$\displaystyle\Bigg{|}\int_{\mathbb{R}^{N}}\prod_{q=2}^{k}\left(\sum_{j=1}^{N}(x_{j}-\textnormal{i})^{q}\right)^{n_{q}}\prod_{j=1}^{N}$
$\displaystyle\frac{e^{-\textnormal{i}\frac{t_{1}}{N}x_{j}}}{\left(1+x_{j}^{2}\right)^{s+N}}\Delta^{2}(\mathbf{x})\mathrm{d}\mathbf{x}\Bigg{|}$
$\displaystyle\leq\int_{\mathbb{R}^{N}}\prod_{q=2}^{k}\left(\sum_{j=1}^{N}\big{(}1+x_{j}^{2}\big{)}^{q/2}\right)^{n_{q}}\prod_{j=1}^{N}\frac{1}{\left(1+x_{j}^{2}\right)^{l+N}}\Delta^{2}(\mathbf{x})\mathrm{d}\mathbf{x}.$

Now, observe that proving (1.9) amounts to showing the equality after the
averages are replaced by these integrals, due to the cancellation of the
normalization constants of $\mathbf{M}_{N}^{(s)}$. Thus, using the bounds for
the integrals above and the fact that $P_{m}^{(s,N)}(t_{1})$ are polynomials
in $s$, we see that both sides satisfy the conditions of Carlson’s lemma.
Therefore, if they agree for $s\in\mathbb{N}$, $s\geq l$, then they agree for
all $s$ with $\Re(s)\geq l$. Hence, noting that both sides are analytic on
$\Re(s)>2^{-1}(\sum_{q=2}^{k}qn_{q}-1)$ and using the identity theorem, we get
the desired result. ∎

Before moving on to the actual proof of Theorem 1.9, we show that it indeed
implies Theorem 1.11. Without using the machinery we developed earlier, making
rigorous the large $N$ limits in the proof below appears to be difficult.

###### Proof of Theorem 1.11 assuming Theorem 1.9.

First, note that by an application of Lemma 4.1, the differential equality
(1.9) holds for any $s\in\mathbb{R}$ with $s>2^{-1}(\sum_{q=2}^{k}qn_{q}-1)$,
so that it now suffices to prove that for all $t\in\mathbb{R}$:

$\displaystyle\frac{1}{N^{\sum_{m=2}^{k}mn_{m}}}\mathbb{E}_{N}^{(s)}\left[e^{-\textnormal{i}\frac{t}{N}\sum_{j=1}^{N}\mathsf{x}^{(N)}_{j}}\prod_{m=2}^{k}\left(\sum_{j=1}^{N}(\mathsf{x}^{(N)}_{j}-\textnormal{i})^{m}\right)^{n_{m}}\right]$
$\displaystyle\xrightarrow[]{N\to\infty}\mathbb{E}\left[e^{-\textnormal{i}t\mathsf{q}_{1}(s)}\prod_{m=2}^{k}\left(\mathsf{q}_{m}(s)\right)^{n_{m}}\right],$
(40)
$\displaystyle\frac{\mathrm{d}^{m}}{\mathrm{d}t^{m}}\mathbb{E}_{N}^{(s)}\left[e^{-\textnormal{i}t\sum_{j=1}^{N}\frac{\mathsf{x}^{(N)}_{j}}{N}}\right]$
$\displaystyle\xrightarrow[]{N\to\infty}\frac{\mathrm{d}^{m}}{\mathrm{d}t^{m}}\mathbb{E}\left[e^{-\textnormal{i}t\mathsf{q}_{1}(s)}\right].$

The second limit follows immediately, for any $m<2s+1$, from our results in
Section 3. To prove the first one, on the other hand, note that if we denote

$\mathrm{p}_{m}(x_{1},x_{2},\ldots,x_{N})=\sum_{j=1}^{N}x_{j}^{m}$

then we have the idenity

$\mathrm{p}_{m}(x_{1},x_{2},\ldots,x_{n})=(-1)^{m-1}m\mathrm{e}_{m}(x_{1},\ldots,x_{n})+\sum_{j=1}^{m-1}(-1)^{m-1+j}\mathrm{e}_{m-j}(x_{1},\ldots,x_{n})\mathrm{p}_{j}(x_{1},\ldots,x_{n}).$

Hence, using the convergence results from Section 3 for
$\frac{1}{N^{k}}\mathrm{e}_{k}(\mathsf{x}_{1}^{(N)},\mathsf{x}_{2}^{(N)},\ldots,\mathsf{x}_{N}^{(N)})$
and applying the identity above iteratively, we see that for all
$m=2,\ldots,k$,

$\frac{\mathrm{p}_{m}\left(\mathsf{x}_{1}^{(N)},\mathsf{x}_{2}^{(N)},\ldots,\mathsf{x}_{N}^{(N)}\right)}{N^{m}}\xrightarrow[]{N\to\infty}\mathsf{q}_{m}(s),$

almost surely and in $L^{n_{m}}$. In particular, the relation above still
holds if $\mathrm{p}_{m}(\mathsf{x}_{1}^{(N)},\ldots,\mathsf{x}_{N}^{(N)})$ is
replaced by $\sum_{j=1}^{N}\left(\mathsf{x}^{(N)}_{j}-i\right)^{m}$. Thus,
arguing as in the proof of Proposition 2.8 we see that (40) indeed holds,
where the polynomials $\mathcal{A}_{m}^{(s)}$ are given by,

$\mathcal{A}_{m}^{(s)}(t)=\lim_{N\to\infty}\frac{P_{m}^{(s,N)}(t)}{N^{\sum_{q=2}^{k}(q-1)n_{q}}},$

and this concludes the proof. ∎

The rest of this section will be devoted to proving Theorem 1.9. Again, we
stress that by virtue of Lemma 4.1 we only need to prove it for
$s\in\mathbb{N}$. The strategy we will adopt can be summarized in the
following steps:

  * •

Step 1: We will write the averages of the form

$\mathcal{L}_{N,s}^{(\ell_{2},\ldots,\ell_{k})}\left(t_{1}\right)\overset{\textnormal{def}}{=}\mathbb{E}_{N}^{(s)}\left[e^{\textnormal{i}\frac{t_{1}}{N}\sum_{j=1}^{N}\mathsf{x}_{j}^{(N)}}\left(\sum_{j=1}^{N}\left(\mathsf{x}_{j}^{(N)}-\textnormal{i}\right)^{2}\right)^{\ell_{2}}\cdots\left(\sum_{j=1}^{N}\left(\mathsf{x}_{j}^{(N)}-\textnormal{i}\right)^{k}\right)^{\ell_{k}}\right],$
(41)

in terms of derivatives of Hankel-determinants. To be more precise, we will
show that the above average can be given as

$\prod_{m=2}^{k}\frac{\partial^{\ell_{m}}}{\partial
t_{m}^{\ell_{m}}}\det_{0\leq i,j\leq
N-1}\left[\psi_{i+j}(t_{1},\ldots,t_{k})\right]\Big{|}_{t_{2},\ldots,t_{k}=0}=\prod_{m=2}^{k}\frac{\partial^{\ell_{m}}}{\partial
t_{m}^{\ell_{m}}}\Psi_{N}(t_{1},\ldots,t_{k})\Big{|}_{t_{2},\ldots,t_{k}=0},$

for some explicit functions $\psi_{\gamma}(t_{1},\ldots,t_{k})$ with
$\Psi_{N}(t_{1},\dots,t_{k})$ denoting their Hankel determinant.

  * •

Step 2: Next, we will introduce the associated functions
$\Psi_{N,\boldsymbol{\lambda}}$ where $\boldsymbol{\lambda}$ are integer
partitions. These are basically generalizations of the Hankel determinant on
the left hand-side of the equation above, such that
$\Psi_{N,\emptyset}=\Psi_{N}$. We will then show that the $t_{m}$-derivatives
of $\Psi_{N}$, where $m=2,\ldots,k$ are given as linear combinations of
$\Psi_{N,\boldsymbol{\lambda}}$ where $\boldsymbol{\lambda}$ ranges over a
specific class of partitions.

  * •

Step 3: Moving on, we will show that the quantities
$\Psi_{N,\boldsymbol{\lambda}}$ satisfy certain recursive relations as
$\boldsymbol{\lambda}$ ranges over a specific class of integer partitions.

