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# Relations between moments for the Jacobi and Cauchy random matrix ensembles

Peter J. Forrester  School of Mathematics and Statistics, ARC Centre of
Excellence for Mathematical and Statistical Frontiers, University of
Melbourne, Victoria 3010, Australia<EMAIL_ADDRESS>and  Anas A. Rahman
School of Mathematics and Statistics, ARC Centre of Excellence for
Mathematical and Statistical Frontiers, University of Melbourne, Victoria
3010, Australia<EMAIL_ADDRESS>
###### Abstract.

We outline a relation between the densities for the $\beta$-ensembles with
respect to the Jacobi weight $(1-x)^{a}(1+x)^{b}$ supported on the interval
$(-1,1)$ and the Cauchy weight
$(1-\mathrm{i}x)^{\eta}(1+\mathrm{i}x)^{\bar{\eta}}$ by appropriate analytic
continuation. This has the consequence of implying that the latter density
satisfies a linear differential equation of degree three for $\beta=2$, and of
degree five for $\beta=1$ and $4$, analogues of which are already known for
the Jacobi weight $x^{a}(1-x)^{b}$ supported on $(0,1)$. We concentrate on the
case $a=b$ (Jacobi weight on $(-1,1)$) and $\eta$ real (Cauchy weight) since
the density is then an even function and the differential equations simplify.
From the differential equations, recurrences can be obtained for the moments
of the Jacobi weight supported on $(-1,1)$ and/or the moments of the Cauchy
weight. Particular attention is paid to the case $\beta=2$ and the Jacobi
weight on $(-1,1)$ in the symmetric case $a=b$, which in keeping with a recent
result obtained by Assiotis et al. for the $\beta=2$ case of the symmetric
Cauchy weight (parameter $\eta$ real), allows for an explicit solution of the
recurrence in terms of particular continuous Hahn polynomials. Also for the
symmetric Cauchy weight with $\eta=-\beta(N-1)/2-1-\alpha$, after
appropriately scaling $\alpha$ proportional to $N$, we use differential
equations to compute terms in the $1/N^{2}$ ($1/N$) expansion of the resolvent
for $\beta=2$ ($\beta=1,4$).

##  1\. Introduction

The study of moments in random matrix theory was initiated by Wigner in 1955
[38]. This was in the context of analysing the limiting spectral density of a
particular class of real symmetric matrices $J_{N}$ of size
$(2N+1)\times(2N+1)$, with diagonal entries set to zero, and the strictly
upper triangular entries independently chosen from the values $\pm w$ with
equal probability. Upon scaling the eigenvalues by $1/\sqrt{N}$, or
equivalently multiplying $J_{N}$ by this value to obtain the scaled matrices
$\bar{J}_{N}$, Wigner considered the trace of even powers of $\bar{J}_{N}$,
and was able to prove that for $p\in\mathbb{Z}_{\geq 0}$,

$\lim_{N\to\infty}{1\over N}\Big{\langle}{\rm
Tr}\,\bar{J}_{N}^{2p}\Big{\rangle}=w^{p}{1\over p+1}\binom{2p}{p}.$ (1.1)

The combinatorial expression on the RHS is recognised as the $p$-th Catalan
number.

The average of the LHS of (1.1) is an example of a linear statistic of the
eigenvalues $\\{\lambda_{j}\\}_{j=1}^{N}$. Indeed, ${\rm
Tr}\,\bar{J}_{N}^{2p}=\sum_{j=1}^{N}\lambda_{j}^{2p}$ is a sum over individual
eigenvalues, which is the defining feature of a linear statistic. Introduce
the scaled eigenvalue density $\bar{\rho}^{(N)}(\lambda)$ corresponding to the
matrices $\bar{J}_{N}$, which by definition has the property that
$N\int_{a}^{b}\bar{\rho}^{(N)}(\lambda)\,d\lambda$ is equal to the expected
number of eigenvalues in the interval $[a,b]$. The fact that ${\rm
Tr}\,\bar{J}_{N}^{2p}$ is a linear statistic tells us that its average can be
expressed as an integral over the scaled eigenvalue density,

${1\over N}\Big{\langle}{\rm
Tr}\,\bar{J}_{N}^{2p}\Big{\rangle}=\int_{-\infty}^{\infty}\lambda^{2p}\bar{\rho}^{(N)}(\lambda)\,d\lambda.$
(1.2)

Consider now the particular density function $\rho^{\rm W}(\lambda)$ supported
on $[-2w,2w]$ specified by the functional form

$\rho^{\rm W}(\lambda)=\frac{1}{2\pi w^{2}}(4w^{2}-\lambda^{2})^{1/2}.$ (1.3)

This is referred to as the Wigner semi-circle. A simple change of variables
and use of knowledge of the Euler beta function in terms of gamma functions
verifies that

$\int_{-2w}^{2w}\lambda^{2p}\rho^{\rm W}(\lambda)\,d\lambda=w^{p}{1\over
p+1}\binom{2p}{p}.$ (1.4)

Thus upon comparing to (1.1) and recalling (1.2), we have

$\lim_{N\to\infty}\int_{-\infty}^{\infty}\lambda^{2p}\bar{\rho}^{(N)}(\lambda)\,d\lambda=\int_{-2w}^{2w}\lambda^{2p}\rho^{\rm
W}(\lambda)\,d\lambda.$ (1.5)

In relation to (1.4), and using too the fact that the odd moments vanish, the
exponential generating function can be formed to obtain

$\int_{-2w}^{2w}e^{is\lambda}\rho^{\rm W}(\lambda)\,d\lambda={J_{1}(2ws)\over
ws},$ (1.6)

where $J_{\nu}$ denotes the Bessel function of order $\nu$. This is analytic
at the origin, telling us that $\rho^{\rm W}(\lambda)$ is uniquely defined by
its moments. This fact is sufficient for the limit formula (1.5) to imply that
the sequence of measures
$\\{\bar{\rho}^{(N)}(\lambda)\,d\lambda\\}_{N=1}^{\infty}$ converges weakly to
$\rho^{\rm W}(\lambda)\,d\lambda$, i.e. to the Wigner semi-circle. This was
the conclusion of Wigner’s analysis. Wigner’s second paper on this topic,
published in 1958 [39], considered a wider class of random real symmetric
matrices. Independence of entries, fixed variance and bounded moments were
shown, by the same technique, to be sufficient conditions for the limiting
spectral density to be given by the Wigner semi-circle.

Knowledge of the limiting form of the spectral density and moments is refined
by knowledge of finite $N$ corrections. For the class of Wigner matrices, i.e.
real symmetric or complex Hermitian random matrices with entries on the
diagonal chosen independently from a particular zero mean distribution, and
upper triangular entries chosen independently from a finite mean, finite
variance distribution, it is known that averages of suitable test functions
$\phi(\lambda)$ over $\bar{\rho}^{(N)}(\lambda)$ permit $1/N$ expansions

$\int_{-\infty}^{\infty}\phi(\lambda)\bar{\rho}^{(N)}(\lambda)\,d\lambda=\int_{-2w}^{2w}\phi(\lambda)\rho^{\rm
W,0}(\lambda)\,d\lambda+\frac{1}{N}\int_{-\infty}^{\infty}\phi(\lambda)\rho^{\rm
W,1}(\lambda)\,d\lambda+\cdots$ (1.7)

where $\rho^{\rm W,0}(\lambda)=\rho^{\rm W}(\lambda)$ is the Wigner semi-
circle (1.3) and $\rho^{\rm W,1}(\lambda)$ depends on the second moment of the
diagonal entries, and moments up to and including the fourth of the off-
diagonal elements [3, 27]; see the recent work [19] for a convenient summary.

The explicit form of the expansion (1.7) is known to higher orders in the case
of real symmetric Wigner matrices with independent normals ${\rm
N}[0,\sqrt{2}w]$ on the diagonal, and ${\rm N}[0,w]$ on the off-diagonal (GOE
matrices), or complex Hermitian Wigner matrices with independent normals ${\rm
N}[0,w]$ on the diagonal, and ${\rm N}[0,w/\sqrt{2}]+i{\rm N}[0,w/\sqrt{2}]$
on the off-diagonal (GUE matrices); see [41]. In general, for a class of
scaled Wigner matrices $\\{X\\}$ with symmetric distributions so that the odd
moments vanish, the $2p$-th power of the trace is a polynomial in $1/N$ of
degree $p$. For the GOE and GUE this polynomial — the constant term of which
is given by the RHS of (1.1) — has a number of unique features among the class
of Wigner matrices which can be traced back to the fact that the spectral
density in these cases can be evaluated in terms of Hermite polynomials; see
[34] and [41] as well as the earlier works [24] and [30]. One of these
features is that the moments can be characterised by a recurrence. The
simplest case is the scaled GUE (we take $w=1/2$ for definiteness), for which
a result first obtained by Ullah [37] tells us that

$\int_{-\infty}^{\infty}e^{ik\lambda}\bar{\rho}^{(N)}(\lambda)\,d\lambda={1\over
N}e^{-k^{2}/8N}L_{N-1}^{(1)}(k^{2}/4N),$ (1.8)

where $L^{(p)}_{n}(x)$ denotes the Laguerre polynomial. Known asymptotics of
the Laguerre polynomials shows that (1.8) limits to (1.6), as it must. As
noted in the recent work [12], this explicit formula can be used to show that
the LHS satisfies a second order differential equation [24, 30], and that
$\bar{\rho}^{(N)}(\lambda)$ satisfies a third order differential equation [29,
23]. From the former a second recurrence satisfied by the sequence of $2p$-th
moments can be deduced [25, 24]. Analogous results can be obtained for the GOE
but with a higher level of complexity. The LHS of (1.8) now satisfies a fourth
order linear differential equation, and the sequence of $2p$-th moments
satisfies a fourth order difference equation [31]. The density satisfies a
fifth order differential equation [41].

The Hermite polynomials expressing the spectral density for the GOE and GUE
(and too the GSE — see e.g. [11, Ch. 6]) correspond to the Gaussian weight in
the theory of classical orthogonal polynomials. We recall that a weight
function is said to be classical if its logarithmic derivative ${d\over
dx}\log w(x)$ can be written in the form $-g(x)/f(x)$ with $f>0$, $f$ and $g$
having no common factors, and the degree of $f$ ($g$) less than or equal to
$2$ $(1)$; see e.g. [11, §5.4.3]. Up to fractional linear transformations, it
is known that the only classical weights with support on the real line are the
Gaussian, the Laguerre weight $x^{a}e^{-x}\chi_{x>0}$ supported on $x>0$, as
well as the Jacobi and Cauchy weights

$(1-x)^{a}(1+x)^{b}\chi_{-1<x<1},\qquad(1-ix)^{\eta}(1+ix)^{\bar{\eta}}.$
(1.9)

The indicator function $\chi_{A}$ is defined to equal $1$ if $A$ is true and
$0$ otherwise. Note that a simple linear change of variables maps the Jacobi
weight as presented in (1.9) to

$x^{a}(1-x)^{b}\chi_{0<x<1}.$ (1.10)

Eigenvalue probability density functions (PDFs) proportional to

$\prod_{l=1}^{N}w(x_{l})\prod_{1\leq j<k\leq N}|x_{k}-x_{j}|^{\beta},$ (1.11)

with $w(x)$ a classical weight, and $\beta=1,2$ or $4$, are prominent in
classical random matrix theory. For example, with the choice of $w(x)$ as a
Gaussian, (1.11) is the eigenvalue PDF for the GOE when $\beta=1$, the GUE
when $\beta=2$ and the GSE when $\beta=4$. Thus the exponent $\beta$ — often
referred to as the Dyson index after the pioneering work [10] — corresponds to
the number of independent real parts in the corresponding number field. Beyond
these three special values of $\beta$, for the classical weights there are
constructions of random matrices with eigenvalue PDF (1.11) for general
$\beta>0$. In the case of the Jacobi weight as written in (1.9), this was
first obtained in [28], while for the Cauchy weight it was obtained in [17];
for a text book treatment see [11, §3.11 & §4.3.4]. For the Gaussian, Laguerre
and Jacobi weights, the theory of Selberg correlation integrals as applied in
[36] tells us that there are integrable structures by way of linear
differential equations for the density of degree $\beta+1$ for all $\beta$
even, and that duality formulas extend this characterisation when $\beta$ is
replaced by $4/\beta$. From the differential equations, difference equations
for the moments can be determined. In the Jacobi case these are of the same
degree, but in the Gaussian and Laguerre cases their degree reduces by one to
now be equal to $\beta$. In practice, beyond the classical value $\beta=2,4$
(and by the duality, $\beta=1$) it was not feasible to make these differential
or difference equations explicit. An exception was $\beta=6$ in the Gaussian
case (and by duality $\beta=2/3$), where the seventh order differential
equation, and sixth order difference equation were presented explicity.

