# Spectral Moments of the real Ginibre ensemble

Sung-Soo Byun  Department of Mathematical Sciences and Research Institute of
Mathematics, Seoul National University, Seoul 151-747, Republic of Korea
<EMAIL_ADDRESS>and  Peter J. Forrester  School of Mathematical and
Statistics, The University of Melbourne, Victoria 3010, Australia
<EMAIL_ADDRESS>
###### Abstract.

The moments of the real eigenvalues of real Ginibre matrices are investigated
from the viewpoint of explicit formulas, differential and difference
equations, and large $N$ expansions. These topics are inter-related. For
example, a third order differential equation can be derived for the density of
the real eigenvalues, and this can be used to deduce a second order difference
equation for the general complex moments $M_{2p}^{\rm r}$. The latter are
expressed in terms of the ${}_{3}F_{2}$ hypergeometric functions, with a
simplification to the ${}_{2}F_{1}$ hypergeometric function possible for $p=0$
and $p=1$, allowing for the large $N$ expansion of these moments to be
obtained. The large $N$ expansion involves both integer and half integer
powers of $1/N$. The three term recurrence then provides the large $N$
expansion of the full sequence $\\{M_{2p}^{\rm r}\\}_{p=0}^{\infty}$. Fourth
and third order linear differential equations are obtained for the moment
generating function and for the Stieltjes transform of the real density,
respectively, and properties of the large $N$ expansion of these quantities
are determined.

###### Key words and phrases:

Real Ginibre ensemble, real eigenvalues, spectral moments, hypergeometric
functions, recurrence relation, asymptotic expansions

###### 2020 Mathematics Subject Classification:

Primary 60B20; Secondary 33C20

Sung-Soo Byun was partially supported by the POSCO TJ Park Foundation (POSCO
Science Fellowship) and by the New Faculty Startup Fund at Seoul National
University. Funding support to Peter Forrester for this research was through
the Australian Research Council Discovery Project grant DP210102887.

##  1\. Introduction

###  1.1. Context

The study of moments of the spectral density for random matrix ensembles hold
a special place in the development of random matrix theory. One landmark was
Wigner’s [30, 31] introduction of what is now referred to as the moment method
as a strategy to prove that a large class of symmetric random matrices have
the same scaled spectral density $\rho^{\rm
W}(x):={2\over\pi}(1-x^{2})\chi_{|x|<1}$. The functional form $\rho^{\rm
W}(x)$ is now known as the Wigner semi-circle. An essential point is that the
$2p$-th even moment $m_{2p}$ of $\rho^{\rm W}(x)$ is equal to $2^{-2p}C_{p}$,
where $C_{p}$ denotes the $p$-th Catalan number, and moreover this moment
sequence uniquely determines $\rho^{\rm W}(x)$. Wigner then used the fact that
in the random matrix setting
$m_{2p}=\lim_{N\to\infty}(\sigma^{2}/2^{2}N)^{p}\mathbb{E}({\rm Tr}\,X^{2p})$
as the starting point for the computation of limiting scaled moments of the
spectral density (here $\sigma^{2}$ is the variance of the off diagonal
entries of $X$, which are assumed too to have mean zero). Subsequent to
Wigner’s work, the method of moments, and its generalisation to cumulants, has
been used extensively in questions beyond the spectral density such as for the
Gaussian fluctuations of linear statistics and pair counting statistics; see
the recent review [29].

Specialise now to complex Hermitian matrices from the Gaussian unitary
ensemble, with unit variance of the modulus of the off-diagonal entries so
that the joint element distribution is proportional to $e^{-{\rm
Tr}\,X^{2}/2}$. Another landmark has been in relation to the interpretation of
the coefficients in the large $N$ of $M_{2p}^{\rm GUE}:=\mathbb{E}\,{\rm
Tr}\,X^{2p}$. This expansion is a terminating series in $1/N^{2}$,

(1.1) $N^{-p-1}M_{2p}^{\rm GUE}=\sum_{g=0}^{\lfloor p/2\rfloor}{\nu_{p,g}^{\rm
GUE}\over N^{2g}}.$

The result of Wigner gives that $\mu_{p,0}=C_{p}$. Using a diagrammatic
interpretation of non-zero terms in the computation of ${\rm Tr}\,X^{2p}$ as
implied by Wick’s theorem, it was shown by Brézin et al. [3] that the
$\nu_{p,g}$ count the number of pairings of the sides of a $2p$-gon which are
dual to a map on a compact orientable Riemann surface of genus $g$.
Equivalently, after the sides are identified in pairs, it is required that a
surface of genus $g$ results. The case $g=0$ is referred to as planar, and
represent the leading order in (1.1).

Motivated by this topological interpretation, Harer and Zagier [22] further
investigated the sequences $\\{M_{2p}^{\rm GUE}\\}$ and $\\{\nu_{p,g}^{\rm
GUE}\\}$. In particular they obtained that $\\{M_{2p}^{\rm GUE}\\}$ obey the
three term recurrence

(1.2) $(p+1)M_{2p}^{\rm GUE}=(4p-2)NM_{2p-2}^{\rm
GUE}+(p-1)(2p-1)(2p-3)M_{2p-4}^{\rm GUE}$

subject to the initial conditions $M_{0}^{\rm GUE}=N$, $M_{2}^{\rm
GUE}=N^{2}$. From this it was shown that $\\{\nu_{p,g}\\}$ obey the two-
variable recurrence

(1.3) $(p+2)\nu_{p+1,g}^{\rm GUE}=(p+1/2)p(p-1/2)\nu_{p-1,g-1}^{\rm
GUE}+(2p+1)\nu_{p,g}^{\rm GUE},$

subject to the initial condition $\mu_{0,0}^{\rm GUE}=1$, and boundary
conditions $\nu_{p,g}^{\rm GUE}=0$ for any of the conditions $k<0,g<0$ or
$g>\lfloor p/2\rfloor$.

The focus of our study relating to random matrix spectral moments in the
present work is an outgrowth of theory underlying and relating the the three
term recurrence (1.2), combined with results from the recent paper [4] by one
of us. The question addressed in [4] is to identify a recurrence relation for
the spectral moments of the real eigenvalues of elliptic GinOE matrices. The
latter is the ensemble of asymmetric real Gaussian matrices defined by

(1.4) $\sqrt{1+\tau\over 2}S_{+}+\sqrt{1-\tau\over 2}S_{-},$

where $S_{\pm}$ are independent random real symmetric and skew-symmetric GOE
matrices, and $0\leq\tau\leq 1$ is a parameter. For $\tau=1$ one sees that
elliptic GinOE reduces to GOE. Earlier, the work of Ledoux [24] had found a
fifth order linear recurrence for $\\{M_{2p}^{\rm GOE}\\}$. Existing
literature due to Goulden and Jackson [20], extending the work of Harer and
Zagier, has given an interpretation of the $M_{2p}^{\rm GOE}$ in terms of
pairings which lead to both nonoriented and orientable surfaces. The analogue
of (1.1) is now

(1.5) $N^{-p-1}M_{2p}^{\rm GOE}=\sum_{l=0}^{p}{\nu_{p,l}^{\rm GOE}\over
N^{l}},$

which in particular involves both odd an even powers of $1/N$. In keeping with
the universality of the Wigner semi-circle law, one again has for the leading
contribution $\nu_{p,0}^{\rm GOE}=C_{p}$.

The other extreme of (1.4) is $\tau=0$, when each entry is identically
distributed as an independent standard real Gaussian, this giving rise to a
random matrix from GinOE. See [5, 6] for recent reviews on the Ginibre
ensembles. Here there was no previous literature on the moment sequence of the
real eigenvalues. The $\tau=0$ limiting case of the in general 11-term linear
recurrence found in [4] for the moments of the density of real eigenvalues was
found to reduce to just a three term recurrence

(1.6) $2(2p+1)M_{2p}^{\rm r,GinOE}=(2p-1)(6p+4N-5)M_{2p-2}^{\rm
r,GinOE}-(2p-3)(2p+N-4)(2p+2N-3)M_{2p-4}^{\rm r,GinOE}.$

Motivated by the relative simplicity of (1.6), and its similarity with the GUE
moment recurrence (1.2), in this work we will carry out a study of the moments
$\\{M_{2p}^{\rm r,GinOE}\\}$ as a stand alone sequence, not viewed as a limit
of moments for the real eigenvalues of elliptic GinOE. In the case of the
recurrence (1.2) it has been known since the work of Haagerup and Thorbjørnsen
[21] that there is a tie in with certain higher order differential equation
and also with certain special function functions, in particular hypergeometric
polynomials. In fact such structures have been shown to also be features of
the spectral moments in a broad range of settings [24, 25, 9, 8, 27, 2, 18,
13, 17, 19]. However GinOE is distinct from the ensembles in these earlier
studies since only a fluctuating fraction of eigenvalues are real.