  * •

Step 4: Finally, we will use these recursive relations to prove the result by
a delicate induction argument.

###  4.2 Joint moments and Hankel determinants

Our starting point is the following proposition.

######  Proposition 4.2.

Let $l,N\geq 1$ and $s>l+\frac{1}{2}$ an integer. Let $\mathsf{U}(a,b;z)$ be
the confluent hypergeometric function of second kind, and let

$\displaystyle\theta_{m}(t_{1})\overset{\textnormal{def}}{=}\exp(-t_{1})\mathsf{U}(1-N-s,2-2N-2s+m;2t_{1}).$
(42)

Then, we have

$\displaystyle\mathcal{L}_{N,s}^{(\ell_{2},\ldots,\ell_{k})}\left(t_{1}\right)=C_{N}^{(s)}(-2\textnormal{i})^{\sum_{m=2}^{k}m\ell_{m}}\prod_{m=2}^{k}\frac{\partial^{\ell_{m}}}{\partial
t_{m}^{\ell_{m}}}\det_{0\leq i,j\leq
N-1}\left[\sum_{m_{2},\ldots,m_{k}=0}^{\infty}\frac{t_{2}^{m_{2}}\cdots
t_{k}^{m_{k}}}{m_{1}!\cdots
m_{k}!}\theta_{i+j+\sum_{l=2}^{k}lm_{l}}(t_{1})\right]\Bigg{|}_{t_{2}=\cdots=t_{k}=0}$

for some constant $C_{N}^{(s)}$, whose explicit value will not play a role in
what follows.

It will be clear, while proving the result above, that the earlier restriction
to $s\in\mathbb{N}$ guarantees that the power sum considered above is well-
defined, and has derivatives of all orders. This will then allow us to use the
recursive relations for $\theta_{\gamma}$ to obtain desired relations for
these series, which will be the main ingredient in proving various relations
between Hankel determinants of concern. Moving on, to prove the Proposition
above, we will need two elementary lemmas.

######  Lemma 4.3.

Let $g(t_{1},\ldots,t_{N})$ be a symmetric function of the form

$g(t_{1},\ldots,t_{N})=\sum_{r=1}^{R}a_{r}\prod_{i=1}^{N}f_{r,i}(t_{i})$

for some functions $f_{r,i}$. Then,

$\prod_{1\leq i<j\leq n}\Bigg{(}\frac{\partial}{\partial
y_{j}}-\frac{\partial}{\partial
y_{i}}\Bigg{)}^{2}g(t_{1},\ldots,t_{N})\;\Big{\rvert}_{t_{1}=\cdots=t_{N}=t}=N!\sum_{r=1}^{R}a_{r}\det_{0\leq
i,j\leq N-1}\left[f_{r,i+1}^{(i+j)}(t)\right].$

######  Lemma 4.4.

Let $N\geq 1$, $s\in\mathbb{N}$, and let $I\subset\mathbb{R}$ be a compact
set. Then, there exists a constant $K=K_{N,I,s}$ so that for all $t\in I$,

$\left|\theta_{m}(t)\right|\leq K_{N,I,s}m^{N+s}$ (43)

where $\theta_{m}$ is defined as in (42).

###### Proof.

Using the well-known relation between the confluent hypergeometric function
$\mathsf{U}$ and Laguerre polynomials (see, for instance, [8, p. 755]), we
have

$\mathsf{U}(1-N-s,2-2N-2s+m;2t_{1})=(-1)^{N+s-1}(N+s-1)!\>\mathsf{L}_{N+s-1}^{(1-2N-2s+m)}(2t_{1}),$

where the Laguerre polynomial $\mathsf{L}_{n}^{(\alpha)}$ is given by

$\mathsf{L}_{n}^{(\alpha)}(z)\overset{\textnormal{def}}{=}\frac{z^{-\alpha}e^{z}}{n!}\frac{\mathrm{d}^{n}}{\mathrm{d}z^{n}}\left(e^{-z}z^{n+\alpha}\right).$

Computing the derivative explicitly gives the desired bound immediately. ∎

###### Proof of Proposition 4.2.

We start by noting that (from here on the constant $C_{N}^{(s)}$ might change
from line to line)

$\displaystyle\mathcal{L}_{N,s}^{(\ell_{2},\ldots,\ell_{k})}(t)$
$\displaystyle=C_{N}^{(s)}e^{Nt}\int_{\mathbb{R}^{N}}\prod_{j=1}^{N}\frac{e^{-\textnormal{i}t(x_{j}-\textnormal{i})}}{\left(1+x_{j}^{2}\right)^{s+N}}\prod_{m=2}^{k}\left(\sum_{j=1}^{N}(x_{j}-\textnormal{i})^{m}\right)^{\ell_{m}}\Delta^{2}(x_{1},\ldots,x_{N})\mathrm{d}\mathbf{x}$
$\displaystyle=C_{N}^{(s)}e^{Nt}\prod_{1\leq i<j\leq
n}\Bigg{(}\frac{\partial}{\partial t_{j}}-\frac{\partial}{\partial
t_{i}}\Bigg{)}^{2}\int_{\mathbb{R}^{N}}\prod_{j=1}^{N}\frac{e^{-\textnormal{i}t_{j}(x_{j}-\textnormal{i})}}{\left(1+x_{j}^{2}\right)^{s+N}}\prod_{m=2}^{k}\left(\sum_{j=1}^{N}(x_{j}-\textnormal{i})^{m}\right)^{\ell_{m}}\mathrm{d}\mathbf{x}\Big{\rvert}_{t_{1}=\cdots=t_{N}=t}.$

Now, expanding the powers of $\sum_{j}(x_{j}-\textnormal{i})^{m}$ in the final
integral above and applying Lemma 4.3 to this symmetric function, we get that

$\mathcal{L}_{N,s}^{(\ell_{2},\ldots,\ell_{k})}(t)=C_{N}^{(s)}e^{Nt}\textnormal{i}^{\sum_{m=2}^{k}m\ell_{m}}\sum_{\begin{subarray}{c}\mu_{2,1}+\cdots+\mu_{2,N}=\ell_{2}\\\\[-4.0pt]
\vdots\\\\[2.0pt]
\mu_{k,1}+\cdots+\mu_{k,N}=\ell_{k}\end{subarray}}\prod_{m=2}^{k}{\ell_{m}\choose{\mu_{m,1},\dots,\mu_{m,N}}}\det_{0\leq
i,j\leq N-1}\left[f^{(i+j+\sum_{m=2}^{k}m\mu_{m,j})}(t)\right]$

where the function $f$ is given by:

$f(t)\overset{\textnormal{def}}{=}\int_{\mathbb{R}}\frac{e^{-\textnormal{i}t(x-\textnormal{i})}}{(1+x^{2})^{s+N}}\mathrm{d}x=B_{N}^{(s)}e^{-2t}\mathsf{U}(1-N-s,2N-2s;2t),$

for some constant $B_{N}^{(s)}$. But then, combining the growth condition from
Lemma 4.4 and the differential relation (see [21, p.119 formula (12)]),

$\frac{\mathrm{d}^{m}}{\mathrm{d}t^{m}}\Big{(}\exp(-2t)\mathsf{U}(1-N-s,2-2N-2s;2t)\Big{)}=(-2)^{m}\exp(-2t)\mathsf{U}(1-N-s,2-2N-2s+m;2t),$

we get the desired result. ∎

###  4.3 Hankel determinants shifted by partitions

Following the strategy sketched earlier we define the Hankel determinants:

$\Psi_{N}(t_{1},\ldots,t_{k})=\det_{0\leq i,j\leq
N-1}\left[\sum_{m_{2},\dots,m_{k}=0}^{\infty}\frac{t_{2}^{m_{2}}\cdots
t_{k}^{m_{k}}}{m_{1}!\cdots
m_{k}!}\theta_{i+j+\sum_{l=2}^{k}lm_{l}}(t_{1})\right].$ (44)

Next, to make the presentation clearer, we fix some notation that will be used
throughout the rest of this section. Firstly, define