Absent from the study [36] was consideration of analogous integrable
properties in the case of the Cauchy weight in (1.11); see [11, §2.5, §3.9,
§4.3.4] for a textbook treatment relating to the latter. With $\beta=2$,
results of the type obtained in [36] for the Jacobi weight (1.10) (see also
[30, 34, 6]) have recently been obtained for the Cauchy weight in the work of
Assiotis et al. [2], with the requirement that the Cauchy parameter $\eta$ in
(1.9) be real. In particular, it was shown that the sum of the $2p$-th and
$(2p+2)$-th moment can be identified as a continuous Hahn polynomial in the
variable $p$. In [6] it has earlier been shown that the difference between
successive moments in the Jacobi case of (1.11) can be identified in terms of
Wilson polynomials from the Askey scheme (for other recent appearances of
polynomials from the Askey scheme in studies in random matrix theory see [14,
13, 22]).

Underpinning the present paper is the observation that integrations with
respect to the Cauchy ensemble can be computed by an analytic continuation in
the parameters $a,b$ of integrations in the Jacobi ensemble with weight
defined on $(-1,1)$. The details of such inter-relations are given in Section
2. From our work [36] giving linear differential equations for the density of
the Jacobi ensemble defined on $(0,1)$ with $\beta=1$, $2$ and $4$, a simple
linear change of variables gives differential equations for the corresponding
Jacobi ensemble defined on $(-1,1)$. Applying the results of Section 2 then
gives us linear differential equations for the density of the Cauchy ensemble
with $\beta=1$, $2$ and $4$, which we present in Section 3. We highlight the
symmetric case $\eta=\bar{\eta}$ and take $\eta$ to be general complex only in
the $\beta=2$ case in order to keep our presentation neat. In the $\beta=2$
case with $\eta$ real, we reclaim the third order differential equation
derived recently in [2]. Moreover, the differential equations for the
densities of the $\beta=1,2,4$ Cauchy ensembles simplify if written with
$(1+x^{2})\rho_{(1)}^{(Cy)}(x)$ as the dependent variable; in the Jacobi case,
the inter-relations tell us that the analogous simplification occurs with
$(1-x^{2})\rho_{(1)}^{(J)}(x)$ as the dependent variable.

Two consequences are further developed. One is to quantify the limiting
density and its moments in the symmetric Cauchy ensemble with
$\eta=-\beta(N-1)/2-1-\hat{\alpha}\beta N/2$. It is also possible to use
differential equations to study $1/N^{2}$ corrections (for $\beta=2$) and
$1/N$ corrections (for $\beta=1,4$) of the type well known in the study of the
GUE [25]. The other is the specification of (three-term for $\beta=2$, five-
term for $\beta=1,4$) recurrences for the differences of successive even
moments of the Jacobi ensembles on $(-1,1)$, which are equivalent (up to minus
signs) to recurrences for the sums of successive even moments of the
corresponding Cauchy ensembles. These recurrences are presented in Section 4,
along with a demonstration of the fact that the recurrence in the $\beta=2$,
$a=b$ Jacobi case on $(-1,1)$ can be solved in terms of continuous Hahn
polynomials. This latter fact is consistent, via the inter-relations of
Section 2, to an equivalent observation of [2] on the associated recurrence in
the Cauchy case.

##  2\. Relating the density for the Cauchy and Jacobi $\beta$-ensembles

The density in the Cauchy $\beta$-ensemble is specified by

$\rho_{(1)}^{(Cy)}(x)=N{w_{\beta}^{(Cy)}(x)\over\mathcal{N}_{N}^{(Cy)}}\int_{-\infty}^{\infty}dx_{2}\cdots\int_{-\infty}^{\infty}dx_{N}\,\prod_{l=2}^{N}w_{\beta}^{(Cy)}(x_{l})|x-x_{l}|^{\beta}\prod_{2\leq
j<k\leq N}|x_{k}-x_{j}|^{\beta},$ (2.1)

where

$w_{\beta}^{(Cy)}(x)=(1-ix)^{\alpha_{\beta,N}}(1+ix)^{\bar{\alpha}_{\beta,N}},\qquad{\alpha}_{\beta,N}=-\beta(N-1)/2-1-\alpha.$
(2.2)

Here, $\alpha\in\mathbb{C}$ and it is required that ${\rm Re}\,\alpha>-1/2$.
The dependence on $\beta$ and $N$ in the exponent $\alpha_{\beta,N}$ has been
chosen so that this latter requirement ensures that the normalisation
$\mathcal{N}_{N}^{(Cy)}$ is finite; see [11, §3.9] for working which gives
insight into this convergence condition. In fact, the normalisation is known
explicitly (see [11, Exercises 4.7 q.4(i)]),

$\mathcal{N}_{N}^{(Cy)}=2^{-\beta N(N-1)/2-2N{\rm
Re}\,\alpha}\pi^{N}\,M_{N}(\alpha,\bar{\alpha},\beta/2),$ (2.3)

where

$M_{N}(a,b,\lambda)=\prod_{j=0}^{N-1}{\Gamma(\lambda
j+a+b+1)\Gamma(\lambda(j+1)+1)\over\Gamma(\lambda j+a+1)\Gamma(\lambda
j+b+1)\Gamma(1+\lambda)}.$ (2.4)

In relation to the Jacobi weight from (1.9), the density for the corresponding
Jacobi $\beta$-ensemble is

$\rho_{(1)}^{(J)}(x)=N{w_{\beta}^{(J)}(x)\over\mathcal{N}_{N}^{(J)}}\int_{-1}^{1}dx_{2}\cdots\int_{-1}^{1}dx_{N}\,\prod_{l=2}^{N}w_{\beta}^{(J)}(x_{l})|x-x_{l}|^{\beta}\prod_{2\leq
j<k\leq N}|x_{k}-x_{j}|^{\beta},$ (2.5)

where

$w_{\beta}^{(J)}(x)=(1-x)^{a}(1+x)^{b}\chi_{-1<x<1}.$ (2.6)

Here, the normalisation is finite for ${\rm Re}\,a,b>-1$ and is given in terms
of the Selberg integral (see [11, §4.1])

$\mathcal{N}_{N}^{(J)}=2^{\beta N(N-1)/2+N(a+b+1)}\,S_{N}(a,b,\beta/2),$ (2.7)

where

$S_{N}(a,b,\lambda)=\prod_{j=0}^{N-1}{\Gamma(a+1+\lambda j)\Gamma(b+1+\lambda
j)\Gamma(1+\lambda(j+1))\over\Gamma(a+b+2+\lambda(N+j-1))\Gamma(1+\lambda)}.$
(2.8)

###  2.1. The symmetric case

Let us now specialise the Cauchy weight to the case $\alpha$ real, and
specialise the Jacobi weight to the case $a=b$. Both weights are then even
functions of $x$. The following relation between multiple integrals over these
weights holds true:

######  Proposition 2.1.

Let $f(x_{1},\dots,x_{N})$ be a multivariable symmetric polynomial of degree
$d$ in each $x_{i}$. For $2\eta<-(d+1)$, define

$I_{N,\eta}^{(Cy)}[f(x_{1},\ldots,x_{N})]:=\int_{-\infty}^{\infty}dx_{1}\,(1+x_{1}^{2})^{\eta}\cdots\int_{-\infty}^{\infty}dx_{N}\,(1+x_{N}^{2})^{\eta}\,f(x_{1},\dots,x_{N}),$
(2.9)

and for $\eta$ outside of this range, define
$I_{N,\eta}^{(Cy)}[f(x_{1},\ldots,x_{N})]$ by its analytic continuation. Also,
in relation to the Jacobi weight with $a=b>-1$, define

$I_{N,a}^{(J)}[f(x_{1},\ldots,x_{N})]:=\int_{-1}^{1}dx_{1}\,(1-x_{1}^{2})^{a}\cdots\int_{-1}^{1}dx_{N}\,(1-x_{N}^{2})^{a}\,f(x_{1},\dots,x_{N}),$
(2.10)

and for $a$ outside of this range, define
$I_{N,a}^{(J)}[f(x_{1},\ldots,x_{N})]$ by its analytic continuation. We have

$I_{N,\eta}^{(Cy)}[f(ix_{1},\dots
ix_{N})]=(\tan\pi\eta)^{N}I_{N,\eta}^{(J)}[f(x_{1},\dots x_{N})].$ (2.11)

###### Proof.

Let $p\in\mathbb{Z}_{\geq 0}$. Suppose $2\eta<-(p+1)$ and $a>-1$. A simple
change of variables (for $k$ even) and use of the Euler beta integral
evaluation (see [11, Exercises 5.4 q.2]) shows

$\int_{-\infty}^{\infty}(1+x^{2})^{\eta}x^{k}\,dx=\begin{cases}0,&k\>{\rm
odd},\\\
(-1)^{k/2}\tan\pi\eta\,{\Gamma(1+\eta)\Gamma((k+1)/2)\over\Gamma((k+3)/2+\eta)},&k\>{\rm
even},\end{cases}$ (2.12)

and

$\int_{-1}^{1}(1-x^{2})^{a}x^{k}\,dx=\begin{cases}0,&k\>{\rm odd},\\\
\,{\Gamma(1+a)\Gamma((k+1)/2)\over\Gamma((k+3)/2+a)},&k\>{\rm
even}.\end{cases}$ (2.13)

The functions $f(x_{1},\ldots,x_{N})$ and $f(ix_{1},\ldots,ix_{N})$ are
polynomials, so the computation of
$I_{N,\eta}^{(Cy)}[f(ix_{1},\ldots,ix_{N})]$ and
$I_{N,a}^{(J)}[f(x_{1},\ldots,x_{N})]$ reduces to the above one-dimensional
integrals. Since, as analytic functions of $\eta$, we read off from the
respective evaluations that

$\int_{-\infty}^{\infty}(1+x^{2})^{\eta}(ix)^{2k}\,dx=\tan\pi\eta\int_{-1}^{1}(1-x^{2})^{\eta}x^{2k}\,dx,$

the stated result (2.11) follows ∎

One immediate consequence is a relation between the normalisations in (2.1)
and (2.5) in specialisations of the parameters that conform with Proposition
2.1.

######  Corollary 2.2.

Let $\alpha$ be real and related to $\alpha_{\beta,N}$ as in (2.2). For
$\beta$ even, we have

$(-1)^{\beta
N(N-1)/4}\mathcal{N}_{N}^{(Cy)}=(-\tan\pi\alpha)^{N}\mathcal{N}_{N}^{(J)}\Big{|}_{a=b=-\beta(N-1)/2-1-\alpha},$
(2.14)

where both sides are to be interpreted as analytic functions in $\alpha$.

###### Proof.

With $\alpha$ and thus $\alpha_{\beta,N}$ real, we have

$\mathcal{N}_{N}^{(Cy)}=\int_{-\infty}^{\infty}dx_{1}(1+x_{1}^{2})^{\alpha_{\beta,N}}\cdots\int_{-\infty}^{\infty}dx_{N}(1+x_{N}^{2})^{\alpha_{\beta,N}}\prod_{1\leq
j<k\leq N}|x_{k}-x_{j}|^{\beta}.$

Also, with $a=b$, we have

$\mathcal{N}_{N}^{(J)}=\int_{-1}^{1}dx_{1}\,(1-x_{1}^{2})^{a}\cdots\int_{-1}^{1}dx_{N}\,(1-x_{N}^{2})^{a}\prod_{1\leq
j<k\leq N}|x_{k}-x_{j}|^{\beta}.$

For $\beta$ even, $\prod_{1\leq j<k\leq N}|x_{k}-x_{j}|^{\beta}$ is a
multivariable symmetric polynomial, so application of Proposition 2.1 gives
(2.14). ∎

######  Remark 2.3.

The explicit form of the analytic continuations in $\alpha$ of both sides of
(2.14) is known from (2.3) and (2.7). In the notation therein, the equality
(2.14) requires

$(-1)^{\beta N(N-1)/4}(2\pi)^{N}M_{N}(\alpha,\alpha,\beta/2)\\\
=(-\tan\pi\alpha)^{N}S_{N}(-\beta(N-1)/2-1-\alpha,-\beta(N-1)/2-1-\alpha,\beta/2).$
(2.15)

Under the assumption that $\beta$ is even, this can be checked upon the
manipulation $j\mapsto N-1-j$ in the product defining $S_{N}$, and then use of
the reflection equation for the gamma functions in that product. Agreement
with the LHS of (2.15) is obtained.