###  1.2. Some known results

Let $G$ be a real Ginibre matrix (GinOE) of size $N$, defined by the
requirement that all entries are independent standard Gaussians. By making use
of knowledge of the Schur function average with respect to GinOE matrices, it
was shown in [28, 16] that for any positive integer $p\geq 1$,

(1.7)
$\mathbb{E}\Big{[}\operatorname{Tr}G^{2p}\Big{]}=2^{p}\frac{\Gamma(N/2+p)}{\Gamma(N/2)}=N(N+2)\dots(N+2p-2).$

But with the eigenvalues of GinOE matrices being in general both real and
complex, this is a result which combines moments relating to the real
eigenvalues, and moments relating to the complex eigenvalues. Specifically,
let $\mathcal{N}_{\mathbb{R}}$ be the number of real eigenvalues and define

(1.8) $M_{2p,N}^{\rm
r}:=\mathbb{E}\bigg{[}\sum_{j=1}^{\mathcal{N}_{\mathbb{R}}}x_{j}^{2p}\bigg{]},\qquad
M_{2p,N}^{\rm
c}:=\mathbb{E}\bigg{[}\sum_{j=1}^{N-\mathcal{N}_{\mathbb{R}}}z_{j}^{2p}\bigg{]}=2\,\mathbb{E}\bigg{[}\sum_{j=1}^{(N-\mathcal{N}_{\mathbb{R}})/2}\operatorname{Re}z_{j}^{2p}\bigg{]},$

where $x_{j}$ $(j=1,\dots,\mathcal{N}_{\mathbb{R}})$ and $z_{j}$
($j=1,\dots,N-\mathcal{N}_{\mathbb{R}}$) are real and complex eigenvalues of
$G$, respectively. Here and in the sequel, we drop the superscript “GinOE”,
cf. (1.6). We have used the convention
$z_{j+(N-\mathcal{N}_{\mathbb{R}})/2}=\bar{z}_{j}.$ Note that by definition,

(1.9) $M_{2p,N}^{\rm r}+M_{2p,N}^{\rm
c}=\mathbb{E}\Big{[}\operatorname{Tr}G^{2p}\Big{]}.$

It has been known for some time [11, 10] that the average densities of real
and complex eigenvalues of the GinOE are given by

(1.10) $\displaystyle\rho_{N}^{\rm r}(x)$
$\displaystyle=\frac{1}{\sqrt{2\pi}(N-2)!}\bigg{(}\Gamma(N-1,x^{2})+2^{(N-3)/2}e^{-\frac{x^{2}}{2}}|x|^{N-1}\gamma\Big{(}\frac{N-1}{2},\frac{x^{2}}{2}\Big{)}\bigg{)},$
(1.11) $\displaystyle\rho_{N}^{\rm c}(x+iy)$
$\displaystyle=\sqrt{\frac{2}{\pi}}|y|\operatorname{erfc}(\sqrt{2}|y|)e^{2y^{2}}\frac{\Gamma(N-1,x^{2}+y^{2})}{\Gamma(N-1)}.$

These relate to the even integer moments of the real and complex eigenvalues
by

(1.12) $M_{2p,N}^{\rm r}=\int_{\mathbb{R}}x^{2p}\rho_{N}^{\rm r}(x)\,dx,\qquad
M_{2p,N}^{\rm c}=\int_{\mathbb{R}^{2}}(x+iy)^{2p}\rho_{N}^{\rm
c}(x+iy)\,dx\,dy.$

The pioneering work [11] relating to the real eigenvalues of GinOE has
provided both an exact evaluation, and an asymptotic expansion, for

$M_{0,N}^{\rm r}=N-M_{0,N}^{\rm c}=\mathbb{E}\mathcal{N}_{\mathbb{R}}.$

Thus, from [11, Cor. 5.1] we have the expression in terms of a particular
Gauss hypergeometric function

(1.13) $M_{0,N}^{\rm
r}=\frac{1}{2}+\sqrt{\frac{2}{\pi}}\frac{\Gamma(N+\frac{1}{2})}{(N-1)!}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{1}{2}}{N}\;\middle|\;\frac{1}{2}\right).$

As an application of this formula, it is shown in [11, Cor. 5.2] that for
$N\to\infty$

(1.14) $M_{0,N}^{\rm r}=\sqrt{2N\over\pi}\bigg{(}1-{3\over 8N}-{3\over
128N^{2}}+{27\over 1024N^{3}}+{499\over 32768N^{4}}+{\rm O}\Big{(}{1\over
N^{5}}\Big{)}\bigg{)}+{1\over 2}.$

As a series, the Gauss hypergeometric function in (1.13) is not terminating.
Nonetheless, by considering the recurrence in $N$ implied by this formula,
such terminating forms were obtained [11, Cor. 5.3]

(1.15) $M_{0,N}^{\rm r}=\begin{cases}\displaystyle
1+\sqrt{2}\sum_{k=1}^{(N-1)/2}\frac{(4k-3)!!}{(4k-2)!!}&N\textup{ odd},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle\sqrt{2}\sum_{k=0}^{N/2-1}\frac{(4k-1)!!}{(4k)!!}&N\textup{
even}.\end{cases}$

(See also [1, Prop. 2.1].) As a consequence, one reads off that for $N$ even
$M_{0,N}^{\rm r}$ is equal to $\sqrt{2}$ times a rational number, while for
$N$ odd it is equal to 1 plus $\sqrt{2}$ times a rational number. As an aside,
we remark that an arithmetic result of this type is also known for the
expected number of real eigenvalues in the case of the product of two size $N$
GinOE matrices, where it is shown in [14, §4.2] to be of the form $\pi$ times
a rational number for $N$ even, and 1 plus $\pi$ times a rational number for
$N$ odd. The special function that appears here is not the Gauss
hypergeometric function but rather a particular Meijer G-function, which was
simplified to a finite series by Kumar [23].

###  1.3. New results

Our first new result generalises (1.14).

######  Theorem 1.1.

Let $m$ be any positive integer. Then as $N\to\infty$, we have

(1.16) $M_{0,N}^{\rm
r}=\sqrt{\frac{2}{\pi}N}\bigg{(}1+\sum_{l=1}^{m-1}\frac{a_{l}}{N^{l}}+{\rm
O}(N^{-m})\bigg{)}+\frac{1}{2},$

where

(1.17)
$a_{l}=-\frac{1}{\sqrt{\pi}}\frac{\Gamma(l-\frac{1}{2})}{l!}\frac{d^{l}}{dt^{l}}\bigg{[}\Big{(}\frac{e^{t}-1}{t}\Big{)}^{-\frac{3}{2}}\frac{e^{2t}}{e^{t}+1}\bigg{]}_{t=0}.$

Furthermore, as $N\to\infty$, we have

(1.18) $N^{-1}M_{2,N}^{\rm
r}=\sqrt{\frac{2}{\pi}N}\bigg{(}\frac{1}{3}+\sum_{l=1}^{m-1}\frac{b_{l}}{N^{l}}+{\rm
O}(N^{-m})\bigg{)}+\frac{1}{2},$

where

(1.19)
$b_{l}=-\frac{1}{2\sqrt{\pi}}\frac{\Gamma(l-\frac{3}{2})}{l!}\frac{d^{l}}{dt^{l}}\bigg{[}\Big{(}\frac{e^{t}-1}{t}\Big{)}^{-\frac{5}{2}}\frac{e^{2t}(e^{t}-3)}{(e^{t}+1)^{2}}\bigg{]}_{t=0}.$

This gives for the first few terms

(1.20) $N^{-1}M_{2,N}^{\rm
r}=\sqrt{\frac{2}{\pi}N}\bigg{(}\frac{1}{3}+\frac{3}{8N}-\frac{43}{384N^{2}}+\frac{29}{1024N^{3}}+\frac{1859}{98304N^{4}}+{\rm
O}\Big{(}\frac{1}{N^{5}}\Big{)}\bigg{)}+\frac{1}{2}.$

The three term recurrence (1.6) then gives that for all $p\geq 0$,

(1.21) $N^{-p}M_{2p,N}^{\rm r}=\sqrt{\frac{2}{\pi}N}\bigg{(}{1\over
2p+1}+\sum_{l=1}^{m-1}{b_{l,p}\over N^{l}}+{\rm O}(N^{-m})\bigg{)}+{1\over
2}+\sum_{l=1}^{p-1}{c_{l,p}\over N^{l}}$

for certain coefficients $b_{l,p}$ and $c_{l,p}.$

We remark that a generalisation of the asymptotic formula (1.16) for the
elliptic GinOE can also be found in [7, Prop. 2.2]. The terminating series
$\sum_{l=1}^{p-1}c_{l,p}N^{-l}$ for the first $p=2,3,4,5$ are given by

(1.22)
$\displaystyle\frac{1}{N},\qquad\frac{3}{N}+\frac{4}{N^{2}},\qquad\frac{6}{N}+\frac{22}{N^{2}}+\frac{24}{N^{3}},\qquad\frac{10}{N}+\frac{70}{N^{2}}+\frac{200}{N^{3}}+\frac{192}{N^{4}},$

respectively. These coefficients can also be derived from the moment
generating function, see (3.25) and (3.26).

