48 | 0.94 | 2.44 | 1.49

$T_{1}$ shows a systematic increase of 1.7% as pressure increases 0.47 bar,
while the final calculated temperature ($\overline{T}$) only varies 0.2% in
the studied pressure range. The pressure measurements are benchmarked against
a pressure transducer ($\overline{P_{trans.}}$). The agreement between
$\overline{P}$ with $\overline{P_{trans.}}$ is better than $1$ %. The
temperature precision improves from 1.36 to 0.94% as pressure increases due to
the increased signal from ICARSτ1. The pressure precision decreases from 0.83
to 2.44% as pressure increases, due to the signal decrease from ICARSτ2.

To demonstrate the utility of the dual-probe HRCARS technique, measurements
were applied to a N2 jet impinging on a surface. The surface is curved along
the z direction (d=300 mm) for easier optical access. Measurements are
performed under two conditions, $\overline{V}$=5.2 m/s (Re=1,420, laminar) and
$\overline{V}$=104.2 m/s (Re=28,510, turbulent). The jet inner diameter is D=4
mm, and the distance between the jet and the surface is H=11 mm, providing a
H/D ratio of 2.75. The 4.5 mm closest to the plate are evaluated for
temperature, pressure, and O2/N2. For every 2D field, twenty 1D measurements
are performed, spaced every 200 $\mu$m, covering the radial direction $\pm$2
mm of the jet impingement. The surface is (4 x 4.5 mm2, $\Delta x$ x $\Delta
y$), and each averaged field contains the analysis of (150 x 20 x 100) 0.3
million spectra.

Figure 4:  2D pressure field from a turbulent (Re=28,510) N2 jet impinging on
a surface: (a) averaged and (b) single-shot.

The laminar pressure field and the laminar and turbulent temperature fields
are uniform, thus not shown here. The pressure and temperature fields obtained
from the laminar jet are used to evaluate the measurements precision. The
spatio-temporal precision for pressure is 0.42% ($\sigma_{P}/\overline{P}$)
and for temperature is 0.62% ($\sigma_{T}/\overline{T}$). Figure 4 shows
averaged and instantaneous turbulent pressure fields. The maximum measured
pressure is $\approx$1.11 bar in both the averaged and single-shot fields.
This maximum pressure is located near the symmetry axis close to the surface
and decays both radially and axially. Spatial pressure gradients of the order
of 0.04 bar/mm are well-captured.

To further illustrate the jet behavior, Fig. 5 shows both averaged and single-
shot O2/N2 ratios for both (a), (b) the laminar and (c), (d) the turbulent
conditions. The stagnating flow field causes no O2 detection close to the
plate. Comparing both O2/N2 averaged fields, the turbulent field has a
O2/N${}_{2}\approx$ 0.06 ratio, a three-fold increase due to enhanced air
entrainment near the jet radius. Differences in mixing mechanisms are captured
in the single-shot measurements: for the laminar jet, the O2/N2 measurements
show smooth O2/N2 gradients typical of mixing dominated mostly by diffusion,
while the turbulent jet shows fine structure of air pockets from air
entrainment, which also demonstrates the exceptional spatial resolution.

Figure 5: 2D O2/N2 field from a N2 jet impinging on a surface: laminar jet
[(a) averaged field, (b) single-shot field] and turbulent jet [(c) averaged
field, (d) single-shot field].

In summary, with dual-probe 1D HRCARS, by extracting information from both the
time and frequency-domains, we have demonstrated simultaneous pressure,
temperature, and O2/N2 ratio measurements. A new methodology to evaluate
pressure leads to high accuracy (<1%) for the pressure range 1-1.5 bar,
remarkable spatio-temporal temperature precision of 0.62%
($\sigma_{T}/\overline{T}$), unmatched spatio-temporal pressure precision of
0.42% ($\sigma_{P}/\overline{P}$), and fine spatial resolution (0.055 x 1.2 x
0.060 mm3). The pressure measurements are able to describe pressure gradients
of the order of 0.04 bar/mm. The O2/N2 measurements also capture the spatial
characteristics from the mixing interface both in laminar and turbulent
conditions. Applying this measurement technique in a more challenging
environment with higher pressures, temperatures, and species gradients will be
the focus of future work. For such studies, anticipated issues such as reduced
signal due to fast collisional dephasing at high pressures, unknown
collisional partners, and choice of optimal probe delays will be investigated.

Funding. Engineering and Physical Sciences Research Council (EP/P020593/1,
EP/P001661/1); European Research Council (759546).

Disclosures. The authors declare no conflicts of interest.

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library/opticsletters


is uniformly bounded in $R$ for any compact $K\Subset M\setminus\Omega$ by
(A.8), we can invoke the basic but very useful [HKM06, Lemma 1.33] to infer
that $u\in L^{p}(M\setminus\overline{\Omega})$ and that ${\rm D}u_{R}$
converges weakly to ${\rm D}u$. In particular, by lower semicontinuity of the
norm we have

$\int\limits_{M\setminus\overline{\Omega}}\lvert{\rm D}u\rvert^{p}{\,\rm
d}\mu\leq\liminf_{R\to\infty}\int\limits_{M\setminus\overline{\Omega}}\lvert{\rm
D}u_{R}\rvert^{p}{\,\rm d}\mu,$

and since the righthand side is uniformly bounded again by (A.8), we conclude
that $u\in W^{1,p}(M\setminus\overline{\Omega})$. Since it vanishes at
infinity, it actually belongs to $W_{0}^{1,p}(M\setminus\overline{\Omega})$.
We can thus argue exactly as done for (A.5) to show that $u$ realises the
$p$-capacity of $\Omega$, that is (4.7). ∎

## Appendix B

We furnish a proof of Proposition 5.6. Although this is very classical, and we
are substantially reporting part of the one given for [Cha84, Chapter IV,
Theorem 2], we were not able to find in literature an explicit dependence on
the isoperimetric constant and a consequent discussion of the extremal case.