We can make use of Corollary 2.2 and further apply Proposition 2.1 to relate
the densities (2.1) and (2.5), along with the more general $k$-point
correlation functions

$\rho_{(k)}^{(Cy)}(x_{1},\ldots,x_{k}):=\frac{N!}{(N-k)!\,\mathcal{N}_{N}^{(Cy)}}\int_{-\infty}^{\infty}dx_{k+1}\cdots\\\
\cdots\int_{-\infty}^{\infty}dx_{N}\,\prod_{l=1}^{N}w_{\beta}^{(Cy)}(x_{l})\prod_{1\leqslant
j<k\leqslant N}|x_{k}-x_{j}|^{\beta},$ (2.16)
$\rho_{(k)}^{(J)}(x_{1},\ldots,x_{k}):=\frac{N!}{(N-k)!\,\mathcal{N}_{N}^{(J)}}\int_{-\infty}^{\infty}dx_{k+1}\cdots\\\
\cdots\int_{-\infty}^{\infty}dx_{N}\,\prod_{l=1}^{N}w_{\beta}^{(J)}(x_{l})\prod_{1\leqslant
j<k\leqslant N}|x_{k}-x_{j}|^{\beta}.$ (2.17)

######  Proposition 2.4.

In the setting of Corollary 2.2, the $k$-point correlation functions (2.16)
and (2.17) above are related by

$\rho_{(k)}^{(Cy)}(ix_{1},\ldots,ix_{k})=(-\cot\,\pi\alpha)^{k}\rho_{(k)}^{(J)}(x_{1},\ldots,x_{k})\Big{|}_{a=b=-\beta(N-1)/2-1-\alpha}.$
(2.18)

In particular, the densities (2.1) and (2.5), themselves being the $1$-point
correlation functions, obey the relation

$\rho_{(1)}^{(Cy)}(ix)=-\cot\pi\alpha\,\rho_{(1)}^{(J)}(x)\Big{|}_{a=b=-\beta(N-1)/2-1-\alpha}.$
(2.19)

###### Proof.

Since $\beta$ is an even integer, inserting the result of Corollary 2.2 into
(2.16) shows that the LHS of (2.18) is given by

$\frac{N!}{(N-k)!}\frac{(-1)^{\beta
N(N-1)/4}(-\cot\,\pi\alpha)^{N}}{\mathcal{N}_{N}^{(J)}\big{|}_{a=b=-\beta(N-1)/2-1-\alpha}}\prod_{l=1}^{k}w_{\beta}^{(Cy)}(ix_{l})\prod_{1\leqslant
h<j\leqslant k}(ix_{j}-ix_{h})^{\beta}\\\
\times\int_{-\infty}^{\infty}dx_{k+1}(1+x_{k+1}^{2})^{\alpha_{\beta,N}}\cdots\int_{-\infty}^{\infty}dx_{N}(1+x_{N}^{2})^{\alpha_{\beta,N}}\\\
\times\prod_{p=1}^{k}\prod_{q=k+1}^{N}(x_{q}-ix_{p})^{\beta}\prod_{k+1\leqslant
h^{\prime}<j^{\prime}\leqslant N}(x_{j^{\prime}}-x_{h^{\prime}})^{\beta}.$
(2.20)

From Proposition 2.1, interpreting the third line of (2.20) as a multivariable
symmetric polynomial in $\\{ix_{l}\\}_{l=k+1}^{N}$ shows that the $(N-k)$-fold
integral in the above is equal to

$\left(-\tan\,\pi\alpha\right)^{N-k}\int_{-1}^{1}dx_{k+1}(1-x_{k+1}^{2})^{\alpha_{\beta,N}}\cdots\int_{-1}^{1}dx_{N}(1-x_{N}^{2})^{\alpha_{\beta,N}}\\\
\times\prod_{p=1}^{k}\prod_{q=k+1}^{N}(ix_{q}-ix_{p})^{\beta}\prod_{k+1\leqslant
h^{\prime}<j^{\prime}\leqslant N}(ix_{j^{\prime}}-ix_{h^{\prime}})^{\beta},$

where we have additionally made the change of variables $x_{k+1}\mapsto-
x_{k+1},\ldots,x_{N}\mapsto-x_{N}$. Substituting this expression into (2.20),
extracting factors of $i$ from the products of differences to cancel the
factor of $(-1)^{\beta N(N-1)/4}$, and observing that
$w_{\beta}^{(J)}(x)\big{|}_{a=b=\alpha_{\beta,N}}=w_{\beta}^{(Cy)}(ix)$ gives
the RHS of (2.18). ∎

######  Remark 2.5.

Although established for $\beta$ even via reasoning based on Proposition 2.1,
an application of Carlson’s theorem as familiar in the theory of the Selberg
integral (see e.g. [11, §4.1]) shows that Proposition 2.4 remains true for
general $\beta>0$.

For the particular values of $\beta$ even, $\beta=2,4$, we will use the
identity (2.19) relating the density of the Cauchy ensemble for $\alpha$ real
to (an analytic continuation of) the density of the Jacobi ensemble supported
on $(-1,1)$ with $a=b$ and the same value of $\beta$ to study properties of
the former. This is presented in Section 3, but first we outline a
relationship between the Cauchy ensemble when ${\rm Im}\,\alpha\neq 0$ (also
known as the non-symmetric case or the generalised Cauchy ensemble) and the
Jacobi ensemble now requiring $a=\bar{b}$.

###  2.2. The non-symmetric case

A classical result of Cauchy [5] gives

$\int_{-\infty}^{\infty}{dt\over(1-it)^{\gamma}(1+it)^{\delta}}=2^{2-\gamma-\delta}\pi{\Gamma(\gamma+\delta-1)\over\Gamma(\gamma)\Gamma(\delta)}$
(2.21)

subject to the requirement that ${\rm Re}\,(\gamma+\delta)>1$; outside of this
range we consider the integral as defined by the analytic continuation given
by the RHS. Use of the reflection equation for the gamma function allows this
to be rewritten

$\int_{-\infty}^{\infty}(1-it)^{\gamma}(1+it)^{\delta}\,dt=2^{2+\gamma+\delta}{\sin\pi\gamma\,\sin\pi\delta\over\sin\pi(\gamma+\delta)}{\Gamma(\gamma+1)\Gamma(\delta+1)\over\Gamma(\gamma+\delta+2)},$
(2.22)

subject now to the requirement ${\rm Re}\,(\gamma+\delta)<-1$ on the LHS.

The form (2.22) is to be compared against the Euler beta function evaluation

$\int_{0}^{1}t^{c}(1-t)^{d}\,dt={\Gamma(c+1)\Gamma(d+1)\over\Gamma(c+d+2)}$
(2.23)

or, equivalently,

$\int_{-1}^{1}(1-t)^{c}(1+t)^{d}\,dt=2^{1+c+d}{\Gamma(c+1)\Gamma(d+1)\over\Gamma(c+d+2)},$
(2.24)

where on the LHS, it is required that ${\rm Re}\,c>-1$ and ${\rm Re}\,d>-1$.
The agreement in the gamma function dependence of both integrals allows for a
relation between multiple integrals analogous to that in Proposition 2.1 to be
derived.

######  Proposition 2.6.

Let $f(x_{1},\dots,x_{N})$ be a multivariable symmetric polynomial of degree
$\tilde{d}$ in each $x_{i}$. For ${\rm Re}\,(\gamma+\delta)<-\tilde{d}-1$
define

$\tilde{I}_{N,\gamma,\delta}^{(Cy)}[f(x_{1},\ldots,x_{N})]:=\int_{-\infty}^{\infty}dx_{1}\,(1-ix_{1})^{\gamma}(1+ix_{1})^{\delta}\cdots\\\
\cdots\int_{-\infty}^{\infty}dx_{N}\,(1-ix_{N})^{\gamma}(1+ix_{N})^{\delta}\,f(x_{1},\dots,x_{N}),$
(2.25)

and for $\gamma,\delta$ outside of this range, define
$\tilde{I}_{N,\gamma,\delta}^{(Cy)}[f(x_{1},\ldots,x_{N})]$ by its analytic
continuation. Also, in relation to the Jacobi weight with $c,d>-1$ define

$\tilde{I}_{N,c,d}^{(J)}[f(x_{1},\ldots,x_{N})]:=\int_{-1}^{1}dx_{1}\,(1-x_{1})^{c}(1+x_{1})^{d}\cdots\\\
\cdots\int_{-1}^{1}dx_{N}\,(1-x_{N})^{c}(1+x_{N})^{d}\,f(x_{1},\dots,x_{N}),$
(2.26)

and for $c,d$ outside of this range, define
$\tilde{I}_{N,c,d}^{(J)}[f(x_{1},\ldots,x_{N})]$ by its analytic continuation.
We have

$\tilde{I}_{N,\gamma,\delta}^{(Cy)}[f(1-ix_{1},\dots,1-ix_{N})]=\left(2{\sin\pi\gamma\,\sin\pi\delta\over\sin\pi(\gamma+\delta)}\right)^{N}\tilde{I}_{N,\gamma,\delta}^{(J)}[f(1-x_{1},\dots
1-x_{N})].$ (2.27)

###### Proof.

For $p\in\mathbb{Z}_{\geq 0}$, it follows from (2.22) and (2.24) upon setting
$c=\gamma$ and $d=\delta$ that

$\int_{-\infty}^{\infty}(1-it)^{\gamma+p}(1+it)^{\delta}\,dt=2^{2+\gamma+\delta+p}{\sin\pi\gamma\,\sin\pi\delta\over\sin\pi(\gamma+\delta)}{\Gamma(\gamma+p+1)\Gamma(\delta+1)\over\Gamma(\gamma+\delta+p+2)},$
(2.28)

and

$\int_{-1}^{1}(1-t)^{\gamma+p}(1+t)^{\delta}\,dt=2^{1+\gamma+\delta+p}{\Gamma(\gamma+p+1)\Gamma(\delta+1)\over\Gamma(\gamma+\delta+p+2)}.$
(2.29)

Hence, in the sense of analytic continuation,

$\int_{-\infty}^{\infty}(1-it)^{\gamma+p}(1+it)^{\delta}\,dt=2{\sin\pi\gamma\,\sin\pi\delta\over\sin\pi(\gamma+\delta)}\int_{-1}^{1}(1-t)^{\gamma+p}(1+t)^{\delta}\,dt.$
(2.30)

The stated result now follows from the assumption that $f(x_{1},\ldots,x_{N})$
in (2.25) and (2.26) is a polynomial and so the evaluation of the multiple
integrals reduces to the one-dimensional integrals (2.28) and (2.29), which
are related by (2.30).

∎

We can use Proposition 2.6 to relate the normalisations (2.3) and (2.7) in the
case that $\alpha$ in (2.2) is complex.

######  Corollary 2.7.

Let $\alpha$ be, in general, complex and related to $\alpha_{\beta,N}$ as in
(2.2). For $\beta$ even,

$(-1)^{\beta
N(N-1)/4}\mathcal{N}_{N}^{(Cy)}=\left(-2{\sin\pi\alpha\,\sin\pi\bar{\alpha}\over\sin\pi(\alpha+\bar{\alpha})}\right)^{N}\mathcal{N}_{N}^{(J)}\Big{|}_{\bar{a}=b=-\beta(N-1)/2-1-\alpha},$
(2.31)

where the RHS is to be regarded as defined by its analytic continuation.

###### Proof.

We observe that for $\beta$ even, the product of differences in the definition
of the normalisations is a polynomial, and moreover,

$(x_{k}-x_{j})^{\beta}=(-1)^{\beta/2}((1-ix_{k})-(1-ix_{j}))^{\beta}.$

The result now follows from the definitions of the normalisations and the
identity (2.27). ∎

######  Remark 2.8.

1\. Analogous to (2.15), the identity (2.31) can be cast in terms of the
integrals in (2.3) and (2.7), giving

$(-1)^{\beta N(N-1)/4}\pi^{N}M_{N}(\alpha,\bar{\alpha},\beta/2)\\\
=\left(-{\sin\pi\alpha\,\sin\pi\bar{\alpha}\over\sin\pi(\alpha+\bar{\alpha})}\right)^{N}S_{N}(-\beta(N-1)/2-1-\alpha,-\beta(N-1)/2-1-\bar{\alpha},\beta/2).$
(2.32)

This can be verified using the same steps as for (2.15).
2\. In the case $\alpha$ is real, (2.31) reduces to (2.14).

We can make use of Corollary 2.7 and a further application of Proposition 2.6
in the definitions of the $k$-point correlation functions (2.16) and (2.17) to
deduce the analogue of Proposition 2.4 in the non-symmetric case. We skip the
proof, which essentially amounts to combining the proofs of Proposition 2.4
and Corollary 2.7.

######  Proposition 2.9.

In the setting of Corollary 2.7,

$\rho_{(k)}^{(Cy)}(ix)=\left(-{\sin\pi(\alpha+\bar{\alpha})\over
2\sin\pi\alpha\,\sin\pi\bar{\alpha}}\right)^{k}\rho_{(k)}^{(J)}(x)\Big{|}_{\bar{a}=b=-\beta(N-1)/2-1-\alpha}.$
(2.33)

######  Remark 2.10.

1\. In the case $\alpha$ is real, (2.33) reduces to (2.19).
2\. For $\beta=1,2$ and $4$, there are expressions in terms of orthogonal
polynomials for both sides of (2.33) independently [1, 15]. These can checked
to be consistent with (2.33), using the fact that the orthogonal polynomials
associated with the Jacobi weight in (1.9) are the Jacobi polynomials
$P_{N}^{(a,b)}(x)$, while those associated with the Cauchy weight in (1.9) are
the Jacobi polynomials $i^{-N}P_{N}^{(\eta,\bar{\eta})}(ix)$.