Recall that the generalised hypergeometric function is given by the Gauss
series

(1.23)
${}_{r}F_{s}\left(\genfrac{.}{.}{0.0pt}{}{c_{1}{,}\mspace{8.0mu}\dots,\mspace{8.0mu}c_{r}}{d_{1}{,}\mspace{8.0mu}\dots,\mspace{8.0mu}d_{s}}\;\middle|\;z\right):=\sum_{k=0}^{\infty}\frac{(c_{1})_{k}\dots(c_{r})_{k}}{(d_{1})_{k}\dots(d_{s})_{k}}\frac{z^{k}}{k!};$

see e.g. [26, Chapter 16], where it is assumed that the parameters are such
that the series converges. Using this function with $r=3$ and $s=2$ we next
give an explicit formula for the even $2p$-th moments of both the real and
complex eigenvalue density. In earlier work the hypergeometric function
${}_{3}F_{2}$ has appeared in the evaluation of spectral moments of certain
Hermitian unitary ensembles — see [9, Eqs.(3.9),(3.10)] — although there with
extra structure of being terminating and furthermore identifiable in terms of
certain orthogonal polynomials in the Askey scheme.

######  Theorem 1.2.

For all positive integers $N$ and $p$ we have

(1.24) $\displaystyle\begin{split}M_{2p,N}^{\rm
r}&=\frac{1}{\sqrt{2\pi}}\frac{2}{2p+1}\frac{\Gamma(N+p-\frac{1}{2})}{(N-2)!}{}_{3}F_{2}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{1}{2}-p{,}\mspace{8.0mu}\frac{1}{2}+p}{\frac{1}{2}{,}\mspace{8.0mu}\frac{3}{2}-N-p}\;\middle|\;\frac{1}{2}\right)+2^{p}\frac{\Gamma(p+N/2)}{\Gamma(N/2)}\mathbbm{1}_{\\{N:\textup{
odd}\\}}\end{split}$

and

(1.25) $\displaystyle\begin{split}M_{2p,N}^{\rm
c}&=-\frac{1}{\sqrt{2\pi}}\frac{2}{2p+1}\frac{\Gamma(N+p-\frac{1}{2})}{(N-2)!}{}_{3}F_{2}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{1}{2}-p{,}\mspace{8.0mu}\frac{1}{2}+p}{\frac{1}{2}{,}\mspace{8.0mu}\frac{3}{2}-N-p}\;\middle|\;\frac{1}{2}\right)+2^{p}\frac{\Gamma(p+N/2)}{\Gamma(N/2)}\mathbbm{1}_{\\{N:\textup{
even}\\}}.\end{split}$

The definition (1.12) of the even integer moment $M_{2p,N}^{\rm r}$ can be
extended to all complex $p$ with Re$(p)>-1/2$ by the rewrite

(1.26) $M_{2p,N}^{\rm r}=\int_{\mathbb{R}}|x|^{2p}\rho_{N}^{\rm
r}(x)\,dx=2\int_{0}^{\infty}x^{2p}\rho_{N}^{\rm r}(x)\,dx.$

The evaluation formula (1.24) again remains valid.

######  Proposition 1.3.

Define $M_{2p,N}^{\rm r}$ for general $\operatorname{Re}(p)>-1/2$ by (1.26).
As for the case of $p$ a non-negative integer, it is evaluated in terms of a
${}_{3}F_{2}$ function by (1.24).

######  Remark 1.4.

Let $p=q+1/2$ for $q\geq 0$ a non-negative integer. The series (1.23) defining
the ${}_{3}F_{2}$ function in (1.24) is ill-defined as the parameters are such
that the indeterminant zero divided by zero occurs. For the series to be well
defined, the limit $q$ approaches a non-negative integer must be taken.

It is possible to deduce that the three term recurrence (1.6) for the moments
is valid not just for the even integer moments, but the complex moments too,
and to use this to deduce a three term recurrence specifically for the
${}_{3}F_{2}$ function appearing in (1.24). Use will be make of a third order
differential equation satisfied by $\rho_{N}^{\rm r}(x)$, which is of
independent interest.

######  Proposition 1.5.

The density $\rho_{N}^{\rm r}$ of real eigenvalues given in (1.10) satisfies
the differential equation

(1.27) $\mathcal{A}_{N}[x]\rho_{N}^{\rm
r}(x)=0,\quad\mathcal{A}_{N}[x]:=\Big{(}x^{2}\partial_{x}^{3}+x(3x^{2}-3N+4)\partial_{x}^{2}+(2x^{2}-2N+1)(x^{2}-N+2)\partial_{x}\Big{)}.$

######  Corollary 1.6.

The three term recurrence (1.6) remains valid for complex values of $p$ such
the terms are well defined. Also, as a function of complex $p$

(1.28)
$\displaystyle\begin{split}&(2N+2p-1)(2N+2p+1){}_{3}F_{2}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{5}{2}-p{,}\mspace{8.0mu}\frac{5}{2}+p}{\frac{1}{2}{,}\mspace{8.0mu}-\frac{1}{2}-N-p}\;\middle|\;\frac{1}{2}\right)\\\
&\qquad=(6p+4N+7)(2N+2p-1){}_{3}F_{2}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{3}{2}-p{,}\mspace{8.0mu}\frac{3}{2}+p}{\frac{1}{2}{,}\mspace{8.0mu}\frac{1}{2}-N-p}\;\middle|\;\frac{1}{2}\right)\\\
&\qquad\quad-2(2p+N)(2N+2p+1){}_{3}F_{2}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{1}{2}-p{,}\mspace{8.0mu}\frac{1}{2}+p}{\frac{1}{2}{,}\mspace{8.0mu}\frac{3}{2}-N-p}\;\middle|\;\frac{1}{2}\right).\end{split}$

Another consequence of (1.27) relates the specification of a differential
equation for the Fourier-Laplace transform, or equivalently for the (positive
integer) moment generating function, as well as for the Stieltjes transform.

######  Corollary 1.7.

Let

(1.29) $u(t):=\int_{\mathbb{R}}e^{tx}\rho_{N}^{\rm r}(x)\,dx.$

This satisfies the fourth order linear differential equation

(1.30) $D_{N}[t]\,u(t)=0,$

where

(1.31)
$D_{N}[t]:=2t\,\partial_{t}^{4}-(3t^{2}-8)\,\partial_{t}^{3}+t(t^{2}-4N-13)\,\partial_{t}^{2}+((3N+2)t^{2}-8N-8)\,\partial_{t}+(2N^{2}+N)t.$

Also, introduce the Stieltjes transform of the density

(1.32) $W(t):=\int_{\mathbb{R}}{\rho_{N}^{\rm r}(x)\over t-x}\,dx,\qquad
t\notin\mathbb{R}.$

With $\mathcal{A}_{N}[t]$ the differential operator specified in (1.27), but
with respect to $t$ rather than $x$, we have that $W(t)$ satisfies the
inhomogeneous differential equation

(1.33) $\mathcal{A}_{N}[t]W(t)=(1+4N-2t^{2})M_{0,N}^{\rm r}-6M_{N,2}^{\rm r}.$

######  Remark 1.8.

In [4, Cor. 1.5], it was found that the moment generating function $u(t)$
satisfies a seventh-order differential equation. Indeed, it can be observed
that the differential equation in [4, Cor. 1.5] can be further simplified to

(1.34)
$\displaystyle\begin{split}0&=(t^{3}\,\partial_{t}^{3}-6t^{2}\,\partial_{t}^{2}+15t\,\partial_{t}-15)\circ\,D_{N}[t]u(t)\\\
&=(t\,\partial_{t}-a)\circ(t\,\partial_{t}-b)\circ(t\,\partial_{t}-c)\circ
D_{N}[t]\,u(t),\end{split}$

where $(a,b,c)$ is a permutation of $(1,3,5)$.

The proofs of the above results are given in Section 2. In Section 3 we link
up the large $N$ asymptotic expansion of the moments Theorem 1.1 with large
$N$ asymptotic expansions that can be deduced for the Fourier-Laplace
transform $u(t)$ and the Stieltjes transform $W(t)$.