$\psi_{\gamma}(t_{1},\ldots,t_{k})\overset{\textnormal{def}}{=}\sum_{m_{2},\ldots,m_{k}=0}^{\infty}\frac{t_{2}^{m_{2}}\cdots
t_{k}^{m_{k}}}{m_{1}!\cdots
m_{k}!}\theta_{\gamma+\sum_{l=2}^{k}lm_{l}}(t_{1}),$

for any $\gamma\in\mathbb{N}$. Note that these functions are all well-defined
due to Lemma 4.4. Next, we define shifted and weighted versions of the matrix
whose determinant defines $\Psi_{N}$ in (44). For each integer partition
$\boldsymbol{\lambda}=(\lambda_{1}\geq\cdots\geq\lambda_{p})$, we let

$\mathbf{A}_{N,\boldsymbol{\lambda}}(t_{1},\ldots,t_{k})\overset{\textnormal{def}}{=}\left[\psi_{i+j+\lambda_{N-j}}(t_{1},\ldots,t_{k})\right]_{i,j=0,\ldots,N-1},$

where if $b\leq N-1$, we let $\lambda_{b+1}=\cdots=\lambda_{N}=0$.
Accordingly, we let
$\Psi_{N,\boldsymbol{\lambda}}=\det[\mathbf{A}_{N,\boldsymbol{\lambda}}]$.
Related to these matrices, we also define their weighted version:

${\mathbf{A}}_{N,\boldsymbol{\lambda}}^{(w)}\overset{\textnormal{def}}{=}\left[(i+j+\lambda_{N-j})\psi_{i+j+\lambda_{N-j}}(t_{1},\ldots,t_{k})\right]_{i,j=0,\ldots,N-1}.$

Finally, we define the main objects of our computation, which will allow us to
obtain recurrences for derivatives of $\psi_{N,\boldsymbol{\lambda}}$.

######  Definition 4.5.

For each integer partition
$\boldsymbol{\lambda}=(\lambda_{1},\ldots,\lambda_{p})$ with $p\leq N$, define
the shifted version:

$\mathcal{S}_{h}\boldsymbol{\lambda}=(\lambda_{1}+h,\ldots,\lambda_{p}+h,h,\ldots,h).$

Hence, we define the functions $\Psi_{N,\boldsymbol{\lambda},h}$ by,

$\Psi_{N,\boldsymbol{\lambda},h}(t_{1},t_{2},\ldots,t_{k})\overset{\textnormal{def}}{=}\mathrm{Tr}\left[\mathrm{adj}(\mathbf{A}_{N,\mathcal{S}_{h}\boldsymbol{\lambda}})\mathbf{A}_{N,\boldsymbol{\lambda}}\right],$

and ${\Psi}_{N,\boldsymbol{\lambda},h}^{(w)}$ by,

${\Psi}_{N,\boldsymbol{\lambda},h}^{(w)}(t_{1},t_{2},\ldots,t_{k})\overset{\textnormal{def}}{=}\mathrm{Tr}\left[\mathrm{adj}\left({{\mathbf{A}}}_{N,\mathcal{S}_{h}\boldsymbol{\lambda}}^{(w)}\right){\mathbf{A}}_{N,\boldsymbol{\lambda}}\right],$

where $\mathrm{adj}$ denotes the adjugate of a matrix.

It turns out that in most of our computations we will have expressions which
involve a certain distinguished partition. For this reason we define:

######  Definition 4.6.

Let $1\leq q\leq k$. Define the partition $\boldsymbol{\lambda}_{k,q}$ to be
the partition $(k-q+1,1,\ldots,1)$ where the number of $1$’s that appear at
the end is $q-1$.

In what follows, we will first show certain recursive and differential
relations satisfied by the functions $\psi_{\gamma}$, and then leverage these
to obtain relations between derivatives of $\Psi_{N}$ and its shifted versions
defined above.

######  Lemma 4.7.

For each $\gamma\geq 0$, we have that

$\frac{\partial\psi_{\gamma}}{\partial t_{1}}=\psi_{\gamma}-2\psi_{\gamma+1},$

and also

$\displaystyle\psi_{\gamma+2}$
$\displaystyle=\frac{N+s-1-\gamma}{2(t_{1}+t_{2})}\psi_{\gamma}+\left(\frac{2-2N-2s+\gamma}{2(t_{1}+t_{2})}+\frac{t_{1}}{t_{1}+t_{2}}\right)\psi_{\gamma+1}$
$\displaystyle+\frac{1}{2(t_{1}+t_{2})}\sum_{q=3}^{k}((q-1)t_{q-1}-qt_{q})\psi_{\gamma+q}+\frac{kt_{k}}{2(t_{1}+t_{2})}\psi_{\gamma+1+k}.$

###### Proof.

This follows from the properties of the confluent hypergeometric functions

$\displaystyle(b-a-1)\mathsf{U}(a,b-1;t)-(b-1+t)\mathsf{U}(a,b;t)+t\mathsf{U}(a,b+1;t)=0,$
(45)
$\displaystyle\frac{\mathrm{d}\mathsf{U}(a,b;t)}{\mathrm{d}t}=\mathsf{U}(a,b;t)-\mathsf{U}(a,b+1;t),$
(46)

which can be found in [54, p.257 formula (5) and p.258 formula (10)]. Indeed,
the first claim is immediate from the second identity above, whereas using the
first one, we get:

$\displaystyle\psi_{\gamma+2}(t_{1},\dots,t_{k})$
$\displaystyle=\frac{N+s-1-\gamma}{2t_{1}}\psi_{\gamma}(t_{1},\dots,t_{k})-\frac{1}{2t_{1}}\sum_{q=2}^{k}qt_{q}\psi_{\gamma+q}(t_{1},\dots,t_{k})$
$\displaystyle+\left(\frac{2-2N-2s+\gamma}{2t_{1}}+1\right)\psi_{\gamma+1}(t_{1},\dots,t_{k})+\frac{1}{2t_{1}}\sum_{q=2}^{k}qt_{q}\psi_{\gamma+1+q}(t_{1},\dots,t_{k}).$

Now, note that the only term $\psi_{\gamma+2}$ appearing on the right-hand
side above is the $q=2$ term of the first sum. Moving that to the left-hand
side we get the desired result. ∎

For ease of notation, we will rescale the functions $\psi_{\gamma}$ by mapping
$t_{i}\to\frac{t_{i}}{N}$, which does not change the method and only makes the
notation simpler. Hence, we denote

$\boldsymbol{\psi}_{\gamma}(t_{1},\ldots,t_{N})\overset{\textnormal{def}}{=}\psi_{\gamma}\left(\frac{t_{1}}{N},\ldots,\frac{t_{N}}{N}\right).$

We also define $\mathbf{\Psi}_{N,\boldsymbol{\lambda}}$,
$\mathbf{\Psi}_{N,\boldsymbol{\lambda},h}$ and
${\mathbf{\Psi}}_{N,\boldsymbol{\lambda},h}^{(w)}$ as their non-boldface
versions but with $\psi_{\gamma}$ now replaced by
$\boldsymbol{\psi}_{\gamma}$.

######  Lemma 4.8.

Let $k\geq 2$ and $m\geq 3$. Let $\boldsymbol{\lambda}$ be a partition. Then
we have:

$\displaystyle\mathbf{\Psi}_{N,\boldsymbol{\lambda},m}(t_{1},\ldots,t_{k})$
$\displaystyle=\frac{N(N+s-1)}{2(t_{1}+t_{2})}\mathbf{\Psi}_{N,\boldsymbol{\lambda},m-2}(t_{1},\ldots,t_{k})-\frac{N}{2(t_{1}+t_{2})}{\mathbf{\Psi}}^{(w)}_{N,\boldsymbol{\lambda},m-2}(t_{1},\ldots,t_{k})$
$\displaystyle+\,\frac{N(1-2N-2s)}{2(t_{1}+t_{2})}\mathbf{\Psi}_{N,\boldsymbol{\lambda},m-1}(t_{1},\ldots,t_{k})+\frac{N}{2(t_{1}+t_{2})}{\mathbf{\Psi}}^{(w)}_{N,\boldsymbol{\lambda},m-1}(t_{1},\ldots,t_{k})$
$\displaystyle+\,\frac{t_{1}}{t_{1}+t_{2}}\mathbf{\Psi}_{N,\boldsymbol{\lambda},m-1}(t_{1},\ldots,t_{k})+\frac{kt_{k}}{2(t_{1}+t_{2})}\mathbf{\Psi}_{N,\boldsymbol{\lambda},m+k-1}(t_{1},\ldots,t_{k})$
$\displaystyle+\frac{1}{2(t_{1}+t_{2})}\sum_{q=3}^{k}\big{(}(q-1)t_{q-1}-qt_{q}\big{)}\mathbf{\Psi}_{N,\boldsymbol{\lambda},m+q-2}(t_{1},\ldots,t_{k}).$
(47)