##  3\. Differential equations

###  3.1. The symmetric case $\alpha$ real

In our previous work [36], a third order linear differential equation was
obtained for the Jacobi ensemble in the case $\beta=2$ defined with weight
supported on $(0,1)$ specified by

$\tilde{w}^{(J)}(x)=x^{a}(1-x)^{b}\chi_{0<x<1},$ (3.1)

and a fifth order differential equation for this version of the Jacobi
ensemble in the case $\beta=4$ or $\beta=1$. We note that the weight (3.1)
maps to the Jacobi weight in (2.6) by the change of variables
$x\mapsto(1-x)/2$. Making this mapping and furthermore setting $b=a$, we can
read off from [36, Thms. 2.1 and 2.2] the corresponding differential equations
satisfied by the density
$\rho_{(1)}^{(J)}(x)|_{a=b}=\rho_{(1)}^{(J)}(x;\beta)|_{a=b}$.

######  Proposition 3.1.

Define

$\mathcal{D}_{2,N}^{(J)}=(1-x^{2})^{3}{d^{3}\over
dx^{3}}-8x(1-x^{2})^{2}{d^{2}\over dx^{2}}\\\
-2(1-x^{2})[3-2N^{2}-4aN+(2(a+N)^{2}-7)x^{2}]{d\over dx}\\\
+4x(a^{2}+1-N^{2}-2aN+(a+N)^{2}x^{2}-x^{2})$ (3.2)

and for $\beta=1$ and $4$,

$\mathcal{D}_{\beta,N}^{(J)}=4(1-x^{2})^{5}{d^{5}\over
dx^{5}}-80x(1-x^{2})^{4}{d^{4}\over
dx^{4}}+(5\tilde{c}^{2}-493)(1-x^{2})^{4}{d^{3}\over dx^{3}}\\\
-4(5\tilde{a}-88)(1-x^{2})^{3}{d^{3}\over
dx^{3}}+16(11\tilde{a}-8)x(1-x^{2})^{2}{d^{2}\over
dx^{2}}-2(19\tilde{c}^{2}-539)x(1-x^{2})^{3}{d^{2}\over dx^{2}}\\\
+(\tilde{c}^{4}-64\tilde{c}^{2}+719)(1-x^{2})^{3}{d\over
dx}-8\left[(\tilde{c}^{2}-45)(\tilde{a}-3)-124\right](1-x^{2})^{2}{d\over
dx}\\\ +16\left[(\tilde{a}-7)^{2}-65\right](1-x^{2}){d\over
dx}-(\tilde{c}^{2}-9)^{2}x(1-x^{2})^{2}\\\
+4\left[4(\tilde{c}^{2}-9)+(3\tilde{c}^{2}-35)\tilde{a}\right]x(1-x^{2})-32\tilde{a}^{2}x,$
(3.3)

with

$\tilde{a}:={a\over\beta/2-1}\left({a\over\beta/2-1}-2\right),\quad\tilde{c}={2a\over\beta/2-1}+4(\beta/2-1)N-1,$

following [36]. Then for $\beta=1,2$ and $4$, we have

$\mathcal{D}_{\beta,N}^{(J)}\,\rho_{(1)}^{(J)}(x;\beta)|_{a=b}=0.$ (3.4)

######  Remark 3.2.

Analogous to the differential equations for the densities of classical
$\beta$-ensembles considered in [36], the differential equations in
Proposition 3.1 have a unique (up to proportionality) solution of the form of
$w_{\beta}^{(J)}(x)$ times a polynomial of degree $\beta(N-1)$. The
coefficients in the latter can be determined by expanding it about infinity,
substituting the assumed form in the DE, and equating appropriate powers.

According to Proposition 2.4, the differential equation satisfied by
$\rho_{(1)}^{(Cy)}(x)$ for $\beta$ even can be obtained from the one satisfied
by $\rho_{(1)}^{(J)}(x)|_{a=b}$, provided we set $a=-\beta(N-1)/2-1-\alpha$
and replace $x$ by $ix$. Doing this in Proposition 3.1 gives us a third order
linear differential equation satisfied by
$\rho_{(1)}^{(Cy)}(x)=\rho_{(1)}^{(Cy)}(x;\beta)$ in the case $\beta=2$, and a
fifth order equation for $\beta=4$. Furthermore, Remark 2.5 tells us that the
latter differential equation is valid for $\beta=1$, after reparametrising
properly.

######  Proposition 3.3.

Define

$\mathcal{D}_{2,N}^{(Cy)}=(1+x^{2})^{3}{d^{3}\over
dx^{3}}+8x(1+x^{2})^{2}{d^{2}\over dx^{2}}\\\
+2(1+x^{2})[3+2N(N+2\alpha)+(7-2\alpha^{2})x^{2}]{d\over
dx}+4x(1+\alpha^{2}+2N(N+2\alpha)+(1-\alpha^{2})x^{2})$ (3.5)

and for $\beta=1$ and $4$,

$\mathcal{D}_{\beta,N}^{(Cy)}=4(1+x^{2})^{5}{d^{5}\over
dx^{5}}+80x(1+x^{2})^{4}{d^{4}\over
dx^{4}}-4(5\tilde{\alpha}-122)(1+x^{2})^{4}{d^{3}\over dx^{3}}\\\
+4(5\tilde{N}-93)(1+x^{2})^{3}{d^{3}\over
dx^{3}}-8(19\tilde{\alpha}-130)x(1+x^{2})^{3}{d^{2}\over
d^{2}x}+16(11\tilde{N}-19)x(1+x^{2})^{2}{d^{2}\over d^{2}x}\\\
+8(2\tilde{\alpha}^{2}-31\tilde{\alpha}+82)(1+x^{2})^{3}{d\over
dx}-32\left[(\tilde{\alpha}-11)(\tilde{N}-4)-31\right](1+x^{2})^{2}{d\over
dx}\\\ +16\left[(\tilde{N}-8)^{2}-65\right](1+x^{2}){d\over
dx}+16(\tilde{\alpha}-2)^{2}x(1+x^{2})^{2}\\\
-16\left[(3\tilde{\alpha}-8)\tilde{N}+\tilde{\alpha}\right]x(1+x^{2})+32(\tilde{N}-1)^{2}x$
(3.6)

with

$\tilde{\alpha}:={\alpha\over\beta/2-1}\left({\alpha\over\beta/2-1}-1\right),\qquad\tilde{N}:=\left(2(\beta/2-1)N+{\alpha\over\beta/2-1}\right)^{2}.$
(3.7)

Then for $\beta=1,2$ and $4$, we have

$\mathcal{D}_{\beta,N}^{(Cy)}\,\rho_{(1)}^{(Cy)}(x;\beta)=0.$ (3.8)

####  3.1.1. Relation to Painlevé equations in the symmetric case

Let $\\{\rho_{(k)}^{(Cy)}\\}$ denote the $k$-point correlations for the Cauchy
ensemble with weight (2.2). It is well known (see e.g. [11, Ch. 9]) that the
generating function $E_{N}^{(Cy)}((s,\infty);\xi)$ for the probabilities
$\\{E_{N}^{(Cy)}(k;(s,\infty))\\}_{k=0}^{N}$ of there being exactly $k$
eigenvalues in the interval $(s,\infty)$ can be written in terms of the
correlations according to

$E_{N}^{(Cy)}((s,\infty);\xi)=1+\sum_{k=1}^{N}{(-\xi)^{k}\over
k!}\int_{s}^{\infty}dx_{1}\cdots\int_{s}^{\infty}dx_{k}\,\rho_{(k)}^{(Cy)}(x_{1},\dots,x_{k}).$
(3.9)

In the case $\beta=2$, it is known [40] that

$\sigma(s):=(1+s^{2}){d\over ds}\log E_{N}^{(Cy)}((s,\infty);\xi)$ (3.10)

satisfies the nonlinear equation (which can be identified in terms of the
$\sigma$-PVI equation [20]; see also (3.29) below)

$(1+s^{2})^{2}(\sigma^{\prime\prime})^{2}+4(1+s^{2})(\sigma^{\prime})^{3}-8s\sigma(\sigma^{\prime})^{2}\\\
+4\sigma^{2}(\sigma^{\prime}-\alpha^{2})+8\alpha^{2}s\sigma\sigma^{\prime}+4[N(N+2\alpha)-\alpha^{2}s^{2}](\sigma^{\prime})^{2}=0.$
(3.11)

Note that (3.11) is independent of the parameter $\xi$ in (3.9).

According to (3.9), to leading order in $\xi$,

$\sigma(s)=\xi r(s)+O(\xi^{2}),\quad r(s):=(1+s^{2})\rho_{(1)}^{(Cy)}(s).$
(3.12)

Substituting in (3.11) and equating terms to the leading order in $\xi$ (which
is $O(\xi^{2})$) shows

$(1+s^{2})^{2}(r^{\prime\prime}(s))^{2}-4\alpha^{2}(r(s))^{2}+8\alpha^{2}sr(s)r^{\prime}(s)+4[N(N+2\alpha)-\alpha^{2}s^{2}](r^{\prime}(s))^{2}=0.$
(3.13)

Upon differentiating with respect to $s$, a factor of $r^{\prime\prime}(s)$
can be cancelled and a third order linear differential equation results,

$(1+s^{2})^{2}r^{\prime\prime\prime}(s)+2s(1+s^{2})r^{\prime\prime}(s)+4[N(N+2\alpha)-\alpha^{2}s^{2}]r^{\prime}(s)+4\alpha^{2}sr(s)=0.$
(3.14)

Recalling the definition of $r(s)$ in terms of $\rho_{(1)}^{(Cy)}(s)$, we see
that (3.14) is equivalent to the third order differential equation given in
Proposition 3.3.

The use of the characterisation of gap probabilities in terms of
$\sigma$-Painlevé transcendents to derive third order differential equations
for the densities of other classical ensembles at $\beta=2$ can be found in
[18, 36].

######  Remark 3.4.

The $\beta=1,4$ analogue of (3.14), obtained by taking
$\rho_{(1)}^{(Cy)}(s)=r(s)/(1+s^{2})$ in (3.6), is

$(1+s^{2})^{4}r^{(5)}(s)+10s(1+s^{2})^{3}r^{(4)}(s)-(5\tilde{\alpha}-22)(1+s^{2})^{3}r^{\prime\prime\prime}(s)\\\
+(5\tilde{N}-13)(1+s^{2})^{2}r^{\prime\prime\prime}(s)-8(\tilde{\alpha}-1)s(1+s^{2})^{2}r^{\prime\prime}(s)\\\
+2(7\tilde{N}+1)s(1+s^{2})r^{\prime\prime}(s)+4\tilde{\alpha}^{2}(1+s^{2})^{2}r^{\prime}(s)-2\left[(4\tilde{\alpha}-1)\tilde{N}+1\right](1+s^{2})r^{\prime}(s)\\\
+4(\tilde{N}-1)^{2}r^{\prime}(s)-4\tilde{\alpha}^{2}s(1+s^{2})r(s)+4\tilde{\alpha}(\tilde{N}-1)sr(s)=0,$
(3.15)

where $\tilde{\alpha}$ and $\tilde{N}$ are as given in Proposition 3.3. In
comparing (3.14) and (3.15) to their counterparts in Proposition 3.3, we see
that the degree of each of the coefficients (which alternate between being
even in $s$, and being odd in $s$) has been reduced by two.

####  3.1.2. The spectrum singularity scaling regime

It is well known that under the stereographic transformation $s=\cot\theta/2$,
the Cauchy ensemble maps to the circular Jacobi ensemble with weight

$|1-e^{i\theta}|^{2\alpha},\quad\theta\in[0,2\pi)$ (3.16)

and thus eigenvalue PDF proportional to

$\prod_{l=1}^{N}|1-e^{i\theta_{l}}|^{2\alpha}\prod_{1\leq j<k\leq
N}|e^{i\theta_{k}}-e^{i\theta_{j}}|^{\beta};$ (3.17)

see e.g. [11, §2.5], [32]. In particular, the densities in the two ensembles
are related by

$\rho_{(1)}^{(Cy)}(s)=2\sin^{2}{\theta\over 2}\,\rho_{(1)}^{(cJ)}(\theta).$
(3.18)

Equivalently, in the notation $r(s)$ of (3.12),

$r(s)=2\rho_{(1)}^{(cJ)}(\theta).$ (3.19)

In the case $\alpha=0$, the weight (3.16) is a constant and so according to
(3.19), $r(s)$ is then also a constant. We can see immediately that this is
consistent with equations (3.14) and (3.15).