##  2\. Proofs

###  2.1. Proof of Theorem 1.1

To begin we recall a particular asymptotic formula of ${}_{2}F_{1}$, telling
us that as $\lambda\to\infty$ ($\lambda\in\mathbb{R}$),

(2.1)
${}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{a{,}\mspace{8.0mu}b}{c+\lambda}\;\middle|\;z\right)=\frac{\Gamma(c+\lambda)}{\Gamma(c-b+\lambda)}\sum_{s=0}^{m-1}q_{s}(z)\frac{\Gamma(b+s)}{\Gamma(b)}\,\lambda^{-s-b}+{\rm
O}(\lambda^{-m-b});$

see [26, Eq.(15.12.3)]. Here $q_{0}(z)=1$ and $q_{s}(z)$ when $s=1,2,\ldots$
are defined by the generating function

(2.2)
$\Big{(}\frac{e^{t}-1}{t}\Big{)}^{b-1}e^{t(1-c)}(1-z+ze^{-t})^{-a}=\sum_{s=0}^{\infty}q_{s}(z)\,t^{s}.$

Using this with $a=1,b=-1/2,c=0$ and $\lambda=N$ in (1.13) gives (1.16).

In preparation for deducing the large $N$ asymptotic form of $M_{2,N}^{\rm r}$
an evaluation formula analogous to (1.13) is required.

######  Proposition 2.1.

We have

(2.3) $\displaystyle\begin{split}M_{2,N}^{\rm
r}&=\sqrt{\frac{2}{\pi}}\frac{\Gamma(N+\frac{3}{2})}{(N-1)!}\bigg{(}\frac{1}{2N}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{2{,}\mspace{8.0mu}-\frac{1}{2}}{N+1}\;\middle|\;\frac{1}{2}\right)+\frac{1}{3}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{3}{2}}{N}\;\middle|\;\frac{1}{2}\right)\bigg{)}+\frac{N}{2}.\end{split}$

###### Proof.

We will assume temporarily the validity of the evaluation formula given by the
$p=1$ case of $M_{2p,N}^{\rm r}$ in Theorem 1.2 (this will be proved for all
non-negative $p$ in the next section). This gives

(2.4) $\begin{split}M_{2,N}^{\rm
r}=\frac{1}{(N-2)!}\sum_{k=0}^{\infty}\frac{1}{2^{k+\frac{1}{2}}}\frac{\Gamma(k+\frac{3}{2})\Gamma(N-k+\frac{1}{2})}{\Gamma(k+\frac{1}{2})\,\Gamma(-k+\frac{5}{2})}+N\,\mathbbm{1}_{\\{N:\textup{
odd}\\}}.\end{split}$

We note that

$\displaystyle\quad\sum_{k=0}^{\infty}\frac{1}{2^{k+\frac{1}{2}}}\frac{\Gamma(k+\frac{3}{2})\Gamma(N-k+\frac{1}{2})}{\Gamma(k+\frac{1}{2})\,\Gamma(-k+\frac{5}{2})}=\sum_{k=0}^{\infty}\frac{1}{2^{k+\frac{1}{2}}}\frac{(k+\frac{1}{2})\Gamma(N-k+\frac{1}{2})}{\Gamma(-k+\frac{5}{2})}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{2^{k+\frac{1}{2}}}\frac{(k+\frac{1}{2})\Gamma(N-k+\frac{1}{2})}{\Gamma(-k+\frac{5}{2})}=\sum_{k=0}^{\infty}\frac{1}{2^{k+\frac{1}{2}}}\frac{((k+\frac{3}{2})-1)\Gamma(N-k+\frac{1}{2})}{\Gamma(-k+\frac{5}{2})}$
$\displaystyle=\frac{\Gamma(N+\tfrac{1}{2})}{6\sqrt{\pi}}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{-\frac{3}{2}{,}\mspace{8.0mu}-N-\frac{1}{2}}{-N+\frac{1}{2}}\;\middle|\;-1\right)+\frac{\Gamma(N-\tfrac{1}{2})}{2\sqrt{\pi}}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{-\frac{1}{2}{,}\mspace{8.0mu}-N-\frac{1}{2}}{-N+\frac{3}{2}}\;\middle|\;-1\right)$
$\displaystyle=\frac{\sqrt{\pi}(-1)^{N}}{6\,\Gamma(-N+\frac{1}{2})}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{-\frac{3}{2}{,}\mspace{8.0mu}-N-\frac{1}{2}}{-N+\frac{1}{2}}\;\middle|\;-1\right)-\frac{\sqrt{\pi}(-1)^{N}}{2\,\Gamma(-N+\tfrac{3}{2})}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{-\frac{1}{2}{,}\mspace{8.0mu}-N-\frac{1}{2}}{-N+\frac{3}{2}}\;\middle|\;-1\right).$

Then by the linear transformation [26, Eq.(15.8.5)]

(2.5)
$\displaystyle\begin{split}\frac{\sin(\pi(c-a-b))}{\pi\,\Gamma(c)}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{a{,}\mspace{8.0mu}b}{c}\;\middle|\;z\right)&=\frac{z^{-a}}{\Gamma(c-a)\Gamma(c-b)\Gamma(a+b-c+1)}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{a{,}\mspace{8.0mu}a-c+1}{a+b-c+1}\;\middle|\;1-\frac{1}{z}\right)\\\
&\quad-\frac{(1-z)^{c-a-b}z^{a-c}}{\Gamma(a)\Gamma(b)\Gamma(c-a-b+1)}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{c-a{,}\mspace{8.0mu}1-a}{c-a-b+1}\;\middle|\;1-\frac{1}{z}\right)\end{split}$

together with (2.17), we have

$\displaystyle\frac{\sqrt{\pi}(-1)^{N}}{6\,\Gamma(-N+\frac{1}{2})}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{-N-\frac{1}{2}{,}\mspace{8.0mu}-\frac{3}{2}}{-N+\frac{1}{2}}\;\middle|\;-1\right)$
$\displaystyle=-\frac{4}{15}\sqrt{\frac{2}{\pi}}\Gamma(N+\tfrac{3}{2}){}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}N+\frac{3}{2}}{\frac{7}{2}}\;\middle|\;2\right),$
$\displaystyle\frac{\sqrt{\pi}(-1)^{N}}{2\,\Gamma(-N+\tfrac{3}{2})}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{-N-\frac{1}{2}{,}\mspace{8.0mu}-\frac{1}{2}}{-N+\frac{3}{2}}\;\middle|\;-1\right)$
$\displaystyle=-\frac{8}{15}\sqrt{\frac{2}{\pi}}\Gamma(N+\tfrac{3}{2}){}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{2{,}\mspace{8.0mu}N+\frac{3}{2}}{\frac{7}{2}}\;\middle|\;2\right).$

Furthermore, again using (2.5) after removing the removable singularities, it
follows that

$\displaystyle\quad-\frac{4}{15}\sqrt{\frac{2}{\pi}}\Gamma(N+\tfrac{3}{2}){}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}N+\frac{3}{2}}{\frac{7}{2}}\;\middle|\;2\right)=-\frac{1}{\sqrt{2}}\frac{\Gamma(N+\tfrac{3}{2})}{\Gamma(\frac{7}{2})}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}N+\frac{3}{2}}{\frac{7}{2}}\;\middle|\;2\right)$
$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{\Gamma(N+\frac{3}{2})}{3(N-1)}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{3}{2}}{N}\;\middle|\;\frac{1}{2}\right)+\frac{(-1)^{N}(N-2)!}{8}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{\frac{5}{2}{,}\mspace{8.0mu}0}{2-N}\;\middle|\;\frac{1}{2}\right)$
$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{\Gamma(N+\frac{3}{2})}{3(N-1)}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{3}{2}}{N}\;\middle|\;\frac{1}{2}\right)+\frac{(-1)^{N}(N-2)!}{8}$

and

$\displaystyle\quad\frac{8}{15}\sqrt{\frac{2}{\pi}}\Gamma(N+\tfrac{3}{2}){}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{2{,}\mspace{8.0mu}N+\frac{3}{2}}{\frac{7}{2}}\;\middle|\;2\right)=\sqrt{2}\frac{\Gamma(N+\tfrac{3}{2})}{\Gamma(\frac{7}{2})}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{2{,}\mspace{8.0mu}N+\frac{3}{2}}{\frac{7}{2}}\;\middle|\;2\right)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\frac{\Gamma(N+\frac{3}{2})}{N(N-1)}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{2{,}\mspace{8.0mu}-\frac{1}{2}}{N+1}\;\middle|\;\frac{1}{2}\right)+\frac{(-1)^{N}(N-1)!}{2}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{\frac{3}{2}{,}\mspace{8.0mu}-1}{1-N}\;\middle|\;\frac{1}{2}\right)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\frac{\Gamma(N+\frac{3}{2})}{N(N-1)}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{2{,}\mspace{8.0mu}-\frac{1}{2}}{N+1}\;\middle|\;\frac{1}{2}\right)+\frac{(-1)^{N}(4N-1)\,(N-2)!}{8}.$