###### Proof.

By combining Lemma 4.7 with the identities

$\displaystyle\mathbf{\Psi}_{N,\emptyset,l}$
$\displaystyle=\sum_{j=1}^{l}(-1)^{j-1}\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{l,j}},$
$\displaystyle{\mathbf{\Psi}}^{(w)}_{N,\emptyset,l}$
$\displaystyle=\sum_{j=1}^{l}(-1)^{j-1}(2N-2j+l+\alpha)\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{l,j}},$
(48)

the result easily follows. ∎

######  Lemma 4.9.

For each $q=2,\ldots,k$ we have that

$\displaystyle\frac{\partial\mathbf{\Psi}_{N,\boldsymbol{\lambda}}}{\partial
t_{q}}=\frac{1}{N}\mathbf{\Psi}_{N,\boldsymbol{\lambda},q}$

and also that

$\displaystyle\mathbf{\Psi}_{N,\boldsymbol{\lambda},1}=-\frac{N}{2}\frac{\partial\mathbf{\Psi}_{N,\boldsymbol{\lambda}}}{\partial{t_{1}}}+\frac{N}{2}\mathbf{\Psi}_{N,\boldsymbol{\lambda}}.$
(49)

###### Proof.

The first claim follows by noting that

$\frac{\partial\boldsymbol{\psi}_{\gamma}}{\partial
t_{q}}=\frac{1}{N}\boldsymbol{\psi}_{\gamma+q}$

whereas the second claim follows from (4.7). ∎

Next, we give recursive formulae for $\mathbf{\Psi}_{N,\boldsymbol{\lambda}}$.
To do this in a compact way, we first need some notation.

######  Definition 4.10.

For each $r,b,p\in\mathbb{N}$ with $p\leq b$ we define

$\mathbf{\Psi}_{N}^{(r)}[b;p]\overset{\textnormal{def}}{=}\left(\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{b,p}},\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{b,p+1}},\ldots,\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{b,b}},0,\ldots,0\right)^{\textnormal{T}},$

where $r$ specifies the number of $0$’s that appear.

Next, we give recursive formulae for the vectors defined above. These
recursions will represent certain vectors as linear combinations of other
allied vectors after they are multiplied by suitable matrices. These matrices
do have explicit expressions, which we give in the Appendix.

######  Proposition 4.11.

Let $N,s,l,k\in\mathbb{N}$ with $k\geq 2,l\geq 3$. Then, there are matrices
$\mathbf{B}^{(l)}$ and $\mathbf{Q}_{m}^{(l)}$ with $m=0,1,\ldots,k+1$, all of
which, except $\mathbf{Q}_{2}^{(l)}$, have all entries independent of $N$ and
$s$, so that:

$\displaystyle\mathbf{\Psi}_{N}^{(0)}$
$\displaystyle[l;1]=\frac{kt_{k}(-1)^{k+1}\mathbf{B}^{(l)}}{2(t_{1}+t_{2})}\left(-\frac{N}{2}\frac{\partial}{\partial
t_{1}}+\frac{N}{2}\right)\mathbf{\Psi}_{N}^{(0)}[l+k-2;k]$
$\displaystyle+\mathbf{B}^{(l)}\Bigg{(}\frac{N}{2}\frac{\partial}{\partial
t_{1}}-\frac{N}{2}+\frac{N}{2(t_{1}+t_{2})}\Bigg{(}\sum_{m=3}^{k}\mathsf{diff}_{m-1,m}\frac{\partial}{\partial
t_{m-1}}+kt_{k}\frac{\partial}{\partial
t_{k}}\Bigg{)}\Bigg{)}\mathbf{\Psi}_{N}^{(1)}[l-1;1]$
$\displaystyle+\frac{1}{2(t_{1}+t_{2})}\sum_{m=1}^{k-2}\mathsf{diff}_{m+1,m+2}(-1)^{m}\mathbf{Q}_{m+2}^{(l)}\mathbf{\Psi}_{N}^{(0)}[l+m;1]$
$\displaystyle+\left(\frac{t_{1}}{t_{1}+t_{2}}\mathbf{Q}_{1}^{(l)}+\frac{N}{2(t_{1}+t_{2})}{\mathbf{Q}}_{0}^{(l)}\right)\mathbf{\Psi}_{N}^{(0)}[l-1;1]+\frac{N}{2(t_{1}+t_{2})}\mathbf{Q}_{2}^{(l)}\mathbf{\Psi}_{N}^{(0)}[l-2;1]$
$\displaystyle-\frac{\mathbf{B}^{(l)}}{2(t_{1}+t_{2})}\sum_{m=0}^{k-3}\mathsf{diff}_{m+2,m+3}(-1)^{m}(-\frac{N}{2}\frac{\partial}{\partial
t_{1}}+\frac{N}{2})\mathbf{\Psi}_{N}^{(0)}[l+m;m+2]$
$\displaystyle+\frac{1}{2(t_{1}+t_{2})}(-1)^{k-1}kt_{k}\mathbf{Q}_{k+1}^{(l)}\mathbf{\Psi}_{N}^{(0)}[l+k-1;1]$
$\displaystyle+\frac{\mathbf{B}^{(l)}Nkt_{k}}{2(t_{1}+t_{2})}\sum_{h=2}^{k-1}(-1)^{k+h}\frac{\partial}{\partial
t_{h}}\mathbf{\Psi}_{N}^{(1)}[l+k-1-h;k+1-h]$
$\displaystyle+\frac{\mathbf{B}^{(l)}N}{2(t_{1}+t_{2})}\sum_{m=4}^{k}(-1)^{m-1}\mathsf{diff}_{m-1,m}\sum_{h=2}^{m-2}(-1)^{h}\frac{\partial}{\partial
t_{h}}\mathbf{\Psi}_{N}^{(1)}[l+m-2-h;m-h],$

where we denote

$\mathsf{diff}_{p,q}=\mathsf{diff}_{p,q}(t_{1},\ldots,t_{k})\overset{\textnormal{def}}{=}pt_{p}-qt_{q}.$

###### Proof.

We start by noting that after a straightforward but lengthy computation, where
we use Lemma 4.8 and [69, Theorem 17], we get:

$\displaystyle\mathbf{\Psi}_{N}^{(0)}[l;1]=$
$\displaystyle\mathbf{B}^{(l)}\Bigg{(}-\mathbf{\Psi}_{N}^{(1)}[l-1;1]+\frac{N(N+s-1)}{2(t_{1}+t_{2})}\mathcal{R}_{2}+\frac{(1-2N-2s)N+2t_{1}}{2(t_{1}+t_{2})}\mathcal{R}_{1}$
$\displaystyle-\frac{N}{2(t_{1}+t_{2})}\mathcal{R}_{2}^{(w)}+\frac{N}{2(t_{1}+t_{2})}\mathcal{R}_{1}^{(w)}+\frac{1}{2(t_{1}+t_{2})}\sum_{m=3}^{k}\mathsf{diff}_{m-1,m}\mathcal{R}_{m}+\frac{kt_{k}}{2(t_{1}+t_{2})}\mathcal{R}_{k+1}\Bigg{)}$

where $\mathcal{R}_{m}$ are vectors given by:

$(\mathcal{R}_{m})_{j}\overset{\textnormal{def}}{=}\begin{cases}0,&j=1.\\\
\vskip
4.0pt\cr\displaystyle\sum_{h=2}^{j}(-1)^{h}\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{l-h,j+1-h},h+\alpha(m)-2}&j=2,\ldots,l-1,\\\
\vskip 4.0pt\cr\mathbf{\Psi}_{N,\emptyset,l+\alpha(m)-2}&j=l,\end{cases}$

with $\alpha(2)=0$, and $\alpha(m)=m$ for all other $m$. We also denote by
${\mathcal{R}}_{1}^{(w)},$ and ${\mathcal{R}}_{2}^{(w)}$ the vectors defined
in the same way as $\mathcal{R}_{1},\mathcal{R}_{2}$ above with
$\mathbf{\Psi}$ replaced by its weighted version ${\mathbf{\Psi}}^{(w)}$.
Next, for $m=3,\ldots,k+1$, we decompose
$\mathcal{R}_{m}=\mathcal{R}_{m}^{\prime}+\mathcal{R}_{m}^{*}$ where these
latter vectors are given as:

$\displaystyle(\mathcal{R}_{m}^{\prime})_{j}$
$\displaystyle\overset{\textnormal{def}}{=}\begin{cases}\displaystyle\sum_{h=2}^{j+m-2}(-1)^{h+m-2}\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{l+m-2-h,j+m-1-h},h}&j=1,\ldots,l-1,\\\
\vskip 4.0pt\cr\mathbf{\Psi}_{N,\emptyset,l+m-2}&j=l,\end{cases}$
$\displaystyle(\mathcal{R}_{m}^{*})_{j}$
$\displaystyle\overset{\textnormal{def}}{=}\begin{cases}\displaystyle\sum_{h=1}^{m-1}(-1)^{h+m-1}\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{l+m-2-h,j+m-1-h},h}&j=1,\ldots,l-1,\\\
\vskip 4.0pt\cr 0&j=l.\end{cases}$

Then, we apply Lemma 4.9 to $\mathcal{R}_{m}^{*}$ with $t_{q}$ being $t_{h}$
for the $h$-th summand in the entries of $\mathcal{R}_{m}^{*}$. For the
remaining terms, by virtue of [69, Theorem 17, Proposition 20 ], after yet
another demanding computation, we obtain the identities:

$\displaystyle\mathbf{B}^{(l)}\mathcal{R}_{1}=\mathbf{Q}_{1}^{(l)}\mathbf{\Psi}_{N}^{(0)}[l-1;1]\>$
$\displaystyle,\;\;\;\;\mathbf{B}^{(l)}{\mathcal{R}}_{1}^{(w)}=\widetilde{\mathbf{Q}}_{1}^{(l)}\mathbf{\Psi}_{N}^{(0)}[l-1;1],$
$\displaystyle\mathbf{B}^{(l)}\mathcal{R}_{2}=\widehat{\mathbf{Q}}_{2}^{(l)}\mathbf{\Psi}_{N}^{(0)}[l-2;1]\>$
$\displaystyle,\;\;\;\;\mathbf{B}^{(l)}{\mathcal{R}}_{2}^{(w)}=\widetilde{\mathbf{Q}}_{2}^{(l)}\mathbf{\Psi}_{N}^{(0)}[l-2;1],$
$\displaystyle(-1)^{m}\mathbf{B}^{(l)}\mathcal{R}_{m}^{\prime}=\>\mathbf{Q}_{m}^{(l)}$
$\displaystyle\mathbf{\Psi}_{N}^{(0)}[l+m-2;1].$ Hence, defining
$\displaystyle{\mathbf{Q}}_{0}^{(l)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}-2N\mathbf{Q}_{1}^{(l)}+\widetilde{\mathbf{Q}}_{1}^{(l)}+(1-2s)\mathbf{Q}_{1}^{(l)},$
$\displaystyle\;\;\;\textnormal{and}\;\;\;\mathbf{Q}_{2}^{(l)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(N+s-1)\widehat{\mathbf{Q}}_{2}^{(l)}-\widetilde{\mathbf{Q}}_{2}^{(l)}$

we get the desired result. We note that even though both
$\widetilde{\mathbf{Q}}_{2}^{(l)}$ and $(N+s-1)\widehat{\mathbf{Q}}_{2}^{(l)}$
contain terms of order $N^{2}$, these exactly cancel out, giving us the
desired growth for the coefficient of $\mathbf{\Psi}_{N}^{(0)}[l-2;1]$ that
will be needed for the rest of the proof. ∎

We will also require suitable initial conditions to be able to use the
recursion above successively.

######  Lemma 4.12.

For each $N\geq 1$, we have that

$\displaystyle\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{2,1}}$
$\displaystyle=\frac{N^{2}}{8}\frac{\partial^{2}\mathbf{\Psi}_{N,\emptyset}}{\partial
t_{1}^{2}}-\frac{N^{2}}{4}\frac{\partial\mathbf{\Psi}_{N,\emptyset}}{\partial{t_{1}}}+\frac{N^{2}}{8}\mathbf{\Psi}_{N,\emptyset}+\frac{N}{2}\frac{\partial\mathbf{\Psi}_{N,\emptyset}}{\partial
t_{2}},$ $\displaystyle\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{2,1}}$
$\displaystyle=\frac{N^{2}}{8}\frac{\partial^{2}\mathbf{\Psi}_{N,\emptyset}}{\partial
t_{1}^{2}}-\frac{N^{2}}{4}\frac{\partial\mathbf{\Psi}_{N,\emptyset}}{\partial{t_{1}}}+\frac{N^{2}}{8}\mathbf{\Psi}_{N,\emptyset}-\frac{N}{2}\frac{\partial\mathbf{\Psi}_{N,\emptyset}}{\partial
t_{2}}.$

###### Proof.

By a similar computation to the initial step in the proof of Proposition 4.11,
one sees that

$\begin{pmatrix}\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{2,1}}\\\
\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{2,2}}\end{pmatrix}=\begin{pmatrix}1/2&1/2\\\
1/2&-1/2\end{pmatrix}\begin{pmatrix}\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{1,1},1}\\\
\mathbf{\Psi}_{N,\emptyset,2}\end{pmatrix}.$

The result then follows by virtue of Lemma 4.9. ∎

We then arrive to the following proposition.

######  Proposition 4.13.

Let $l\geq 3$ and $1\leq q\leq l$. Let $k\geq 2$ and
$i_{2},n_{3},\ldots,n_{k}$ be non-negative integers. Then define,

$\displaystyle\mathcal{L}_{N,l,q}^{(i_{2},n_{3},\ldots,n_{k})}(t_{1})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\partial^{i_{2}}}{\partial
t_{2}^{i_{2}}}\frac{\partial^{n_{3}}}{\partial
t_{3}^{n_{3}}}\cdots\frac{\partial^{n_{k}}}{\partial
t_{k}^{n_{k}}}\mathbf{\Psi}_{N,\boldsymbol{\lambda}_{l,q}}\Bigg{|}_{t_{2}=\cdots=t_{k}=0}.$
(50)

Let $m\geq 0$ be an integer. Denote

$\displaystyle\textbf{L}_{l}^{(i_{2},n_{3},\ldots,n_{k})}=\begin{pmatrix}\mathcal{L}_{N,l,1}^{(i_{2},n_{3},\ldots,n_{k})}(t_{1}),\ldots,\mathcal{L}_{N,l,l}^{(i_{2},n_{3},\ldots,n_{k})}(t_{1})\end{pmatrix}^{\textnormal{T}}$

and

$\displaystyle\hat{\textbf{L}}_{l,m}^{(i_{2},n_{3},\ldots,n_{k})}=\begin{pmatrix}\mathcal{L}_{N,l+m,m+2}^{(i_{2},n_{3},\ldots,n_{k})}(t_{1}),\ldots,\mathcal{L}_{N,l+m,l+m}^{(i_{2},n_{3},\ldots,n_{k})}(t_{1}),\>0\end{pmatrix}^{\textnormal{T}}.$

Then, we have that

$\displaystyle\textbf{L}_{l}^{(i_{2},n_{3},\ldots,n_{k})}=\mathbf{B}^{(l)}\left(\frac{N}{2}\frac{\mathrm{d}}{\mathrm{d}t_{1}}-\frac{N}{2}\right)\begin{pmatrix}\textbf{L}_{l-1}^{(i_{2},n_{3},\ldots,n_{k})}\\\
0\end{pmatrix}+\sum_{n_{2}=0}^{i_{2}}\frac{(-1)^{i_{2}-n_{2}}\binom{i_{2}}{i_{2}-n_{2}}(i_{2}-n_{2})!}{2t_{1}^{i_{2}-n_{2}+1}}\textbf{V}_{l}^{(n_{2},n_{3},\ldots,n_{k})}.$
(51)