Suppose now $\theta$ is scaled by writing $\theta=2X/N$. The circular Jacobi
ensemble then admits a well defined scaling limit as $N\to\infty$, giving rise
to what is termed a spectrum singularity at $\theta=0$ [11, §3.9]. In view of
(3.19), in the cases $\beta=1,2$ and $4$, differential equations for the
corresponding density $\rho_{(1)}^{(s.s.)}(X)$ can be obtained by setting

$s=N/X$ (3.20)

and $r(s)=\rho_{(1)}^{(s.s.)}(X)$ in (3.14) and (3.15), and then equating
terms at leading order in $N$. Specifically for $\beta=2$, we therefore have
that $R(X)=\rho_{(1)}^{(s.s.)}(X)$ satisfies the third order linear
differential equation

$X^{2}R^{\prime\prime\prime}(X)+4XR^{\prime\prime}(X)+(2-4\alpha^{2}+4X^{2})R^{\prime}(X)-{4\alpha^{2}\over
X}R(X)=0.$ (3.21)

This is consistent with the known exact formula [35], [11, Eq.(7.49) with
$\pi\rho=1$]

$\rho_{(1)}^{(s.s.)}(x)={x\over
2}\Big{(}(J_{\alpha-1/2}(x))^{2}+(J_{\alpha+1/2}(x))^{2}-{2\alpha\over
x}J_{\alpha-1/2}(x)J_{\alpha+1/2}(x)\Big{)},$ (3.22)

as can be checked using computer algebra.

####  3.1.3. The global scaling of the symmetric Cauchy density

The scaling (3.20) corresponds to a spacing of order unity for eigenvalues in
the neighbourhood of the spectrum singularity. This is in contrast to the
global scaling of the density,

$\rho_{(1)}^{(glob.\,Cy)}(s):=\lim_{N\to\infty}{1\over
N}\rho_{(1)}^{(Cy)}(s)\Big{|}_{\alpha=\hat{\alpha}N},\quad\hat{\alpha}\textrm{
constant in }N,$ (3.23)

where the spacing between eigenvalues is of order $1/N$. Upon replacing
$\alpha$ by $\hat{\alpha}N$ in (3.14), equating the leading order terms in $N$
gives

$((1+2\hat{\alpha})-\hat{\alpha}^{2}s^{2})r^{\prime}(s)+\hat{\alpha}^{2}sr(s)=0.$
(3.24)

Solving this first order equation, and making use of the definition of $r(s)$
from (3.12), we conclude for $\beta=2$ that

$\rho_{(1)}^{(glob.\,Cy)}(s)=A{\sqrt{1+2\hat{\alpha}-\hat{\alpha}^{2}s^{2}}\over
1+s^{2}}\chi_{|s|<\sqrt{1+2\hat{\alpha}}/\hat{\alpha}},\quad A={1\over\pi},$
(3.25)

where the scalar $A$ has been determined by the requirement that the RHS
integrate to unity. Equation (3.25) is expected to hold universally for
$\beta>0$, as the limiting densities of the other classical ensembles are
known to be independent of $\beta$. This is in fact known from the equivalence
(3.18) of the Cauchy ensemble to the circular Jacobi ensemble upon the
stereographic transformation $s=\cot\theta/2$. Thus, it has been established
[4, 26] that for all $\beta>0$, the limiting equilibrium density for the
circular Jacobi ensemble with $\alpha=\hat{\alpha}\beta N/2$ is given by the
stereographic transformation of (3.25),

$\rho_{(1)}^{(glob.\,cJ)}(\theta)={\hat{\alpha}\over
2\pi}\sqrt{\cot^{2}{\theta_{c}\over 2}-\cot^{2}{\theta\over
2}}\,\chi_{\theta_{c}<\theta<2\pi-\theta_{c}}$ (3.26)

where

$\theta_{c}=\min_{\theta\in[0,2\pi)}\\{\theta\,:\,\cot^{2}\theta/2=(1+2\hat{\alpha})/\hat{\alpha}^{2}\\}.$

###  3.2. The non-symmetric case Im$\,\alpha\neq 0$

In the symmetric case, differential equations for $\beta=1,2$ and $4$ were
considered. This is again possible in the non-symmetric case; however, the
complexity of the case $\beta=4$ (equivalently $\beta=1$) increases to the
extent that it becomes cumbersome to present (for the Jacobi ensemble defined
on $(0,1)$ its explicit form is given in [36, Th. 2.2]), so we will restrict
attention to the case $\beta=2$. The first task is to change variables
$x\mapsto(1-x)/2$ in the known third order linear differential equation for
the density in the Jacobi ensemble on $(0,1)$ with $\beta=2$ [36, Th. 2.1].
Unlike the symmetric case considered above, we no longer set $a=b$, so this
does not change the differential equation of [36] in any essential way, and
thus the result of this calculation will not be recorded here. However, its
explicit form is required in what comes next. Specifically, starting from the
explicit form and setting $a=-N-\bar{\alpha}$, $b=-N-\alpha$, and replacing
$x$ by $ix$, it follows from (2.33) that we obtain the differential equation
satisfied by $\rho_{(1)}^{(Cy)}(x)$.

######  Proposition 3.5.

Define the third order differential operator

$\tilde{\mathcal{D}}_{2,N}^{(Cy)}=(1+x^{2})^{3}{d^{3}\over
dx^{3}}+8x(1+x^{2})^{2}{d^{2}\over dx^{2}}\\\
+(1+x^{2})[6+4N(N+\alpha+\bar{\alpha})+(\alpha-\bar{\alpha})^{2}+2i(\alpha-\bar{\alpha})(2N+\alpha+\bar{\alpha})x+(14-(\alpha+\bar{\alpha})^{2})x^{2}]{d\over
dx}\\\
+[4+8N(N+\alpha+\bar{\alpha})+3\alpha^{2}-2\alpha\bar{\alpha}+3\bar{\alpha}^{2}+(4-(\alpha+\bar{\alpha})^{2})x^{2}]x\\\
+i(\alpha-\bar{\alpha})(2N+\alpha+\bar{\alpha})(3x^{2}-1).$ (3.27)

For $\beta=2$, we have

$\tilde{\mathcal{D}}_{2,N}^{(Cy)}\,\rho_{(1)}^{(Cy)}(x)=0.$ (3.28)

####  3.2.1. Relation to Painlevé equations in the non-symmetric case

After a simple change of variables, the Jimbo-Miwa-Okamoto $\sigma$-form of
the Painlevé differential equation reads [20, Eq. (1.32)]

$h^{\prime}\Big{(}(1+t^{2})h^{\prime\prime}\Big{)}^{2}+4\Big{(}h^{\prime}(h-th^{\prime})-ib_{1}b_{2}b_{3}b_{4}\Big{)}^{2}+4\prod_{k=1}^{4}(h^{\prime}+b_{k}^{2})=0.$
(3.29)

Let $\mathbf{b}=(b_{1},b_{2},b_{3},b_{4})$ and define
$e_{2}^{\prime}[\mathbf{b}]$, $e_{2}[\mathbf{b}]$ as the elementary symmetric
polynomials of degree two in $\\{b_{1},b_{3},b_{4}\\}$ and
$\\{b_{1},b_{2},b_{3},b_{4}\\}$, respectively. Set

$U_{N}^{(Cy)}(t;(\alpha_{1},\alpha_{2});\xi)=(t^{2}+1){d\over
dt}\log\Big{(}(it-1)^{e_{2}^{\prime}[\mathbf{b}]-e_{2}[\mathbf{b}]/2}(it+1)^{e_{2}[\mathbf{b}]/2}E_{N}^{(Cy)}((t,\infty);\xi))\Big{)},$
(3.30)

where $E_{N}^{(Cy)}$ is specified by (3.9) in the non-symmetric case with
$\alpha=\alpha_{1}+i\alpha_{2}$ ($\alpha_{2}\neq 0$). We have from [20, Prop.
15] that $U_{N}^{(Cy)}$ satisfies the transformed $\sigma$-PVI equation (3.29)
with parameters

$\mathbf{b}=(-\alpha_{1},-i\alpha_{2},N+\alpha_{1},\alpha_{1}).$ (3.31)

Analogous to (3.12), we have from (3.9), (3.30) and (3.31) that

$U_{N}^{(Cy)}(t;(\alpha_{1},\alpha_{2});\xi)=(N+\alpha_{1})\alpha_{2}-\alpha_{1}^{2}t+\xi
r(t)+O(\xi^{2}),\qquad r(t):=(1+t^{2})\rho_{(1)}^{(Cy)}(t).$ (3.32)

Substituting (3.32) in (3.29) and equating terms of leading order in $\xi$,
which as for (3.12) occurs at order $\xi^{2}$, then differentiating and
cancelling a factor of $r^{\prime\prime}$ shows

$(1+t^{2})^{2}r^{\prime\prime\prime}+2t(1+t^{2})r^{\prime\prime}+4[(\alpha_{1}^{2}t+(N+\alpha_{1})\alpha_{2})(r-tr^{\prime})\\\
+((N+\alpha_{1})^{2}-(N+\alpha_{1})\alpha_{2}t-\alpha_{1}^{2}-\alpha_{2}^{2})r^{\prime}]=0.$
(3.33)

In the case $\alpha_{2}=0$, this agrees with (3.14). Now, substituting for
$r(t)$ in terms of $\rho_{(1)}^{(Cy)}(t)$ as specified in (3.32) reclaims
(3.27) and (3.28).

####  3.2.2. The global scaling of the non-symmetric Cauchy density

The global scaling of the density $\rho_{(1)}^{(glob.\,Cy)}(s)$ specified by
(3.23) with $\hat{\alpha}=\hat{\alpha}_{1}+i\hat{\alpha}_{2}$, generalising
the working of §3.1.3, is deduced from (3.33) by setting
$\alpha_{1}=\hat{\alpha}_{1}N$, $\alpha_{2}=\hat{\alpha}_{2}N$, equating terms
of leading order in $N$, solving the resulting first order differential
equation, and finally substituting for $r(t)$ in terms of
$\rho_{(1)}^{(Cy)}(t)$ as specified in (3.32). The final result is

$\rho_{(1)}^{(glob.\,Cy)}(s)=A{\sqrt{(u_{+}-s)(s-u_{-})}\over
1+s^{2}}\chi_{u_{-}<s<u_{+}},$ (3.34)

where

$u_{\pm}={-(1+\hat{\alpha}_{1})\hat{\alpha}_{2}\pm\sqrt{(\hat{\alpha}_{1}^{2}+\hat{\alpha}_{2}^{2})(1+2\hat{\alpha}_{1})}\over\hat{\alpha}_{1}^{2}},\quad
A={\hat{\alpha}_{1}\over\pi}.$ (3.35)

As expected, setting $\hat{\alpha}_{2}=0$ in (3.34) reclaims (3.25). Also, as
with (3.25), the transformation of (3.34) to the unit circle is known in the
context of the study of the circular Jacobi ensemble [4, Eq. (5.6)].

##  4\. Moments

###  4.1. Moment recurrences for the symmetric cases

Restricting attention at first to non-negative integer moments, in the
symmetric Jacobi ensemble, only the even moments are non-zero. The change of
variables $x\mapsto 1-2x$ shows that in this setting,

$\Big{\langle}\sum_{l=1}^{N}x_{l}^{2k}\Big{\rangle}_{(-1,1)}^{(J)}=\Big{\langle}\sum_{l=1}^{N}(1-2x_{l})^{2k}\Big{\rangle}_{(0,1)}^{(J)}=\sum_{s=0}^{2k}\binom{2k}{s}(-2)^{s}\Big{\langle}\sum_{l=1}^{N}x_{l}^{s}\Big{\rangle}_{(0,1)}^{(J)},$
(4.1)

where in the second and third expression, the Jacobi ensemble is specified by
the weight (3.1) supported on $(0,1)$ (as indicated by the notation) and with
$a=b$; the second equality follows from the binomial expansion. The averages
on the RHS for $s=0$ and $s=1$ are immediate,

$\Big{\langle}\sum_{l=1}^{N}x_{l}^{s}\Big{\rangle}_{(0,1)}^{(J)}\Big{|}_{s=0}=N,\qquad\Big{\langle}\sum_{l=1}^{N}x_{l}^{s}\Big{\rangle}_{(0,1)}^{(J)}\Big{|}_{s=1}={N\over
2},$ (4.2)

as follows from the normalisation and symmetry of the distribution about
$x_{l}=1/2$, respectively. For $s\geq 2$ the averages on the RHS of (4.1) are
known from Jack polynomial theory; see [33, Eq. (53)] and [21, Eqns.
(33)-(35)]. In particular, we read off from Eq. (189) of [21] (must multiply
this equation by $N$ and replace $\beta^{2}$ by $-\beta/2$ for the present
setting) and Eq. (B.7b) of [33] that

$\Big{\langle}\sum_{l=1}^{N}x_{l}^{2}\Big{\rangle}_{(0,1)}^{(J)}=N\\\
\times{2a^{2}+a[-(\beta/2)(6-5N)+6]+(\beta/2)^{2}(N-1)(3N-4)-(\beta/2)(9-7N)+4\over
2(2a-(\beta/2)(3-2N)+2)(2a+\beta(N-1)+3)}.$ (4.3)

For integer values of $k$, the Jack polynomial formula leading to (4.3)
reveals that the $2k$-th moment is a rational function in $N$. The recent work
[6] has drawn attention to analogous features of the moments as a function of
$k$ in the complex plane for the Gaussian, Laguerre and non-symmetric Jacobi
(the latter defined on $(0,1)$) classical ensembles. Moreover, for $\beta=2$,
it was found that this function of $k$ can be factored in terms of some gamma
functions times a particular hypergeometric polynomial of degree $N-1$ from
the Askey table. In a subsequent work [2], these considerations were extended
to the symmetric Cauchy ensemble with $\beta=2$, which involved the continuous
Hahn polynomials. Taking the viewpoint of Section 2 that integrations over the
Cauchy ensemble can be obtained as corollaries of integrations over the Jacobi
ensemble defined on $(-1,1)$, our aim in this section is to give a self-
contained derivation of results of this type for the symmetric Jacobi ensemble
with $\beta=2$. The essential idea is that the differential equations of
Section 3 lead to recurrences for the moments.