Therefore we obtain

$\displaystyle\quad\sqrt{\frac{2}{\pi}}\frac{4}{15}\frac{\Gamma(N+\tfrac{3}{2})}{(N-2)!}\bigg{(}2\,{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{2{,}\mspace{8.0mu}N+\frac{3}{2}}{\frac{7}{2}}\;\middle|\;2\right)-{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}N+\frac{3}{2}}{\frac{7}{2}}\;\middle|\;2\right)\bigg{)}$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\frac{\Gamma(N+\frac{3}{2})}{N!}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{2{,}\mspace{8.0mu}-\frac{1}{2}}{N+1}\;\middle|\;\frac{1}{2}\right)+\sqrt{\frac{2}{\pi}}\frac{\Gamma(N+\frac{3}{2})}{3(N-1)!}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{3}{2}}{N}\;\middle|\;\frac{1}{2}\right)+(-1)^{N}\frac{N}{2}$
$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{\Gamma(N+\frac{3}{2})}{(N-1)!}\bigg{(}\frac{1}{2N}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{2{,}\mspace{8.0mu}-\frac{1}{2}}{N+1}\;\middle|\;\frac{1}{2}\right)+\frac{1}{3}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{3}{2}}{N}\;\middle|\;\frac{1}{2}\right)\bigg{)}+(-1)^{N}\frac{N}{2},$

which completes the proof. ∎

Focusing now on (2.3) we note from (2.1) with $a=2,b=-1/2,c=1$ that

$\displaystyle\frac{1}{\sqrt{2\pi}}\frac{\Gamma(N+\frac{3}{2})}{N!}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{2{,}\mspace{8.0mu}-\frac{1}{2}}{N+1}\;\middle|\;\frac{1}{2}\right)$
$\displaystyle\sim
N\sqrt{\frac{2}{\pi}N}\sum_{s=1}^{m-1}\frac{\alpha_{s-1}}{2}\frac{\Gamma(-3/2+s)}{\Gamma(-1/2)}N^{-s},$

where

(2.6)
$4\Big{(}\frac{e^{t}-1}{t}\Big{)}^{-3/2}(1+e^{-t})^{-2}=\sum_{s=0}^{\infty}\alpha_{s}\,t^{s}.$

Also, using (2.1) with $a=1,b=-3/2,c=0$, we have

$\displaystyle\sqrt{\frac{2}{\pi}}\frac{\Gamma(N+\frac{3}{2})}{3(N-1)!}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{3}{2}}{N}\;\middle|\;\frac{1}{2}\right)\sim
N\sqrt{\frac{2}{\pi}N}\sum_{s=0}^{m-1}\frac{\beta_{s}}{3}\frac{\Gamma(-3/2+s)}{\Gamma(-3/2)}N^{-s}$

where

(2.7)
$2\Big{(}\frac{e^{t}-1}{t}\Big{)}^{-5/2}e^{t}(1+e^{-t})^{-1}=\sum_{s=0}^{\infty}\beta_{s}\,t^{s}.$

Therefore we have shown that

$\displaystyle\quad\sqrt{\frac{2}{\pi}}\frac{\Gamma(N+\frac{3}{2})}{(N-1)!}\bigg{(}\frac{1}{2N}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{2{,}\mspace{8.0mu}-\frac{1}{2}}{N+1}\;\middle|\;\frac{1}{2}\right)+\frac{1}{3}{}_{2}F_{1}\left(\genfrac{.}{.}{0.0pt}{}{1{,}\mspace{8.0mu}-\frac{3}{2}}{N}\;\middle|\;\frac{1}{2}\right)\bigg{)}$
$\displaystyle\ \sim
N\sqrt{\frac{2}{\pi}N}\bigg{(}\frac{\beta_{0}}{3}+\sum_{s=1}^{m-1}\frac{\beta_{s}-\alpha_{s-1}}{3}\frac{\Gamma(-3/2+s)}{\Gamma(-3/2)}N^{-s}\bigg{)}.$

Substituting in (2.3) the expansion (1.18) follows. $\square$

###  2.2. Proof of Theorem 1.2

In relation to Theorem 1.2, note that the spectral moments (1.25) of complex
eigenvalues follow from (1.7), (1.9) and their real counterparts (1.24). Thus
it suffices to show (1.24). The more general setting of Proposition 1.3 we
will be assumed, requiring only that ${\rm Re}\,p>-1/2$.

By using [11, Cor. 4.1], we have

(2.8) $\sum_{N=1}^{\infty}\rho_{N}^{\rm r}(x)z^{N}=\mathcal{F}(z,x),$

where

(2.9)
$\mathcal{F}(z,x):=\frac{z}{\sqrt{2\pi}}\bigg{(}e^{-\frac{x^{2}}{2}}+\frac{z}{1-z}e^{(z-1)x^{2}}\bigg{)}+\frac{z^{2}|x|}{2}e^{\frac{(z^{2}-1)x^{2}}{2}}\bigg{(}\operatorname{erf}\Big{(}\frac{z|x|}{\sqrt{2}}\Big{)}+\operatorname{erf}\Big{(}\frac{(1-z)|x|}{\sqrt{2}}\Big{)}\bigg{)}.$

It then follows that

(2.10) $\sum_{N=0}^{\infty}M_{2p,N}^{\rm
r}\,z^{N}=\int_{\mathbb{R}}|x|^{2p}\mathcal{F}(z,x)\,dx.$

Since

$\int_{\mathbb{R}}|x|^{2p}e^{-\frac{x^{2}}{2}}\,dx=2^{p+\frac{1}{2}}\,\Gamma\Big{(}p+\frac{1}{2}\Big{)},\qquad\int_{\mathbb{R}}|x|^{2p}e^{(z-1)x^{2}}\,dx=(1-z)^{-p-\frac{1}{2}}\Gamma\Big{(}p+\frac{1}{2}\Big{)},$

where in the second integral it is assumed $|z|<1$, we have

(2.11)
$\displaystyle\frac{z}{\sqrt{2\pi}}\int_{\mathbb{R}}|x|^{2p}\bigg{(}e^{-\frac{x^{2}}{2}}+\frac{z}{1-z}e^{(z-1)x^{2}}\bigg{)}\,dx$
$\displaystyle=\frac{z}{\sqrt{2\pi}}\bigg{(}2^{p+\frac{1}{2}}+\frac{z}{(1-z)^{p+\frac{3}{2}}}\bigg{)}\Gamma\Big{(}p+\frac{1}{2}\Big{)}.$

On the other hand, by using the expansion [26, Eq.(7.6.2)]

(2.12)
$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}}e^{-z^{2}}\sum_{k=0}^{\infty}\frac{2^{k}z^{2k+1}}{(2k+1)!!},$

we have

(2.13)
$\displaystyle\begin{split}&\int_{\mathbb{R}}|x|^{2p}\frac{z^{2}|x|}{2}e^{\frac{(z^{2}-1)x^{2}}{2}}\bigg{(}\operatorname{erf}\Big{(}\frac{z|x|}{\sqrt{2}}\Big{)}+\operatorname{erf}\Big{(}\frac{(1-z)|x|}{\sqrt{2}}\Big{)}\bigg{)}\,dx\\\
&\qquad=\frac{z^{2}}{2}\sqrt{\frac{2}{\pi}}\sum_{k=0}^{\infty}\int_{\mathbb{R}}\frac{|x|^{2p+2k+2}}{(2k+1)!!}\bigg{(}z^{2k+1}e^{-\frac{x^{2}}{2}}+(1-z)^{2k+1}e^{(z-1)x^{2}}\bigg{)}\,dx\\\
&\qquad=\frac{z^{2}}{2}\sqrt{\frac{2}{\pi}}\sum_{k=0}^{\infty}\frac{1}{(2k+1)!!}\bigg{(}z^{2k+1}2^{p+k+\frac{3}{2}}+(1-z)^{k-p-\frac{1}{2}}\bigg{)}\Gamma\Big{(}p+k+\frac{3}{2}\Big{)}.\end{split}$