Here $\textbf{V}_{l}^{(n_{2},n_{3},\ldots,n_{k})}$ is given by the complicated
explicit expression

$\displaystyle\textbf{V}_{l}^{(n_{2},n_{3},\ldots,n_{k})}=\mathbf{B}^{(l)}N\Bigg{(}\sum_{m=3}^{k}(m-1)n_{m-1}\begin{pmatrix}\textbf{L}_{l-1}^{(n_{2},n_{3},\ldots,n_{k})}\\\
0\end{pmatrix}-\sum_{m=3}^{k}mn_{m}\begin{pmatrix}\textbf{L}_{l-1}^{(n_{2},\ldots,n_{m-1}+1,n_{m}-1,\ldots,n_{k})}\\\
0\end{pmatrix}$
$\displaystyle+kn_{k}\begin{pmatrix}\textbf{L}_{l-1}^{(n_{2},n_{3},\ldots,n_{k})}\\\
0\end{pmatrix}\Bigg{)}+(2t_{1}\mathbf{Q}_{1}^{(l)}+N{\mathbf{Q}}_{0}^{(l)})\textbf{L}_{l-1}^{(n_{2},n_{3},\ldots,n_{k})}+N\mathbf{Q}_{2}^{(l)}\textbf{L}_{l-2}^{(n_{2},n_{3},\ldots,n_{k})}$
$\displaystyle+\sum_{m=1}^{k-2}(m+1)n_{m+1}(-1)^{m}\mathbf{Q}_{m+2}^{(l)}\textbf{L}_{l+m}^{(n_{2},\ldots,n_{m+1}-1,\ldots,n_{k})}+(-1)^{k-1}kn_{k}\mathbf{Q}_{k+1}^{(l)}\textbf{L}_{l+k-1}^{(n_{2},n_{3},\ldots,n_{k}-1)}$
$\displaystyle-(m+2)n_{m+2}(-1)^{m}\mathbf{Q}_{m+2}^{(l)}\textbf{L}_{l+m}^{(n_{2},\ldots,n_{m+2}-1,\ldots,n_{k})}-\mathbf{B}^{(l)}\Bigg{(}-\frac{N}{2}\frac{\mathrm{d}}{\mathrm{d}t_{1}}+\frac{N}{2}\Bigg{)}\Bigg{\\{}\sum_{m=0}^{k-3}(m+2)n_{m+2}(-1)^{m}$
$\displaystyle\hat{\textbf{L}}_{l,m}^{(n_{2},\ldots,n_{m+2}-1,\ldots,n_{k})}-(m+3)n_{m+3}(-1)^{m}\hat{\textbf{L}}_{l,m}^{(n_{2},\ldots,n_{m+3}-1,\ldots,n_{k})}\Bigg{\\}}$
$\displaystyle+\mathbf{B}^{(l)}N\Bigg{\\{}\sum_{m=4}^{k}(-1)^{m-1}(m-1)n_{m-1}\sum_{h=2}^{m-2}(-1)^{h}\hat{\textbf{L}}_{l,m-2-h}^{(n_{2},\ldots,n_{h}+1,\ldots,n_{m-1}-1,\ldots,n_{k})}$
$\displaystyle-(-1)^{m-1}mn_{m}\sum_{h=2}^{m-2}(-1)^{h}\hat{\textbf{L}}_{l,m-2-h}^{(n_{2},\ldots,n_{h}+1,\ldots,n_{m}-1,\ldots,n_{k})}\Bigg{\\}}+\mathbf{B}^{(l)}Nkn_{k}\sum_{h=2}^{k-1}(-1)^{k+h}\hat{\textbf{L}}_{l,k-1-h}^{(n_{2},\ldots,n_{h}+1,\ldots,n_{k}-1)},$

where, for conventional purposes, we assume
$\mathcal{L}_{N,l,j}^{(n_{2},n_{3},\ldots,n_{k})}=0$ whenever there is some
$m\in\\{2,3,\ldots,k\\}$ with $n_{m}<0$. Furthermore, the following initial
conditions are satisfied, where we drop dependence on $t_{1}$ in the notation,

$\displaystyle\mathcal{L}_{N,1,1}^{(n_{2},\ldots,n_{k})}$
$\displaystyle=-\frac{N}{2}\frac{\mathrm{d}\mathcal{L}_{N}^{(n_{2},\ldots,n_{k})}}{\mathrm{d}t_{1}}+\frac{N\mathcal{L}_{N}^{(n_{2},\ldots,n_{k})}}{2},$
$\displaystyle\mathcal{L}_{N,2,1}^{(n_{2},\ldots,n_{k})}$
$\displaystyle=\frac{N^{2}}{8}\frac{\mathrm{d}^{2}\mathcal{L}_{N}^{(n_{2},\ldots,n_{k})}}{\mathrm{d}t_{1}^{2}}-\frac{N^{2}}{4}\frac{\mathrm{d}\mathcal{L}_{N}^{(n_{2},\ldots,n_{k})}}{\mathrm{d}t_{1}}+\frac{N^{2}\mathcal{L}_{N}^{(n_{2},\ldots,n_{k})}}{8}+\frac{N\mathcal{L}_{N}^{(n_{2}+1,\ldots,n_{k})}}{2},$
$\displaystyle\mathcal{L}_{N,2,2}^{(n_{2},\ldots,n_{k})}$
$\displaystyle=\frac{N^{2}}{8}\frac{\mathrm{d}^{2}\mathcal{L}_{N}^{(n_{2},\ldots,n_{k})}}{\mathrm{d}t_{1}^{2}}-\frac{N^{2}}{4}\frac{\mathrm{d}\mathcal{L}_{N}^{(n_{2},\ldots,n_{k})}}{\mathrm{d}t_{1}}+\frac{N^{2}\mathcal{L}_{N}^{(n_{2}+1,\ldots,n_{k})}}{8}-\frac{N\mathcal{L}_{N}^{(n_{2}+1,\ldots,n_{k})}}{2}.$
(52)

###### Proof.

The result follows via a direct but long explicit computation where we
differentiate both sides of the recursion in Proposition 4.11 and the initial
conditions in Lemma 4.12 with respect to $t_{2},\ldots,t_{k}$ and letting
$t_{2}=\cdots=t_{k}=0$. ∎

As discussed earlier, our main goal is to obtain an ”invertible” way to go
between the $t_{2},\ldots,t_{q}$-derivatives of $\mathbf{\Psi}_{N}$ and
$\mathbf{\Psi}_{N,\boldsymbol{\lambda}}$. To write their relation in a more
compact way, we now introduce the following functions:

$\displaystyle\mathsf{D}_{i}(t_{1},\ldots,t_{k})=N\left\\{(N+s-1)\mathbf{\Psi}_{N,\emptyset,i}-{\mathbf{\Psi}}^{(w)}_{N,\emptyset,i}+(1-2N-2s)\mathbf{\Psi}_{N,\emptyset,i+1}+{\mathbf{\Psi}}^{(w)}_{N,\emptyset,i+1}+2\frac{t_{1}}{N}\mathbf{\Psi}_{N,\emptyset,i+1}\right\\}.$

As before, we will be considering the $t_{2},\ldots,t_{k}$ derivatives of
these functions evaluated at $t_{2}=\cdots=t_{k}=0$. Thus, we define

$\mathsf{D}_{i}^{(n_{2},\ldots,n_{k})}(t_{1})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\partial^{n_{2}}}{\partial
t_{2}^{n_{2}}}\cdots\frac{\partial^{n_{k}}}{\partial
t_{k}^{n_{k}}}\mathsf{D}_{i}(t_{1},\ldots,t_{k})\Big{|}_{t_{2}=\cdots=t_{k}=0},$

where we adopt the same convention described in Proposition 4.13 regarding
integer tuples that contain a negative term. Then, using the identities in
(4.3), we then have the expressions