First, note from (1.11) that for a general $\beta$-ensemble with $\beta$ even,
we have

$\rho_{(1)}(x)=w_{\beta}(x)q_{\beta(N-1)/2}(x^{2})$ (4.4)

for some polynomial $q_{\beta(N-1)/2}(y)$ of degree $\beta(N-1)/2$.
Specialising now to the symmetric Jacobi weight (2.6) with $a=b$, and making
use of the integration formula (2.13), it then follows that

$\int_{-1}^{1}|x|^{2k}\rho_{(1)}^{(J)}(x)\,dx={\Gamma(k+1/2)\over\Gamma(k+\beta(N-1)/2+a+3/2)}Q_{\beta(N-1)/2}(k)$
(4.5)

for some polynomial $Q_{\beta(N-1)/2}(y)$ of degree $\beta(N-1)/2$.

It turns out that the recurrences for the symmetric Jacobi ensemble’s moments
simplify if written in terms of the differences of successive even moments

$\mu_{k}^{(J)}=m_{2k+2}^{(J)}-m_{2k}^{(J)},\qquad
m_{2k}^{(J)}:=\int_{-1}^{1}|x|^{2k}\rho_{(1)}^{(J)}(x)|_{a=b}\,dx.$ (4.6)

Note that as a function of $k$, this is well defined for ${\rm Re}\,k>-1/2$.
The equivalent quantities in the symmetric Cauchy case are the sums of
successive even moments

$\mu_{k}^{(Cy)}=m_{2k+2}^{(Cy)}+m_{2k}^{(Cy)},\qquad
m_{2k}^{(Cy)}:=\int_{-\infty}^{\infty}|x|^{2k}\rho_{(1)}^{(Cy)}(x)\,dx.$ (4.7)

For $\mu_{k}^{(Cy)}$ to be well defined, we require $-1/2<{\rm
Re}\,k<\alpha-1/2$, the upper bound now necessary due to the domain of
integration being non-compact. The respective differences and sums are
equivalent in the sense that, via analytic continuation,

$\mu_{k}^{(J)}\Big{|}_{a\mapsto-\beta(N-1)/2-1-\alpha}=(-1)^{k-1}\mu_{k}^{(Cy)},$
(4.8)

as is consistent with Proposition 2.1.

In relation to the difference (4.6), the natural quantity is

$r^{(J)}(x)=(1-x^{2})\rho_{(1)}^{(J)}(x)|_{a=b}.$ (4.9)

It follows from Proposition 3.1 that for $\beta=2$, $r^{(J)}$ satisfies the
differential equation

$(1-x^{2})^{2}r^{\prime\prime\prime}(x)-2x(1-x^{2})r^{\prime\prime}(x)+4[N(N+2a)-(N+a)^{2}x^{2}]r^{\prime}(x)+4(a+N)^{2}xr(x)=0,$
(4.10)

while for $\beta=1$ and $4$, $r^{(J)}$ instead satisfies

$4(1-x^{2})^{4}r^{(5)}(x)-40x(1-x^{2})^{3}r^{(4)}+(5\tilde{c}^{2}-93)(1-x^{2})^{3}r^{\prime\prime\prime}(x)\\\
-4(5\tilde{a}-8)(1-x^{2})^{2}r^{\prime\prime\prime}(x)-8(\tilde{c}^{2}+5)x(1-x^{2})^{2}r^{\prime\prime}(x)+8(7\tilde{a}+18-10x^{2})x(1-x^{2})r^{\prime\prime}(x)\\\
+(\tilde{c}^{2}-1)^{2}(1-x^{2})^{2}r^{\prime}(x)-8\left[\tilde{c}^{2}(\tilde{a}+1)-2\tilde{a}-1\right](1-x^{2})r^{\prime}(x)\\\
+16\tilde{a}^{2}r^{\prime}(x)+(\tilde{c}^{2}-1)\left[(\tilde{c}^{2}-1)(1-x^{2})-4\tilde{a}\right]xr(x)=0,$
(4.11)

with $\tilde{a}$ and $\tilde{c}$ as given in Proposition 3.1. These
differential equations can formally be obtained from (3.14) and (3.15) by the
mappings $x\mapsto ix$, $\alpha\mapsto-\beta(N-1)/2-1-a$, or equivalently, one
can simply take $r^{(J)}(x)=\rho_{(1)}^{(J)}(x)/(1-x^{2})$ in equations
(3.2)–(3.4). Note the simplification relative to (3.2) and (3.3). From these
differential equations, second ($\beta=2$) and fourth ($\beta=1,4$) order
recurrences can be derived for $\\{\mu_{k+j}^{(J)}\\}_{j\in\mathbb{Z}}$. The
method, which is based on integration by parts, requires that the symmetric
Jacobi parameter $a$ be greater than zero, although this condition is not
necessary, and the recurrence correctly specifies the sequence for all values
that it is well defined.

######  Proposition 4.1.

Define $\mu_{k}^{(J)}$ by (4.6) and recall the definitions of $\tilde{a}$ and
$\tilde{c}$ given in Proposition 3.1. For $\beta=2$, the sequence
$\\{\mu_{k+j}^{(J)}\\}_{j\in\mathbb{Z}}$ satisfies the recurrence

$(2k+4)[(2k+3)^{2}-4(a+N)^{2}]\mu_{k+1}-2(2k+1)[(2k+2)^{2}-2N(N+2a)]\mu_{k}\\\
+(2k+1)(2k)(2k-1)\mu_{k-1}=0.$ (4.12)

The initial condition determining the sequence
$\\{\mu_{k}^{(J)}|_{\beta=2}\\}_{k=0}^{\infty}$ is

$\mu_{0}^{(J)}={2N(a+N)(2a+N)\over 1-4(a+N)^{2}}.$ (4.13)

For $\beta=1$ and $4$, the sequence $\\{\mu_{k+j}^{(J)}\\}_{j\in\mathbb{Z}}$
satisfies the recurrence

$\sum_{l=-2}^{2}f_{l}\,\mu_{k+l}^{(J)}=0,$ (4.14)

where

$\displaystyle f_{-2}$ $\displaystyle:=-4(2k+1)(2k)(2k-1)(2k-2)(2k-3),$
$\displaystyle f_{-1}$
$\displaystyle:=(2k+1)(2k)(2k-1)\left[20\tilde{a}-5\tilde{c}^{2}+16k(4k+5)+77\right],$
$\displaystyle f_{0}$
$\displaystyle:=(2k+1)\Big{[}8\tilde{a}(5k+8)(2k+3)-\left(\tilde{c}^{2}-4\tilde{a}-30k^{2}-67k-42\right)^{2}$
$\displaystyle\qquad+516k^{4}+2292k^{3}+3653k^{2}+2534k+673\Big{]},$
$\displaystyle f_{1}$
$\displaystyle:=\tilde{c}^{2}\left[(\tilde{c}^{2}-4\tilde{a})(4k+7)-120k^{3}-656k^{2}-1210k-746\right]$
$\displaystyle\quad+\tilde{a}\left[160k^{3}+736k^{2}+1128k+580\right]$
$\displaystyle\quad+512k^{5}+4480k^{4}+16056k^{3}+29360k^{2}+27270k+10243,$
$\displaystyle f_{2}$
$\displaystyle:=-2(k+3)(\tilde{c}+4k+11)(\tilde{c}+2k+4)(\tilde{c}-2k-4)(\tilde{c}-4k-11).$

The initial conditions determining the sequences
$\\{\mu_{k}^{(J)}|_{\beta=1,4}\\}_{k=0}^{\infty}$ are

$\displaystyle\mu_{0}^{(J)}$
$\displaystyle=\frac{(\tilde{c}-1)\left(\tilde{c}+\tfrac{2a}{\beta/2-1}-3\right)\left(\tilde{c}-\tfrac{2a}{\beta/2-1}+1\right)}{8\tilde{c}(\tilde{c}-3)(1-\beta/2)},$
(4.15) $\displaystyle\mu_{1}^{(J)}$
$\displaystyle=\frac{(\tilde{c}^{2}-5)(\tilde{c}-7)-4\tilde{a}(\tilde{c}-1)}{4(\tilde{c}^{2}-4)(\tilde{c}-7)}\mu_{0}^{(J)}.$
(4.16)

###### Proof.

We first give the details for the $\beta=2$ recurrence (4.12). Begin by
multiplying (3.28) by $x|x|^{2k}$ and then integrate over $x$ from $-1$ to
$1$. From the definition (4.6), the final term of the differential equation
transforms to $-4(a+N)^{2}\mu_{k+1}.$ For the other terms, integration by
parts is required to reduce the integrations to the form of (4.6). Due to the
absolute value sign in the multiplying term $x|x|^{2k}$, this requires
considering the intervals $(0,1)$ and $(-1,0)$ separately. The endpoints do
not contribute to the integration. At the origin this is due to the term
$x|x|^{2k}$ with $k>-1/2$, and at $\pm 1$, there is no contribution since
$\rho_{(1)}^{(J)}(x)|_{a=b}$ is then zero for $a>0$. After simplification,
(4.12) results.

To obtain recurrence (4.14), apply these same steps to equation (4.11). For
both recurrences, the value of $\mu_{0}^{(J)}$ follows from (4.1)–(4.3), while
$\mu_{1}^{(J)}$ can be computed using MOPS [8] and equation (4.1). ∎

######  Remark 4.2.

1\. The analogue of (4.10) for the $\beta=2$ non-symmetric Jacobi case can be
obtained from (3.33) by the mappings $t\mapsto-ix$,
$\alpha_{1}\mapsto-N-(a+b)/2$ and $\alpha_{2}\mapsto(b-a)i/2$. Repeating the
working which gave (4.12) leads to a recurrence which also involves
$\mu_{k+1/2}$. In fact, this recurrence is given in [30, Eq. (31)], which
reduces to (4.12) upon setting $a=b$. For $\beta=1$ and $4$, one may take
existing moment recurrences for the non-symmetric Jacobi ensemble on $(0,1)$
[36] and apply equation (4.1).
2\. Scaling $x\mapsto x/\sqrt{2a}$ in the definition (4.6) of $m_{2k}^{(J)}$
shows

$m_{2k}^{(J)}={1\over(2a)^{k+1/2}}\int_{-\sqrt{a}}^{\sqrt{a}}|x|^{2k}\rho_{(1)}^{(J)}(x/\sqrt{2a})\Big{|}_{a=b}\,dx.$

Now, from the elementary limit $(1-x^{2}/2a)^{a}\to e^{-x^{2}/2}$ as
$a\to\infty$, it follows from the definition of $\mu_{k}^{(J)}$ that

$\lim_{a\to\infty}(2a)^{k+1/2}\mu_{k}^{(J)}=-m_{2k}^{{\rm GUE}^{*}},$

where GUE∗ refers to the ensemble (1.11) with weight $w(x)=e^{-x^{2}/2}$ and
$\beta=2$. Multiplying (4.12) by $(2a)^{k-1/2}$ and taking $a\to\infty$ thus
implies the recurrence for $\\{m_{2k}^{{\rm GUE}^{*}}\\}$,

$-(k+2)m_{2k+2}^{{\rm GUE}^{*}}+2N(2k+1)m_{2k}^{{\rm
GUE}^{*}}+k(2k+1)(2k-1)m_{2k-2}^{{\rm GUE}^{*}}=0,$ (4.17)

found originally by Harer and Zagier [25]. Likewise, multiplying (4.14) by
$(2a)^{k-3/2}$ and taking $a\to\infty$ recovers the recurrences for the GOE
and GSE moments given in [31, Thm. 2], [41, Thms. 11 and 17].

As a simple consequence of the relation (4.8), Proposition 4.1 leads to
recurrences on the sums of successive even moments $\mu_{k}^{(Cy)}$ of the
$\beta=1,2$ and $4$ Cauchy ensembles.