Combining the above, we have

$\displaystyle\int_{\mathbb{R}}|x|^{2p}\mathcal{F}(z,x)\,dx$
$\displaystyle=\frac{z}{\sqrt{2\pi}}\bigg{(}2^{p+\frac{1}{2}}+\frac{z}{(1-z)^{p+\frac{3}{2}}}\bigg{)}\Gamma\Big{(}p+\frac{1}{2}\Big{)}$
$\displaystyle\quad+\frac{z^{2}}{\sqrt{2\pi}}\sum_{k=0}^{\infty}\frac{1}{(2k+1)!!}\bigg{(}z^{2k+1}2^{p+k+\frac{3}{2}}+(1-z)^{k-p-\frac{1}{2}}\bigg{)}\Gamma\Big{(}p+k+\frac{3}{2}\Big{)},$

which can be written as

(2.14)
$\int_{\mathbb{R}}|x|^{2p}\mathcal{F}(z,x)\,dx=\frac{z^{2}}{\sqrt{2\pi}}\sum_{k=0}^{\infty}\frac{1}{(2k-1)!!}\bigg{(}z^{2k-1}2^{p+k+\frac{1}{2}}+(1-z)^{k-p-\frac{3}{2}}\bigg{)}\Gamma\Big{(}p+k+\frac{1}{2}\Big{)}.$

By using

$\displaystyle(1-z)^{k-p-\frac{3}{2}}=\sum_{l=0}^{\infty}\binom{k-p-3/2}{l}(-z)^{l}=\sum_{l=0}^{\infty}\frac{\Gamma(k-p-\frac{1}{2})}{l!\,\Gamma(k-p-l-\frac{1}{2})}(-z)^{l},$

we obtain

(2.15)
$\displaystyle\begin{split}\int_{\mathbb{R}}|x|^{2p}\mathcal{F}(z,x)\,dx&=\frac{1}{\sqrt{2\pi}}\sum_{k=0}^{\infty}\frac{\Gamma(k+p+\frac{1}{2})2^{p+k+\frac{1}{2}}}{(2k-1)!!}z^{2k+1}\\\
&\quad+\frac{1}{\sqrt{2\pi}}\sum_{N=0}^{\infty}\bigg{(}\sum_{k=0}^{\infty}\frac{(-1)^{N}\Gamma(k+p+\frac{1}{2})\Gamma(k-p-\frac{1}{2})}{(2k-1)!!\,(N-2)!\,\Gamma(k-p-N+\frac{3}{2})}\bigg{)}z^{N}.\end{split}$

This gives rise to

(2.16) $\begin{split}M_{2p,N}^{\rm
r}&=\frac{1}{\sqrt{2\pi}}\frac{(-1)^{N}}{(N-2)!}\sum_{k=0}^{\infty}\frac{\Gamma(k+p+\frac{1}{2})\Gamma(k-p-\frac{1}{2})}{(2k-1)!!\,\Gamma(k-p-N+\frac{3}{2})}+\frac{1}{\sqrt{2\pi}}\frac{\Gamma(p+N/2)2^{p+N/2}}{(N-2)!!}\mathbbm{1}_{\\{N:\textup{
odd}\\}}\\\
&=\frac{1}{(N-2)!}\sum_{k=0}^{\infty}\frac{1}{2^{k+\frac{1}{2}}}\frac{\Gamma(k+p+\frac{1}{2})\Gamma(N+p-k-\frac{1}{2})}{\Gamma(k+\frac{1}{2})\,\Gamma(p-k+\frac{3}{2})}+2^{p}\frac{\Gamma(p+N/2)}{\Gamma(N/2)}\mathbbm{1}_{\\{N:\textup{
odd}\\}}.\end{split}$

Note here that

$\displaystyle\frac{1}{\sqrt{2\pi}}\frac{\Gamma(p+N/2)2^{p+N/2}}{(N-2)!!}$
$\displaystyle=\frac{\Gamma(p+N/2)2^{p}}{\Gamma(N/2)}$

and that

$\displaystyle\frac{1}{\sqrt{2\pi}}\frac{\Gamma(k+p+\frac{1}{2})\Gamma(k-p-\frac{1}{2})}{(2k-1)!!\,\Gamma(k-p-N+\frac{3}{2})}$
$\displaystyle=\frac{1}{2^{k+\frac{1}{2}}}\frac{\Gamma(k+p+\frac{1}{2})\Gamma(k-p-\frac{1}{2})}{\Gamma(k+\frac{1}{2})\,\Gamma(k-p-N+\frac{3}{2})}$
$\displaystyle=\frac{(-1)^{N}}{2^{k+\frac{1}{2}}}\frac{\Gamma(k+p+\frac{1}{2})\Gamma(N+p-k-\frac{1}{2})}{\Gamma(k+\frac{1}{2})\,\Gamma(p-k+\frac{3}{2})},$

where to obtain the final line use has been made of the reflection formula of
Gamma function

(2.17) $\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z).$

Now the expression (1.24) follows from the definition (1.23) of the
generalised hypergeometric function. $\square$

###  2.3. Proof of Proposition 1.3

We follow a strategy introduced in [32, Proofs of Thms. 4 and 9] in relation
to the differential equations satisfied by the eigenvalue density for the GUE
and GOE. Let

$\displaystyle
f(x)=\frac{d}{dx}\bigg{[}\Gamma(N-1,x^{2})+2^{(N-3)/2}e^{-\frac{x^{2}}{2}}x^{N-1}\gamma\Big{(}\frac{N-1}{2},\frac{x^{2}}{2}\Big{)}\bigg{]},$

so that according to (1.10), the function $f(x)$ is proportional to
$\rho_{N}(x)$. In terms of $f(x)$, the differential equation of the
proposition reads

(2.18)
$x^{2}f^{\prime\prime}(x)+x(3x^{2}-3N+4)f^{\prime}(x)+(2x^{2}-2N+1)(x^{2}-N+2)f(x)=0.$

Using

$\displaystyle\frac{d}{dx}\Gamma(N-1,x^{2})=-x^{2N-3}e^{-x^{2}},\qquad
2^{(N-3)/2}e^{-\frac{x^{2}}{2}}\frac{d}{dx}\gamma\Big{(}\frac{N-1}{2},\frac{x^{2}}{2}\Big{)}=x^{N-2}e^{-x^{2}},$

we have

(2.19)
$\displaystyle\begin{split}f(x)&=-x^{2N-3}e^{-x^{2}}+2^{(N-3)/2}e^{-\frac{x^{2}}{2}}x^{N-2}\Big{(}-x^{2}+N-1\Big{)}\gamma\Big{(}\frac{N-1}{2},\frac{x^{2}}{2}\Big{)}\\\
&=:-a(x)+(-x^{2}+N-1)b(x).\end{split}$

Similarly, it follows that

(2.20)
$\displaystyle\begin{split}f^{\prime}(x)&=x^{2N-3}\Big{(}x-\frac{N-2}{x}\Big{)}e^{-x^{2}}\\\
&\quad+2^{(N-3)/2}e^{-\frac{x^{2}}{2}}x^{N-2}\Big{(}x^{3}-(2N-1)x+\frac{(N-1)(N-2)}{x}\Big{)}\gamma\Big{(}\frac{N-1}{2},\frac{x^{2}}{2}\Big{)}\\\
&=\Big{(}x-\frac{N-2}{x}\Big{)}a(x)+\Big{(}x^{3}-(2N-1)x+\frac{(N-1)(N-2)}{x}\Big{)}b(x),\end{split}$

and

(2.21)
$\displaystyle\begin{split}f^{\prime\prime}(x)&=x^{2N-3}\Big{(}-x^{2}+(2N-5)-\frac{(N-2)(N-3)}{x^{2}}\Big{)}e^{-x^{2}}\\\
&\quad+2^{(N-3)/2}e^{-\frac{x^{2}}{2}}x^{N-2}\Big{(}-x^{4}+3N\,x^{2}-3(N-1)^{2}+\frac{(N-1)(N-2)(N-3)}{x^{2}}\Big{)}\gamma\Big{(}\frac{N-1}{2},\frac{x^{2}}{2}\Big{)}\\\
&=\Big{(}-x^{2}+(2N-5)-\frac{(N-2)(N-3)}{x^{2}}\Big{)}a(x)\\\
&\quad+\Big{(}-x^{4}+3N\,x^{2}-3(N-1)^{2}+\frac{(N-1)(N-2)(N-3)}{x^{2}}\Big{)}b(x).\end{split}$

The idea now is to use (2.19) and (2.20) regarded as a linear system to solve
for $a(x)$, $b(x)$ in terms of $f(x)$ and $f^{\prime}(x)$. With this done,
substituting in (2.21) gives the desired differential equation (2.18). The
intermediate working is best carried out using computer algebra for efficiency
and accuracy. $\square$

###  2.4. Proof of Corollary 1.6

To deduce the three term recurrence (1.2) with $p$ in general complex, we
multiply the differential equation by $x^{p}$ and integrate over
$x\in(0,\infty)$. Carrying out the integration using integration by parts
gives (1.2), provided Re$(p)$ is large enough so that all terms requiring
evaluation at $x=0$ vanish. Analytic continuation removes the need for such a
restriction.