$\displaystyle\mathsf{D}_{0}^{(n_{2},n_{3},\ldots,n_{k})}=\frac{N^{2}s}{2}\mathcal{L}_{N}^{(n_{2},\ldots,n_{k})}-sN\mathcal{L}_{N,1,1}^{(n_{2},n_{3},\ldots,n_{k})}+t_{1}\mathcal{L}_{N,1,1}^{(n_{2},n_{3},\ldots,n_{k})}$
(53)

and, for $j\geq 1$,

$\displaystyle\mathsf{D}_{j}^{(n_{2},n_{3},\ldots,n_{k})}=\sum_{i=1}^{j+1}(-1)^{i-1}\Bigg{(}\frac{(2-2s+j-2i)N}{2}\mathcal{L}_{N,j+1,i}^{(n_{2},\ldots,n_{k})}+t_{1}\mathcal{L}_{N,j+1,i}^{(n_{2},\ldots,n_{k})}\Bigg{)}$
$\displaystyle+\sum_{i=1}^{j}(-1)^{i-1}\frac{(-N+s+2i-j-1)N}{2}\mathcal{L}_{N,j,i}^{(n_{2},\ldots,n_{k})}.$
(54)

######  Lemma 4.14.

Let $k\geq 2$ and $2\leq q\leq k$. Let $m\geq 1$. Then

$\displaystyle N\frac{\partial\mathbf{\Psi}_{N}}{\partial
t_{q}}=\sum_{i=2}^{q-1}t_{i}R_{i}(t_{1},\ldots,t_{k})+\frac{\mathsf{D}_{q-2}}{2(t_{1}+t_{2})}+\sum_{i=2}^{m}\frac{1}{(2(t_{1}+t_{2}))^{i}}\sum_{j=0}^{i-1}\binom{i-1}{j}(kt_{k})^{j}$
$\displaystyle\sum_{\begin{subarray}{c}h_{2}+\cdots+h_{k}=i-1-j\\\
h_{2}=\cdots=h_{q-1}=0\\\ h_{q}\geq 0,\ldots,h_{k}\geq
0\end{subarray}}\prod_{n=q}^{k}(nt_{n})^{h_{n}}\Bigg{(}\sum_{\begin{subarray}{c}h_{3}^{\prime}\leq
h_{3}\\\ \ldots\\\ h_{k-1}^{\prime}\leq
h_{k-1}\end{subarray}}a_{h_{2},h_{3}^{\prime},\ldots,h_{k-1}^{\prime},h_{k}}^{(i,j)}\mathsf{D}_{q+\sum_{n=1}^{k-2}nh_{n+1}+(k-2)h_{k}-\sum_{n=3}^{k-1}h_{n}^{\prime}+(k-1)j-2}\Bigg{)}+\frac{1}{(2(t_{1}+t_{2}))^{m}}$
$\displaystyle\;\;\;\;\sum_{j=0}^{m}\binom{m}{j}(kt_{k})^{j}\sum_{\begin{subarray}{c}h_{2}+\cdots+h_{k}=m-j\\\
h_{2}=\cdots=h_{q-1}=0\\\ h_{q}\geq 0,\ldots,h_{k}\geq
0\end{subarray}}\prod_{n=q}^{k}(nt_{n})^{h_{n}}\Bigg{(}\sum_{\begin{subarray}{c}h_{3}^{\prime}\leq
h_{3}\\\ \ldots\\\ h_{k-1}^{\prime}\leq
h_{k-1}\end{subarray}}a_{h_{2},h_{3}^{\prime},\ldots,h_{k-1}^{\prime},h_{k}}^{(i,j)}\mathbf{\Psi}_{N,q+\sum_{n=1}^{k-2}nh_{n+1}+(k-2)h_{k}-\sum_{n=3}^{k-1}h_{n}^{\prime}+(k-1)j}\Bigg{)}$

where $R_{2},\ldots,R_{q-1}$ are functions of $t_{1},\ldots,t_{k}$ and

$\displaystyle
a_{h_{2},h_{3}^{\prime},\ldots,h_{k-1}^{\prime},h_{k}}^{(i,j)}=\frac{(i-1-j)!(-1)^{h_{3}^{\prime}+\cdots+h_{k-1}^{\prime}+h_{k}}}{(h_{2}+h_{3}^{\prime})!(h_{k-1}-h_{k-1}^{\prime}+h_{k})!\prod_{n=3}^{k-2}(h_{n}-h_{n}^{\prime}+h_{n+1}^{\prime})!}.$
(55)

###### Proof.

We set $\boldsymbol{\lambda}=\emptyset$ and iteratively replace
$\mathbf{\Psi}_{N,\boldsymbol{\lambda},m+q-2}$ via the formula (4.8). Then, we
obtain for $q=2,\ldots,k$ and for $m\geq 1$,

$\displaystyle\frac{\partial\mathbf{\Psi}_{N}}{\partial t_{q}}$
$\displaystyle=$ $\displaystyle\mathbf{\Psi}_{N,\emptyset,q}$ $\displaystyle=$
$\displaystyle\frac{\mathsf{D}_{q-2}}{2(t_{1}+t_{2})}+\sum_{i=2}^{m}\frac{1}{(2(t_{1}+t_{2}))^{i}}\sum_{j=0}^{i-1}\binom{i-1}{j}(kt_{k})^{j}$
$\displaystyle\sum_{l_{1}=1}^{k-2}\sum_{l_{2}=1}^{k-2}\cdots\sum_{l_{i-1-j}=1}^{k-2}\Big{(}\prod_{n=1}^{i-1-j}\big{(}(l_{n}+1)t_{l_{n}+1}-(l_{n}+2)t_{l_{n}+2}\big{)}\Big{)}\mathsf{D}_{q+l_{1}+\cdots+l_{i-1-j}+(k-1)j-2}$
$\displaystyle+\frac{1}{(2(t_{1}+t_{2}))^{m}}\sum_{j=0}^{m}\binom{m}{j}(kt_{k})^{j}$
$\displaystyle\sum_{l_{1}=1}^{k-2}\sum_{l_{2}=1}^{k-2}\cdots\sum_{l_{m-j}=1}^{k-2}\Big{(}\prod_{n=1}^{m-j}\big{(}(l_{n}+1)t_{l_{n}+1}-(l_{n}+2)t_{l_{n}+2}\big{)}\Big{)}\mathbf{\Psi}_{N,\emptyset,q+l_{1}+\cdots+l_{m-j}+(k-1)j}.$

Here, we use the convention that when $j=m$,

$\displaystyle\sum_{l_{1}=1}^{k-2}\sum_{l_{2}=1}^{k-2}\cdots\sum_{l_{m-j}=1}^{k-2}\Big{(}\prod_{n=1}^{m-j}\big{(}(l_{n}+1)t_{l_{n}+1}-(l_{n}+2)t_{l_{n}+2}\big{)}\Big{)}\mathbf{\Psi}_{N,\emptyset,q+l_{1}+\cdots+l_{m-j}+(k-1)j}=\mathbf{\Psi}_{N,\emptyset,q+(k-1)m},$

and when $j\leq(m-1)$ and $k=2$,

$\displaystyle\sum_{l_{1}=1}^{k-2}\sum_{l_{2}=1}^{k-2}\cdots\sum_{l_{m-j}=1}^{k-2}\prod_{n=1}^{m-j}\big{(}(l_{n}+1)t_{l_{n}+1}-(l_{n}+2)t_{l_{n}+2}\big{)}\mathbf{\Psi}_{N,\emptyset,q+l_{1}+\cdots+l_{m-j}+(k-1)j}=0.$