######  Corollary 4.3.

Define $\mu_{k}^{(Cy)}$ by (4.7) and retain the definitions of
$f_{-2},\ldots,f_{2}$ given in Proposition 4.1. For $\beta=2$, the sequence
$\\{\mu_{k+j}^{(Cy)}\\}_{j\in\mathbb{Z}}$ satisfies the recurrence

$(2k+4)\left[(2k+3)^{2}-4\alpha^{2}\right]\mu_{k+1}^{(Cy)}+2(2k+1)\left[(2k+2)^{2}+2N(N+2\alpha)\right]\mu_{k}^{(Cy)}\\\
+(2k+1)(2k)(2k-1)\mu_{k-1}^{(Cy)}=0.$ (4.18)

The initial condition determining the sequence
$\\{\mu_{k}^{(Cy)}|_{\beta=2}\\}_{k=0}^{\infty}$ is

$\mu_{0}^{(Cy)}={2N\alpha(N+2\alpha)\over(2\alpha-1)(2\alpha+1)},\quad\alpha>1/2.$
(4.19)

For $\beta=1$ and $4$, the sequence $\\{\mu_{k+j}^{(Cy)}\\}_{j\in\mathbb{Z}}$
satisfies the recurrence

$\sum_{l=-2}^{2}g_{l}\,\mu_{k+l}^{(Cy)}=0,$ (4.20)

where $g_{l}:=(-1)^{l-1}f_{l}\big{|}_{a\mapsto-\beta(N-1)/2-1-\alpha}$ with
initial conditions given by parsing equations (4.15) and (4.16) through (4.8).

######  Remark 4.4.

Let $\tilde{m}_{k}^{(J)}=\tilde{m}_{k}^{(J)}(N,\beta,a,b)$ denote the $k$-th
moment of the spectral density for the Jacobi $\beta$-ensemble defined on
$(0,1)$. It is known [9, 21, 16] that

$\tilde{m}_{k}^{(J)}(N,\beta,a,b)=-(2/\beta)\tilde{m}_{k}^{(J)}(-\beta
N/2,4/\beta,-2a/\beta,-2b/\beta).$ (4.21)

It follows from (4.1) that this remains true for the moments $m_{k}^{(J)}$ of
the spectral density for the Jacobi $\beta$-ensemble defined on $(-1,1)$.
Suppose now we set

$a=-\beta(N-1)/2-1-\bar{\alpha},\qquad b=-\beta(N-1)/2-1-\alpha.$

Then we know from Proposition 2.6 that $m_{k}^{(J)}$ equals
$(-1)^{k/2}m_{k}^{(Cy)}(N,\beta,\alpha,\bar{\alpha})$. Relating the RHS of
(4.21) to $m_{k}^{(Cy)}$ for renormalised parameters shows

$m_{k}^{(Cy)}(N,\beta,\alpha,\bar{\alpha})=-(2/\beta)m_{k}^{(Cy)}(-\beta
N/2,4/\beta,-2\alpha/\beta,-2\bar{\alpha}/\beta).$ (4.22)

It can be checked that recurrence (4.20) respects the duality (4.22).

###  4.2. Continuous Hahn polynomials

The $\beta=2$ recurrence (4.12) simplifies upon introducing the rescaling

$\mu_{k}^{(J)}={\Gamma(k+1/2)\over\Gamma(k+\beta(N-1)/2+a+5/2)}\tilde{\mu}_{k}^{(J)},$
(4.23)

as motivated by (4.5). According to this rescaling, we have

$\displaystyle\mu_{k+1}^{(J)}$ $\displaystyle=-{k+1/2\over
N+a+k+3/2}{\Gamma(k+1/2)\over\Gamma(N+a+k+3/2)}\tilde{\mu}_{k+1}^{(J)},$
$\displaystyle\mu_{k-1}^{(J)}$ $\displaystyle=-{N+a+k+1/2\over
k-1/2}{\Gamma(k+1/2)\over\Gamma(N+a+k+3/2)}\tilde{\mu}_{k-1}^{(J)}.$

Substituting these expressions into (4.12) gives a three term recurrence with
coefficients that are quadratic rather than cubic in $k$. (We remark that this
simplification does not extend to the $\beta=1,4$ recurrence (4.14).)

######  Corollary 4.5.

Define $\tilde{\mu}_{k}^{(J)}$ by (4.23) and (4.6). For $\beta=2$, the
sequence $\\{\tilde{\mu}_{k+j}^{(J)}\\}_{j\in\mathbb{Z}}$ satisfies the
recurrence

$(2k+4)[(2k+3)-2(a+N)]\tilde{\mu}_{k+1}+2[(2k+2)^{2}-2N(N+2a)]\tilde{\mu}_{k}\\\
+(2k)(2(N+a+k)+1)\tilde{\mu}_{k-1}=0.$ (4.24)

######  Remark 4.6.

For a given value of $\tilde{\mu}_{0}$, the recurrence (4.24) uniquely
specifies $\\{\tilde{\mu}_{k}^{(J)}\\}_{k=1}^{\infty}$. With $\tilde{\mu}_{0}$
defined by (4.13) and (4.23), we have that

$\tilde{\mu}_{0}^{(J)}={\Gamma(N+a+3/2)\over\Gamma(1/2)}\Big{(}{2N(a+N)(2a+N)\over
1-4(a+N)^{2}}\Big{)}.$ (4.25)

At this stage, following [2], introduce the continuous Hahn polynomials

$S_{n}(x;a,b,c,d)=i^{n}{(a+c)_{n}(a+d)_{n}\over
n!}\,\,{}_{3}F_{2}\bigg{(}\begin{matrix}-n,n+a+b+c+d-1,a+ix\\\
a+c,a+d\end{matrix}\,\bigg{|}\,1\bigg{)},$ (4.26)

where on the RHS the notation is standard as for hypergeometric functions.
Further specialise these polynomials by writing
$s_{n}(x;a,b)=S_{n}(x;a,b,\bar{a},\bar{b})$. The latter polynomials satisfy
the difference equation in $x$ [7, 18.22.13–18.22.15],

$A(x)p_{n}(x+i)-(A(x)+C(x)-n(n+2{\rm Re}\,(a+b)-1))p_{n}(x)+C(x)p_{n}(x-i)=0,$
(4.27)

where

$A(x)=(x+i\bar{a})(x+i\bar{b}),\qquad C(x)=(x-i{a})(x-i{b}).$ (4.28)

######  Proposition 4.7.

Let $\tilde{\mu}_{0}^{(J)}$ be given by (4.25). For general ${\rm Re}\,k>-1/2$
and $\beta=2$, we have

$\tilde{\mu}_{k}^{(J)}={\tilde{\mu}_{0}^{(J)}i^{1-N}\over
N(3/2-(a+N))_{N-1}}s_{N-1}\Big{(}i(k+1);1,{1\over 2}-(a+N)\Big{)},$ (4.29)

and thus ${\mu}_{k}^{(J)}$ with $\beta=2$ is given in terms of the continuous
Hahn polynomials according to

${\mu}_{k}^{(J)}={\tilde{\mu}_{0}^{(J)}i^{1-N}\Gamma(k+1/2)\over
N\Gamma(k+N+a+3/2)(3/2-(a+N))_{N-1}}s_{N-1}\Big{(}i(k+1);1,{1\over
2}-(a+N)\Big{)}.$ (4.30)

###### Proof.

Comparing (4.27), (4.28) with (4.24) we see that

$C_{N,a}s_{N-1}\Big{(}i(k+1);1,{1\over 2}-(a+N)\Big{)},$ (4.31)

where $C_{N,a}$ is independent of $k$, satisfies the recurrence (4.24). This
is also a polynomial of degree $(N-1)$ in $k$, which upon choosing

$C_{N,a}={\tilde{\mu}_{0}^{(J)}\over s_{N-1}\Big{(}i;1,{1\over
2}-(a+N)\Big{)}}={\tilde{\mu}_{0}^{(J)}i^{1-N}\over N(3/2-(a+N))_{N-1}},$
(4.32)

where the second equality follows from (4.26), agrees with
$\tilde{\mu}_{k}^{(J)}$ for $k=0$. The recurrence (4.24) then gives that this
polynomial is equal to $\tilde{\mu}_{k}^{(J)}$ at each positive integer value
of $k$, and thus the polynomials must in fact be identical, giving (4.29).

The formula (4.30) follows from (4.29) using (4.23) with $\beta=2$. ∎

######  Remark 4.8.

For $\beta$ even, we know from the discussion leading to (4.23) that
$\mu_{k}^{(J)}$ permits a factorisation involving a polynomial part
$\tilde{\mu}_{k}^{(J)}$. For $\beta=2$, this polynomial is given in terms of
particular continuous Hahn polynomials according to (4.29). As noted in [2] in
the context of the moments of the symmetric Cauchy ensemble (the exact
evaluation of $\mu_{k}^{(Cy)}$ in terms of continuous Hahn polynomials implied
by (4.8), (4.30) agrees with that given therein), the fact that $s_{n}(x;a,b)$
has all real zeros implies that the zeros of $\tilde{\mu}_{k}|_{\beta=2}$ are
all on the line ${\rm Re}\,k=-1$. However, this property does not carry over
to other values of $\beta$. Thus, it is a simple exercise to compute
explicitly $\tilde{\mu}_{k}^{(J)}$ with $\beta$ even in the case $N=2$.
Further specialising to $\beta=4$, when the polynomial is a quadratic, it is
found that the zeros are distinct, and lie on the negative real axis at
positions varying with $a$.

###  4.3. Resolvent for the symmetric Cauchy case

Accompanying the definition (4.7) of the sum of successive even moments
$\mu_{k}^{(Cy)}$ of the Cauchy ensemble is the requirement that $-1/2<{\rm
Re}\,k<\alpha-1/2$. This implies that $\mu_{k}^{(Cy)}$, and thus
$m_{k}^{(Cy)}$, is well defined for only finitely many positive integers $k$,
which, in turn, prevents us from making sense of their generating functions.
This complication vanishes in the large $N$ limit when we set
$\alpha=\hat{\alpha}\beta N/2$ as seen in §3.1.3. We divide this subsection
into three. In the first part, we study the generating functions of the large
$N$ limiting forms of the $\mu_{k}^{(Cy)}$ and $m_{k}^{(Cy)}$. We then study
the finite-$N$ analogues of these generating functions, which are to be
understood in a formal sense through analytic continuation. In the third and
final part, we use the differential equations of Section 3 to study the $1/N$
expansions of the aforementioned formal generating functions.

####  4.3.1. Large $N$ limit

Here, we will make a study of the moments in the large $N$ limit to give a
complementary viewpoint to the results of §3.1.3. For this purpose, define

$\hat{\mu}_{k}^{(Cy)}=\lim_{N\to\infty}{1\over
N}\mu_{k}^{(Cy)}\Big{|}_{\alpha=\hat{\alpha}\beta
N/2},\qquad\hat{m}_{k}^{(Cy)}=\lim_{N\to\infty}{1\over
N}m_{2k}^{(Cy)}\Big{|}_{\alpha=\hat{\alpha}\beta N/2}$

and introduce the generating functions

$H(x)=\sum_{k=0}^{\infty}{\hat{\mu}_{k}^{(Cy)}\over x^{2k}},\qquad
G(x)=\sum_{k=0}^{\infty}{\hat{m}_{2k}^{(Cy)}\over x^{2k}}.$ (4.33)

The definition (4.7) shows that the latter are related by

$G(x)={H(x)+x^{2}\over 1+x^{2}}.$ (4.34)

Furthermore, with $\hat{\rho}_{(1)}^{(Cy)}$ denoting the LHS of (3.25), and
$I$ denoting its support, for $x\notin I$, we have

$G(x)=\int_{I}{\hat{\rho}_{(1)}^{(Cy)}(y)\over 1-(y/x)^{2}}\,dy,$ (4.35)

and so by the Sokhotski-Plemelj formula,

$s\hat{\rho}_{(1)}^{(Cy)}(s)={1\over\pi}\lim_{\epsilon\to 0^{+}}{\rm
Im}\,G(x)\Big{|}_{x=s-i\epsilon}.$ (4.36)

We will now show how to use (4.36) to re-derive (3.25). To begin, with
$\alpha=\hat{\alpha}N$ in (4.18), (4.19), equating leading powers in $N$ shows

$-(2k+4)\hat{\alpha}^{2}\hat{\mu}_{k+1}^{(Cy)}+(2k+1)(1+2\hat{\alpha})\hat{\mu}_{k}^{(Cy)}=0$
(4.37)

subject to the initial condition

$\hat{\mu}_{0}^{(Cy)}={1+2\hat{\alpha}\over 2\hat{\alpha}}.$ (4.38)

Hence, by iterating (4.37), it follows that

$\hat{\mu}_{k}^{(Cy)}={(1+2\hat{\alpha})^{k+1}\over
2\hat{\alpha}^{2k+1}}{(1/2)_{k}\over(2)_{k}}={(1+2\hat{\alpha})^{k+1}\over(2\hat{\alpha})^{2k+1}}C_{k},$
(4.39)

where $(u)_{k}$ is the (rising) Pochhammer symbol, and $C_{k}$ denotes the
$k$-th Catalan number (recall (1.1)). This substituted in the first equation
of (4.33) tells us that

$\displaystyle H(x)={(1+2\hat{\alpha})\over
2\hat{\alpha}}\,{}_{2}F_{1}(1,1/2;2;(1+2\hat{\alpha})/(\hat{\alpha}x)^{2})={(1+2\hat{\alpha})\over\hat{\alpha}}{1-\sqrt{1-z}\over
z}\Big{|}_{z=(1+2\hat{\alpha})/(\hat{\alpha}x)^{2}}.$ (4.40)

Substituting in (4.34) and then substituting the result in (4.36), we reclaim
(3.25). It can be checked that (4.39) satisfies (4.20) with
$\alpha=\hat{\alpha}\beta N/2$, owing to $\hat{\rho}^{(Cy)}_{(1)}(s)$ then
being $\beta$ independent.