In relation to the three term recurrence for the ${}_{3}F_{2}$ function
(1.28), we notice by direct computations that

$\displaystyle 2(2p+5)2^{p+2}\frac{\Gamma(p+2+N/2)}{\Gamma(N/2)}$
$\displaystyle=(2p+3)(6p+4N+7)2^{p+1}\frac{\Gamma(p+1+N/2)}{\Gamma(N/2)}$
$\displaystyle\quad-(2p+1)(2p+N)(2p+2N+1)2^{p}\frac{\Gamma(p+N/2)}{\Gamma(N/2)}.$

Therefore by (1.24), one can observe that the recursion formula (1.2) implies
(1.28). $\square$

###  2.5. Proof of Corollary 1.7

Note that

$\displaystyle\frac{d^{k}}{dt^{k}}u(t)=\int_{\mathbb{R}}e^{tx}x^{k}\rho_{N}^{\rm
r}(x)\,dx.$

Then the differential equation (1.30) follows from Proposition 1.3, after
multiplying by $e^{tx}$ and integrating over $x\in\mathbb{R}$ using
integrating by parts. For the latter, we note

$\displaystyle\frac{d}{dx}\Big{[}(2x^{2}-2N+1)(x^{2}-N+2)e^{tx}\Big{]}-\frac{d^{2}}{dx^{2}}\Big{[}x(3x^{2}-3N+4)e^{tx}\Big{]}+\frac{d^{3}}{dx^{3}}\Big{[}x^{2}e^{tx}\Big{]}$
$\displaystyle\qquad=\Big{(}2tx^{4}-(3t^{2}-8)x^{3}+t(t^{2}-4N-13)x^{2}+((3N+2)t^{2}-8N-8)x+(2N^{2}+N)t\Big{)}e^{tx}.$

In relation to the Stieltjes transform, we first separate out the coefficients
of $\partial_{x}^{2}$ and $\partial_{x}$ in $\mathcal{A}_{N}[x]$ so that the
differential equation (1.27) reads

$\Big{(}x^{2}\partial_{x}^{3}+R(x)\partial_{x}^{2}+S(x)\partial_{x}\big{)}\rho_{N}^{\rm
r}(x)=0,$

with

$R(x)=x(3x^{2}-3N+4),\qquad S(x)=(2x^{2}-2N+1)(x^{2}-N+2).$

Next we manipulate this equation so that it takes the form

$\Big{(}t^{2}\partial_{x}^{3}+R(t)\partial_{x}^{2}+S(t)\partial_{x}\Big{)}\rho_{N}^{\rm
r}(x)=\Big{(}(t^{2}-x^{2})\partial_{x}^{3}+(R(t)-R(x))\partial_{x}^{2}+(S(t)-S(x))\partial_{x}\Big{)}\rho_{N}^{\rm
r}(x).$

Multiplying through by $1/(t-x)$ and integrating over $x\in\mathbb{R}$, the
LHS is readily identified as $\mathcal{A}_{N}[t]$ after integration by parts.
On the RHS the term $1/(t-x)$ can be cancelled with factors in the
coefficients, which then reduce to polynomials symmetric in $t$ and $x$.
Integration by parts requires that these polynomials be differentiated with
respect to $x$ a suitable number of times, with the result giving the RHS of
(1.33). $\square$

##  3\. Links between large N expansions

###  3.1. Asymptotic expansion of the moment generating function

Let us define the rescaled moment generating function

(3.1)
$\widetilde{u}(t):=\frac{1}{\sqrt{N}}u\Big{(}\frac{t}{\sqrt{N}}\Big{)}=\int_{\mathbb{R}}e^{tx}\rho_{N}^{\rm
r}(\sqrt{N}x)\,dx.$

Then (1.30) gives rise to

(3.2)
$\bigg{(}\mathcal{D}_{0}[t]+\frac{\mathcal{D}_{1}[t]}{N}+\frac{\mathcal{D}_{2}[t]}{N^{2}}\bigg{)}\widetilde{u}(t)=0,$

where

(3.3)
$\displaystyle\begin{split}\mathcal{D}_{0}[t]&:=2t\,\partial_{t}^{4}+8\,\partial_{t}^{3}-4t\,\partial_{t}^{2}-8\,\partial_{t}+2t\end{split}$
(3.4) $\displaystyle\mathcal{D}_{1}[t]$
$\displaystyle:=-3t^{2}\,\partial_{t}^{3}-13t\,\partial_{t}^{2}+(3t^{2}-8)\,\partial_{t}+t,$
(3.5) $\displaystyle\mathcal{D}_{2}[t]$
$\displaystyle:=t^{3}\,\partial_{t}^{2}+2t^{2}\,\partial_{t}.$

As is consistent with the expansion of the even integer moments Theorem 1.1,
introduce the large $N$ expansion

(3.6)
$\widetilde{u}(t)=\sum_{k=0}^{\infty}\bigg{(}\frac{\widetilde{u}_{(k)}(t)}{N^{k}}+\frac{\widetilde{u}_{(k+1/2)}(t)}{N^{k+1/2}}\bigg{)},$

where with $\rho_{N}^{\rm r}(\sqrt{N}x)$ expanded in the sense of distribution
according to

(3.7) $\rho_{N}^{\rm
r}(\sqrt{N}x)=\sum_{k=0}^{\infty}\bigg{(}\frac{{r}_{(k)}(t)}{N^{k}}+\frac{{r}_{(k+1/2)}(t)}{N^{k+1/2}}\bigg{)},$

one has

(3.8)
$\widetilde{u}_{(k)}(t)=\int_{\mathbb{R}}e^{tx}r_{(k)}(x)\,dx,\qquad\widetilde{u}_{(k+1/2)}(t)=\int_{\mathbb{R}}e^{tx}r_{(k+1/2)}(x)\,dx.$

We know from [6, pp. 33–34]) that

(3.9) $r_{(0)}(x)=\frac{1}{\sqrt{2\pi}}\,\mathbbm{1}_{(-1,1)}(x),\qquad
r_{(1/2)}(x)=\frac{1}{4}\Big{(}\delta(x-1)+\delta(x+1)\Big{)}$

and so

(3.10)
$\widetilde{u}_{(0)}(t):=\sqrt{\frac{2}{\pi}}\frac{\sinh(t)}{t},\qquad\widetilde{u}_{(1/2)}(t):=\frac{\cosh(t)}{2}.$

Note also that by (1.16) and (1.18), for $k\geq 1$,

(3.11)
$\widetilde{u}_{(k)}(0)=\sqrt{\frac{2}{\pi}}\,a_{k},\qquad\widetilde{u}^{\prime\prime}_{(k)}(0)=\sqrt{\frac{2}{\pi}}\,b_{k}$

and

(3.12)
$\widetilde{u}_{(k+1/2)}(0)=\widetilde{u}^{\prime\prime}_{(k+1/2)}(0)=0.$

Scaling (3.2), $t\mapsto t/\sqrt{N}$, and substituting (3.6) shows

(3.13)
$\mathcal{D}_{0}[t]\,\widetilde{u}_{(k)}(t)+\mathcal{D}_{1}[t]\,\widetilde{u}_{(k-1)}(t)+\mathcal{D}_{2}[t]\,\widetilde{u}_{(k-2)}(t)=0$

and

(3.14)
$\mathcal{D}_{0}[t]\,\widetilde{u}_{(k+1/2)}(t)+\mathcal{D}_{1}[t]\,\widetilde{u}_{(k-1/2)}(t)+\mathcal{D}_{2}[t]\,\widetilde{u}_{(k-3/2)}(t)=0,$

with the convention that $\widetilde{u}_{j}\equiv 0$ if $j<0.$ One observes
that the differential operator $\mathcal{D}_{0}[t]$ has the factorisations

(3.15)
$\mathcal{D}_{0}[t]=2(t\,\partial_{t}^{2}+4\,\partial_{t}-t)\circ(\partial_{t}^{2}-1)=2(\partial_{t}^{2}-1)\circ(t\,\partial_{t}^{2}+2\,\partial_{t}-t).$

From the explicit functional forms of $\widetilde{u}_{(0)}(t)$ and
$\widetilde{u}_{(1/2)}(t)$ (3.10) it is observed that both are annihilated by
$\mathcal{D}_{0}[t]$. Indeed, the general even solution to
$\mathcal{D}_{0}[t]f(t)=0$ is of the form