###### Proof of Proposition 5.6.

Let $V:[0,T]\to\mathbb{R}$, where $T=\max_{\overline{\Omega}}f$, the function
defined by $V(t)=\lvert\\{f\geq t\\}\rvert.$ This function is easily seen to
be continuous and actually $\mathscr{C}^{\infty}$ on regular values of $f$.
Consider $B_{t}=\mathbb{B}_{\lvert\\{f\geq t\\}\rvert}$, that is, the ball in
$\mathbb{R}^{n}$ with (Euclidean) volume equal to $V(t)$. This defines a
bijective function $\rho:[0,T]\to[0,\rho(0)]$ such that $V(t)=\lvert
B(\rho(t))\rvert_{\mathbb{R}^{n}}$, where $B(\rho(t))$ is any ball of radius
$\rho(t)$. Define finally $F:\overline{\mathbb{B}_{v}}\to\mathbb{R}$ by
$F=\rho^{-1}\circ r$, where $r$ is the distance from the center of
$\mathbb{B}_{v}$. It is shown exactly as in [Cha84] that with this definition
the identity in (5.20) is satisfied. By coarea formula, we have

$\int_{\Omega}\lvert{\rm D}f\rvert^{2}{\,\rm
d}\mu=\int\limits_{0}^{T}\int\limits_{\\{f=t\\}}\lvert{\rm D}f\rvert{\,\rm
d}\sigma{\,\rm
d}t\geq\int\limits_{0}^{T}\lvert\\{f=t\\}\rvert^{2}\int\limits_{\\{f=t\\}}\frac{1}{\lvert{\rm
D}f\rvert}{\,\rm d}\sigma{\,\rm d}t$ (B.1)

On the other hand, by (5.19),

$\lvert\\{f=t\\}\rvert^{2}\,\geq\,\lvert\\{f\geq
t\\}\rvert^{2\frac{n-1}{n}}\lvert\mathbb{S}^{n-1}\rvert^{2}\,\lvert\mathbb{B}^{n}\rvert^{2\frac{n-1}{n}}\mathrm{C}_{g}^{\frac{2}{n}}\,=\,\lvert
B_{t}\rvert_{{\mathbb{R}^{n}}}^{2\frac{n-1}{n}}\,\lvert\mathbb{S}^{n-1}\rvert^{2}\,\lvert\mathbb{B}^{n}\rvert^{2\frac{n-1}{n}}\mathrm{C}_{g}^{\frac{2}{n}}\,=\,\lvert\partial
B_{t}\rvert_{{\mathbb{R}^{n}}}^{2}\mathrm{C}_{g}^{{2}/{n}},$ (B.2)

where in the last equality we used the fact that balls in $\mathbb{R}^{n}$
satisfy equality in the Isoperimetric Inequality. Using that

$\rho^{\prime}(t)=-\frac{1}{\lvert\partial
B_{t}\rvert_{\mathbb{R}^{n}}}\int\limits_{\\{f=t\\}}\frac{1}{\lvert{\rm
D}f\rvert}{\,\rm
d}\sigma\qquad\quad\left(\rho^{-1}\right)^{\prime}(\rho(t))=\frac{1}{\rho^{\prime}(t)},$

we get, plugging the outcome of (B.2) in (B.1) and using again the coarea
formula,

$\int_{\Omega}\lvert{\rm D}f\rvert^{2}{\,\rm
d}\mu\,\geq\,\mathrm{C}_{g}^{{2}/{n}}\int\limits_{0}^{T}\left[\left(\rho^{-1}\right)^{\prime}\right]^{2}\lvert\partial
B_{t}\rvert_{\mathbb{R}^{n}}\rho^{\prime}(t){\,\rm
d}t\,=\,\mathrm{C}_{g}^{{2}/{n}}\int_{\mathbb{B}_{v}}\lvert{\rm
D}F\rvert^{2}_{\mathbb{R}^{n}}{\,\rm d}\mu_{\mathbb{R}^{n}},$ (B.3)

as desired.

Assume now that equality holds in the inequality of (5.20). Then, we deduce
from (B.3) and (B.2) that

$\int\limits_{0}^{T}\lvert\\{f=t\\}\rvert^{2}-\lvert\partial
B_{t}\rvert_{{\mathbb{R}^{n}}}^{2}\mathrm{C}_{g}^{\frac{2}{n}}{\,\rm d}t=0,$
(B.4)

that, by the nonnegativity of the integrand, implies that the sets $\\{f\geq
t\\}$ satisfy equality in the Isoperimetric Inequality of $(M,g)$ for any
regular value $t$. Letting $t\to 0^{+}$, that is possible by Sard’s Theorem,
we conclude that $\Omega$ satisfy equality in (5.19). ∎

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