####  4.3.2. Finite $N$

At this point, let us stress that the $\mu_{k}^{(Cy)}$ and their recurrences
have formal interpretations through analytic continuation without the need for
taking $N$ large. Hence, we are able to complement the above computation with
the $1/N$ correction to $\hat{\rho}^{(Cy)}_{(1)}(s)$. To this end, introduce
the formal sums

$W_{1}(x)=\sum_{k=0}^{\infty}\frac{m_{k}^{(Cy)}}{x^{k+1}},\qquad\hat{W}_{1}(x)=W_{1}(x)\Big{|}_{\alpha=\hat{\alpha}\beta
N/2}$ (4.41)

so that for $\beta=2$, $\lim_{N\to\infty}x\hat{W}_{1}(x)=G(x)$. It is known
for the Gaussian, Laguerre and Jacobi (with $(0,1)$ as support) ensembles that
the corresponding resolvents defined analogously to $W_{1}(x)$ above satisfy
the same differential equations as the densities $\rho_{(1)}(x)$ with
additional inhomogeneous terms. The same is true in the Cauchy case, seen by a
tweaking of the reasoning in [36, Appendix A]. Thus, replacing
$\rho^{(Cy)}_{(1)}(x)$ in the differential equation (3.8) by the expression
for $W_{1}(x)$ given in (4.41) yields inhomogeneous differential equations
satisfied by $W_{1}(x)$.

######  Proposition 4.9.

In the setting of Proposition 3.3, we have

$\mathcal{D}^{(Cy)}_{\beta,N}\,\frac{1}{N}W_{1}(x)=\begin{cases}4(N+\alpha)(N+2\alpha),&\beta=2,\\\
h(x;N,\alpha),&\beta=4,\\\ h(x;-N/2,-2\alpha),&\beta=1,\end{cases}$ (4.42)

where

$h(x;N,\alpha)=8(2N+2\alpha-1)\Big{[}(2N+1)(8N^{2}+x^{2}-1)+4N\alpha(6N+x^{2}+3)\\\
+\alpha^{2}(3x^{2}-4N(x^{2}-2)+5)-2\alpha(1+x^{2}\alpha^{2})\Big{]}.$ (4.43)

Comparing to Corollary 4.3, one should interpret
$\mathcal{D}_{\beta,N}^{(Cy)}$ as encoding the moment recurrences therein,
while the right-hand side of (4.42) encodes the initial conditions.

####  4.3.3. Expansion in $1/N$

Considering the moment recurrences and initial conditions of Corollary 4.3
directly, it is evident that $m_{k}^{(Cy)}/N$ admits a $1/N$ expansion upon
setting $\alpha=\hat{\alpha}\beta N/2$. Thus, we may take as ansatz

$\frac{1}{N}\hat{W}_{1}(x)=\sum_{l=0}^{\infty}\frac{\hat{W}_{1,l}(x)}{N^{l}}.$
(4.44)

Setting $\alpha=\hat{\alpha}\beta N/2$ in (4.42) results in differential
equations for $\hat{W}_{1}(x)/N$. Substituting the above ansatz into these
differential equations and equating powers of $N$ then gives first order
differential equations for the expansion coefficients $\hat{W}_{1,l}(x)$.

######  Proposition 4.10.

Set $\beta=2$. The leading term $\hat{W}_{1,0}(x)$ in (4.44) satisfies the
differential equation

$(1+x^{2})(\hat{\alpha}^{2}x^{2}-2\hat{\alpha}-1)\hat{W}_{1,0}^{\prime}(x)+(\hat{\alpha}^{2}x^{2}-\hat{\alpha}^{2}-4\hat{\alpha}-2)x\hat{W}_{1,0}(x)=-(\hat{\alpha}+1)(2\hat{\alpha}+1).$
(4.45)

Next, $\hat{W}_{1,1}(x)$ satisfies the homogeneous differential equation
corresponding to (4.45). Finally, for $l\geqslant 2$, we have

$4(1+x^{2})(\hat{\alpha}^{2}x^{2}-2\hat{\alpha}-1)\hat{W}_{1,l}^{\prime}(x)+4(\hat{\alpha}^{2}x^{2}-\hat{\alpha}^{2}-4\hat{\alpha}-2)x\hat{W}_{1,l}(x)\\\
=(1+x^{2})^{3}\hat{W}_{1,l-2}^{\prime\prime\prime}(x)+8(1+x^{2})^{2}x\hat{W}_{1,l-2}^{\prime\prime}(x)\\\
+2(1+x^{2})(7x^{2}+3)\hat{W}_{1,l-2}^{\prime}(x)+4(1+x^{2})x\hat{W}_{1,l-2}(x).$
(4.46)

######  Remark 4.11.

1\. The equation (4.45) has general solution

$\hat{W}_{1,0}(x)=\frac{(\hat{\alpha}+1)x}{1+x^{2}}+C\frac{\sqrt{\hat{\alpha}^{2}x^{2}-2\hat{\alpha}-1}}{1+x^{2}}.$
(4.47)

The integration constant $C$ is to be set equal to $-1$. Then
$\hat{W}_{1,0}(x)\sim 1/x$ as $x\to\infty$, in line with $m_{0}^{(Cy)}=N$.
This shows $x\hat{W}_{1,0}(x)=G(x)$, as expected.
2\. The fact that $\hat{W}_{1,1}(x)$ satisfies the homogeneous differential
equation corresponding to (4.45) is consistent with the requirement that
$\hat{W}_{1,l}(x;\beta=2)=0$ for odd $l$, since the moments $m_{k}^{(Cy)}$ are
odd functions of $N$ when $\beta=2$.
3\. To uniquely determine $\hat{W}_{1,2p}(x)$ for $p\geqslant 1$ an integer,
we use (4.19) to compare the $1/N^{2}$ expansion of
$\frac{1}{N}m_{2}^{(Cy)}|_{\alpha=\hat{\alpha}N}$ to (4.44) and consequently
see that

$\hat{W}_{1,2p}(x)\underset{x\to\infty}{\sim}\frac{1+2\hat{\alpha}}{(2\hat{\alpha})^{2p+1}x^{3}}.$
(4.48)

Thus, the integration constant present in the solution of (4.46) is zero. In
particular, solving (4.46) with $l=2$ gives

$\hat{W}_{1,2}(x)=\frac{\hat{\alpha}^{2}(1+2\hat{\alpha})(1+x^{2})}{8(\hat{\alpha}^{2}x^{2}-2\hat{\alpha}-1)^{5/2}}.$
(4.49)

Hence by the Sokhotski-Plemelj formula,

$\frac{1}{N}\rho_{(1)}^{(Cy)}(x)\Big{|}_{\alpha=\hat{\alpha}N}=\hat{\rho}_{(1)}^{(Cy)}(x)+\frac{1}{N^{2}}\frac{\hat{\alpha}^{2}(1+2\hat{\alpha})(1+x^{2})}{8\pi(1+2\hat{\alpha}-\hat{\alpha}^{2}x^{2})^{5/2}}\chi_{|x|<\sqrt{1+2\hat{\alpha}}/\hat{\alpha}}+{\rm
O}\left(\frac{1}{N^{4}}\right).$ (4.50)

In the cases $\beta=1$ and $4$, the analogue of Proposition 4.10 is extracted
from equation (4.42) using the procedure described above. We do not present it
here for brevity, but simply note that in these cases, the first order
differential equation satisfied by $\hat{W}_{1,l}(x)$ has inhomogeneous terms
dependent on $\hat{W}_{1,p}(x)$ for $1\leqslant p\leqslant 4$. This is in
contrast to the $\beta=2$ case, where the differential equation for
$\hat{W}_{1,l}(x)$ does not have $\hat{W}_{1,l-1}(x)$ present in the
inhomogeneous term. Skipping these details, we now supplement equations (4.49)
and (4.50) with their $\beta=1,4$ analogues.

######  Proposition 4.12.

Set $\beta=1$ or $4$. Then, $\hat{W}_{1,0}(x)$ is given by (4.47) with $C=-1$
(recall the $\beta$-independence of $\hat{\rho}_{(1)}^{(Cy)}(x)$). In
addition,

$\displaystyle\hat{W}_{1,1}(x)$
$\displaystyle=\frac{(1-2/\beta)\hat{\alpha}}{2}\left[\frac{1}{\sqrt{\hat{\alpha}^{2}x^{2}-2\hat{\alpha}-1}}-\frac{\hat{\alpha}x}{\hat{\alpha}^{2}x^{2}-2\hat{\alpha}-1}\right],$
(4.51) $\displaystyle\hat{W}_{1,2}(x)$
$\displaystyle=\frac{(1-2/\beta)^{2}}{2}\left[\frac{\hat{\alpha}^{2}(1+2\hat{\alpha}+\hat{\alpha}^{2})x^{2}}{(\hat{\alpha}^{2}x^{2}-2\hat{\alpha}-1)^{5/2}}+\frac{\hat{\alpha}^{2}(1+2\hat{\alpha})(1+x^{2})}{4(\hat{\alpha}^{2}x^{2}-2\hat{\alpha}-1)^{5/2}}-\frac{\hat{\alpha}(1+2\hat{\alpha}+\hat{\alpha}^{2})x}{(\hat{\alpha}^{2}x^{2}-2\hat{\alpha}-1)^{2}}\right]$
$\displaystyle\qquad+\frac{\hat{\alpha}^{2}(1+2\hat{\alpha})(1+x^{2})}{4\beta(\hat{\alpha}^{2}x^{2}-2\hat{\alpha}-1)^{5/2}}.$
(4.52)

Applying the Sokhotski-Plemelj formula shows

$\frac{1}{N}\rho_{(1)}^{(Cy)}(x)\Big{|}_{\alpha=\hat{\alpha}\beta
N/2}=\hat{\rho}_{(1)}^{(Cy)}(x)+\frac{1}{N}(1-2/\beta)\left[\frac{\hat{\alpha}}{2\pi\sqrt{1+2\hat{\alpha}-\hat{\alpha}^{2}x^{2}}}\chi_{|x|<\sqrt{1+2\hat{\alpha}}/\hat{\alpha}}\right.\\\
\left.-\frac{1}{4}\delta(x-\sqrt{1+2\hat{\alpha}}/\hat{\alpha})-\frac{1}{4}\delta(x+\sqrt{1+2\hat{\alpha}}/\hat{\alpha})\right]+{\rm
O}\left(\frac{1}{N^{2}}\right),$ (4.53)

where $\delta(x)$ is the Dirac delta.

Although derived for $\beta=1,4$, Remark 4.11 tells us that both (4.51) and
(4.52) are also valid for $\beta=2$. In fact, adopting the viewpoint of a loop
equation analysis (see [41, §3.1], [16]), it is expected that
$\beta^{p}\hat{W}_{1,2p}$ is an even polynomial in the variable
$(\sqrt{\beta}-2/\sqrt{\beta})$. If this was to be assumed, it would follow
that the results of Proposition 4.12 are in fact valid for general $\beta>0$.

More parallels between the Cauchy ensemble resolvent and their Gaussian and
Laguerre ensemble analogues can be drawn from the references [41], [16]. For
one, the structure exhibited in (4.50) of the $1/N^{2}$ correction term having
a nonintegrable singularity at each endpoint of the support, which diverges
like an inverse $5/2$ power, is shared by the analogous expansion for the GUE
and the LUE (the latter with the Laguerre parameter proportional to $N$).
Moreover, when $\beta\neq 2$, the correction term given in these references is
of order $1/N$ with Dirac delta contributions at the endpoints of support.
This property is shared with the correction term in (4.53). As a final remark,
we note that the aforementioned references also show how integration of the
correction terms against monomials, so as to compute the corresponding
moments, can be regularised using integration by parts.

### Acknowledgements

This research is part of the program of study supported by the Australian
Research Council Centre of Excellence ACEMS. The work of PJF was also
partially supported by the Australian Research Council Grant DP210102887, and
that of AAR by the Australian Government Research Training Program
Scholarship.

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