$f(t)=c_{0}\cosh(t)+c_{1}\frac{\sinh(t)}{t},\qquad c_{0},c_{1}\in\mathbb{R}.$

By taking $k=1$ in (3.13),

$\displaystyle\mathcal{D}_{0}[t]\,\widetilde{u}_{1}(t)+\mathcal{D}_{1}[t]\,\widetilde{u}_{0}(t)=\mathcal{D}_{0}[t]\,\widetilde{u}_{1}(t)-6\sqrt{\frac{2}{\pi}}\sinh(t)=0.$

By solving this differential equation with the initial condition (3.11), we
have

(3.16)
$\displaystyle\widetilde{u}_{(1)}(t)=\sqrt{\frac{2}{\pi}}\frac{3}{8}\bigg{(}t\sinh(t)-\cosh(t)\bigg{)}.$

Similarly, it follows that

(3.17) $\displaystyle\widetilde{u}_{(2)}(t)$
$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{1}{384}\bigg{(}(23t^{2}+9)\,t\,\sinh(t)-(26t^{2}+9)\cosh(t)\bigg{)},$
(3.18) $\displaystyle\widetilde{u}_{(3)}(t)$
$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{1}{15360}\bigg{(}(91t^{4}-285t^{2}-405)\,t\,\sinh(t)-5(t^{4}-84t^{2}-81)\cosh(t)\bigg{)}.$

In general, one can observe that $\widetilde{u}_{k}$ is of the form

(3.19)
$\widetilde{u}_{(k)}(t)=\sqrt{\frac{2}{\pi}}\bigg{(}P_{k,1}(t)\,t\,\sinh(t)+P_{k,2}(t)\,\cosh(t)\bigg{)},$

where $P_{k,1}$ and $P_{k,2}$ are some even polynomials of degree $k+1.$ The
corresponding quantities in the expansion (3.6) are then

(3.20)
${r}_{(k)}(x)=\frac{1}{\sqrt{2\pi}}\bigg{(}P_{k,1}(-\partial_{x})(-\partial_{x})\Big{(}\delta(x-1)-\delta(x+1)\Big{)}+P_{k,2}(-\partial_{x})\Big{(}\delta(x-1)+\delta(x+1)\Big{)}\bigg{)}$

as can be checked from (3.8).

Regarding the half integer coefficients in (3.6), we also have

(3.21) $\displaystyle\widetilde{u}_{(3/2)}(t)$
$\displaystyle=\frac{1}{8}\bigg{(}t^{2}\,\cosh(t)-t\,\sinh(t)\bigg{)},$ (3.22)
$\displaystyle\widetilde{u}_{(5/2)}(t)$
$\displaystyle=\frac{1}{192}\bigg{(}3t^{2}(t^{2}-1)\cosh(t)-t(2t^{2}-3)\sinh(t)\bigg{)}.$

For general $k,$ we have

(3.23)
$\widetilde{u}_{(k+1/2)}(t)=\widehat{P}_{k,1}(t)\,t\,\cosh(t)+\widehat{P}_{k,2}(t)\,\sinh(t),$

where $\widehat{P}_{k,1}$ and $\widehat{P}_{k,2}$ are certain odd polynomials
of degree $k+1$, from which it follows

(3.24)
${r}_{(k+1/2)}(x)=\frac{1}{2}\bigg{(}\widehat{P}_{k,1}(-\partial_{x})(-\partial_{x})\Big{(}\delta(x-1)+\delta(x+1)\Big{)}+\widehat{P}_{k,2}(-\partial_{x})\Big{(}\delta(x-1)-\delta(x+1)\Big{)}\bigg{)};$

cf. (3.20). Notice also that

(3.25)
$\displaystyle\widetilde{u}_{(3/2)}(0)=\widetilde{u}_{(3/2)}^{\prime\prime}(0)=0,\qquad\widetilde{u}_{(3/2)}^{\prime\prime\prime\prime}(0)=1,\qquad\widetilde{u}_{(3/2)}^{(6)}(0)=3,\qquad\widetilde{u}_{(3/2)}^{(8)}(0)=6,\qquad\widetilde{u}_{(3/2)}^{(10)}(0)=10.$

These coincide with the coefficients of the ${\rm O}(1/N)$ term in (1.22).
Along the same lines, we have

(3.26)
$\displaystyle\widetilde{u}_{(5/2)}(0)=\widetilde{u}_{(5/2)}^{\prime\prime}(0)=\widetilde{u}_{(5/2)}^{\prime\prime\prime\prime}(0)=0,\qquad\widetilde{u}_{(5/2)}^{(6)}(0)=4,\qquad\widetilde{u}_{(5/2)}^{(8)}(0)=22,\qquad\widetilde{u}_{(5/2)}^{(10)}(0)=70$

which also correspond to the coefficients of the ${\rm O}(1/N^{2})$ term in
(1.22). To be consistent with the fact that the final sum in (1.21)
terminates, for general $k$ we must have
${\partial_{t}^{2j}}\tilde{u}_{(k+1/2)}(t)|_{t=0}=0$ for $j=0,\dots,k$.

###  3.2. Asymptotic expansion of the Stieltjes transform

Let us write

(3.27) $\widetilde{W}(t):=\int_{\mathbb{R}}\frac{\rho_{N}^{\rm
r}(\sqrt{N}x)}{t-x}\,dx=W(\sqrt{N}t)$

for the Stieltjes transform of the rescaled density. Then (1.33) can be
rewritten as

(3.28)
$\bigg{(}\widehat{\mathcal{D}}_{0}[t]+\frac{\widehat{\mathcal{D}}_{1}[t]}{N}+\frac{\widehat{\mathcal{D}}_{2}[t]}{N^{2}}\bigg{)}\widetilde{W}(t)=\Big{(}4-2t^{2}+\frac{1}{N}\Big{)}\frac{M_{0,N}^{\rm
r}}{N^{1/2}}-6\frac{M_{N,2}^{\rm r}}{N^{3/2}},$

where

(3.29)
$\displaystyle\widehat{\mathcal{D}}_{0}[t]:=2(t^{2}-1)^{2}\partial_{t},$
(3.30)
$\displaystyle\widehat{\mathcal{D}}_{1}[t]:=(t^{2}-1)(3t\,\partial_{t}^{2}+5\,\partial_{t}),$
(3.31)
$\displaystyle\widehat{\mathcal{D}}_{2}[t]:=t^{2}\,\partial_{t}^{3}+4t\,\partial_{t}^{2}+2\,\partial_{t}.$

On the other hand, by Theorem 1.1,

(3.32) $\Big{(}4-2t^{2}+\frac{1}{N}\Big{)}\frac{M_{0,N}^{\rm
r}}{N^{1/2}}-6\frac{M_{N,2}^{\rm
r}}{N^{3/2}}\sim\sqrt{\frac{2}{\pi}}\bigg{(}2-2t^{2}+\sum_{l=1}^{\infty}\frac{(4-2t^{2})a_{l}+a_{l-1}-6b_{l}}{N^{l}}\bigg{)}-\frac{t^{2}+1}{N^{1/2}}+\frac{1}{2N^{3/2}}.$

Then as before, by recursively solving this system of differential equations
with the initial condition $\widetilde{W}(t)={\rm O}(1/t)$ as $t\to\infty$,
one can derive the expansion

(3.33)
$\widetilde{W}(t)=\sum_{k=0}^{\infty}\bigg{(}\frac{\widetilde{W}_{(k)}(t)}{N^{k}}+\frac{\widetilde{W}_{(k+1/2)}(t)}{N^{k+1/2}}\bigg{)}.$

For instance, we have

(3.34)
$\displaystyle\widetilde{W}_{(0)}(t)=\frac{1}{\sqrt{2\pi}}\log\Big{(}\frac{t+1}{t-1}\Big{)},\qquad\widetilde{W}_{(1/2)}(t)=\frac{t}{2(t^{2}-1)},$

which are consistent with (3.9), and

(3.35)
$\widetilde{W}_{(1)}(t)=-\frac{1}{\sqrt{2\pi}}\frac{3t(t^{2}-3)}{4(t^{2}-1)^{2}},\qquad\widetilde{W}_{(3/2)}(t)=\frac{t}{(t^{2}-1)^{3}}.$

In particular, one reads off that as $t\to\infty$,
$\widetilde{W}_{(1)}(t)\asymp t^{-1}$, whereas $\widetilde{W}_{(3/2)}(t)\asymp
t^{-3}$. Generally it is required that for $t\to\infty$,
$\widetilde{W}_{(k)}(t)\asymp t^{-1}$, whereas
$\widetilde{W}_{(2k+1)/2}(t)\asymp{t^{-2k-1}}$, so as to be consistent with
(1.21).

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