It is not hard to check that

$\displaystyle\sum_{l_{1}=1}^{k-2}\sum_{l_{2}=1}^{k-2}\cdots\sum_{l_{i-1-j}=1}^{k-2}\Big{(}\prod_{n=1}^{i-1-j}\big{(}(l_{n}+1)t_{l_{n}+1}-(l_{n}+2)t_{l_{n}+2}\big{)}\Big{)}\mathsf{D}_{q+l_{1}+\cdots+l_{i-1-j}+(k-1)j-2}$
$\displaystyle=\sum_{\begin{subarray}{c}h_{2}+\cdots+h_{k}=i-1-j\\\ h_{2}\geq
0,\ldots,h_{k}\geq
0\end{subarray}}\prod_{n=2}^{k}(nt_{n})^{h_{n}}\Bigg{(}\sum_{\begin{subarray}{c}h_{3}^{\prime}\leq
h_{3}\\\ \ldots\\\ h_{k-1}^{\prime}\leq
h_{k-1}\end{subarray}}a_{h_{2},h_{3}^{\prime},\ldots,h_{k-1}^{\prime},h_{k}}^{(i,j)}\mathsf{D}_{q+\sum_{n=1}^{k-2}nh_{n+1}+(k-2)h_{k}-\sum_{n=3}^{k-1}h_{n}^{\prime}+(k-1)j-2}\Bigg{)},$

and also that

$\displaystyle\sum_{l_{1}=1}^{k-2}\sum_{l_{2}=1}^{k-2}\cdots\sum_{l_{m-j}=1}^{k-2}\Big{(}\prod_{n=1}^{m-j}\big{(}(l_{n}+1)t_{l_{n}+1}-(l_{n}+2)t_{l_{n}+2}\big{)}\Big{)}\mathbf{\Psi}_{N,\emptyset,q+l_{1}+\cdots+l_{m-j}+(k-1)j}$
(56) $\displaystyle=\sum_{\begin{subarray}{c}h_{2}+\cdots+h_{k}=m-j\\\
h_{2}\geq 0,\ldots,h_{k}\geq
0\end{subarray}}\prod_{n=2}^{k}(nt_{n})^{h_{n}}\Bigg{(}\sum_{\begin{subarray}{c}h_{3}^{\prime}\leq
h_{3}\\\ \ldots\\\ h_{k-1}^{\prime}\leq
h_{k-1}\end{subarray}}a_{h_{2},h_{3}^{\prime},\ldots,h_{k-1}^{\prime},h_{k}}^{(i,j)}\mathbf{\Psi}_{N,\emptyset,q+\sum_{n=1}^{k-2}nh_{n+1}+(k-2)h_{k}-\sum_{n=3}^{k-1}h_{n}^{\prime}+(k-1)j}\Bigg{)},$
(57)

where $a_{h_{2},h_{3}^{\prime},\ldots,h_{k-1}^{\prime},h_{k}}^{(i,j)}$ is
given as (55). Combining these observations, the result follows. ∎

######  Proposition 4.15.

Let $a_{h_{2},h_{3}^{\prime},\ldots,h_{k-1}^{\prime},h_{k}}^{(i,j)}$ be given
as in (55). Then, whenever $n_{2}\geq 1$, we have that:

$\mathcal{L}_{N}^{(n_{2},\ldots,n_{k})}(t_{1})=\frac{1}{2}\sum_{i_{2}=0}^{n_{2}-1}(-1)^{n_{2}-1-i_{2}}\frac{(n_{2}-1)!}{i_{2}!}\frac{\mathsf{D}_{0}^{(i_{2},n_{3},\ldots,n_{k})}(t_{1})}{t_{1}^{n_{2}-i_{2}}}\\\
+\sum_{i=1}^{n_{2}+\cdots+n_{k}-1}\frac{1}{2^{i+1}}\sum_{j=0}^{i}\binom{i}{j}\sum_{h_{2}+\cdots+h_{k}=i-j}\binom{n_{2}-1}{h_{2}}\binom{n_{k}}{h_{k}+j}\binom{n_{3}}{h_{3}}\ldots\binom{n_{k-1}}{h_{k-1}}(h_{k}+j)!\\\
k^{j+h_{k}}\prod_{n=2}^{k-1}n^{h_{n}}h_{n}!\sum_{i_{2}=0}^{n_{2}-1-h_{2}}\binom{n_{2}-1-h_{2}}{i_{2}}\frac{(i+n_{2}-1-h_{2}-i_{2})!(-1)^{n_{2}-1-h_{2}-i_{2}}}{t_{1}^{i+n_{2}-h_{2}-i_{2}}i!}\\\
\sum_{\begin{subarray}{c}h_{3}^{\prime}\leq h_{3}\\\ \ldots\\\
h_{k-1}^{\prime}\leq
h_{k-1}\end{subarray}}a_{h_{2},h_{3}^{\prime},\ldots,h_{k-1}^{\prime},h_{k}}^{(i+1,j)}\mathsf{D}_{\sum_{n=1}^{k-2}nh_{n+1}+(k-2)h_{k}-\sum_{n=3}^{k-1}h_{n}^{\prime}+(k-1)j}^{(i_{2},n_{3}-h_{3},\ldots,n_{k-1}-h_{k-1},n_{k}-j-h_{k})}(t_{1}).$
(58)

When $n_{2}=\cdots=n_{q-1}=0$ and $n_{q}\geq 1$ for some $3\leq q\leq k-1$, we
have:

$\displaystyle\mathcal{L}_{N}^{(0,\ldots,0,n_{q},\ldots,n_{k})}(t_{1})=$
$\displaystyle\frac{\mathsf{D}_{q-2}^{(0,\ldots,0,n_{q}-1,\ldots,n_{k})}(t_{1})}{2Nt_{1}}+\sum_{i=1}^{n_{q}+\cdots+n_{k}-1}\frac{1}{(2t_{1})^{i+1}N}\sum_{j=0}^{i}\binom{i}{j}$
$\displaystyle\sum_{h_{q}+\cdots+h_{k}=i-j}\binom{n_{q}-1}{h_{q}}\binom{n_{k}}{h_{k}+j}\binom{n_{q+1}}{h_{q+1}}\cdots\binom{n_{k-1}}{h_{k-1}}k^{j+h_{k}}(h_{k}+j)!\prod_{n=q}^{k-1}n^{h_{n}}h_{n}!$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\sum_{\begin{subarray}{c}h_{q}^{\prime}\leq
h_{q}\\\ \cdots\\\ h_{k-1}^{\prime}\leq
h_{k-1}\end{subarray}}a_{0,\ldots,0,h_{q}^{\prime},\ldots,h_{k-1}^{\prime},h_{k}}^{(i+1,j)}\mathsf{D}_{q-2+\sum_{n=q-1}^{k-2}nh_{n+1}+(k-2)h_{k}-\sum_{n=q}^{k-1}h_{n}^{\prime}+(k-1)j}^{(0,\ldots,0,n_{q}-1-h_{q},n_{q+1}-h_{q+1},\ldots,n_{k-1}-h_{k-1},n_{k}-j-h_{k})}(t_{1}).$

When $n_{2}=\ldots=n_{k-1}=0$ and $n_{k}\geq 1$, we have

$\displaystyle\mathcal{L}_{N}^{(0,\ldots,0,n_{k})}(t_{1})=\frac{\mathsf{D}_{k-2}^{(0,\ldots,0,n_{k}-1)}(t_{1})}{2Nt_{1}}+\sum_{i=1}^{n_{k}-1}\frac{k^{i}}{(2t_{1})^{i+1}N}\frac{(n_{k}-1)!}{(n_{k}-1-i)!}\sum_{j=0}^{i}\binom{i}{j}\>a_{0,\ldots,0,i-j}^{(i+1,j)}\mathsf{D}_{k-2+(k-1)i-(i-j)}^{(0,\ldots,0,n_{k}-1-i)}.$

###### Proof.

The result follows, in each case, by first using Proposition 4.14 with
$m=n_{2}+\cdots+n_{k}$, and $q$ being the smallest integer so that $n_{q}>0$;
and then evaluating the appropriate derivatives at $t_{2}=\cdots=t_{k}=0$. ∎

###  4.4 Proof of Theorem 1.9

We will use induction on $k$, the number of variables $t_{2},\ldots,t_{k}$, to
prove Theorem 1.9. But before doing so, we will require some preliminary
lemmas to set up the induction. To this end, let us first fix some notation to
make the presentation clearer. Note that our end goal amounts to showing

$\mathcal{L}_{N}^{(n_{2},\ldots,n_{k})}(t_{1})=\frac{1}{t_{1}^{\sum_{q=2}^{k}qn_{q}-1}}\sum_{m=0}^{\sum_{q=2}^{k}(q-1)n_{q}}P_{m}^{(s,N)}(t_{1})\frac{\mathrm{d}^{m}}{\mathrm{d}t_{1}^{m}}\mathbf{\Psi}_{N}(t_{1},0,\ldots,0),$
(59)

for polynomials $P_{m}^{(s,N)}$ as described in the statement of Theorem 1.9.
The following definition is then crucial for our inductive argument.

######  Definition 4.16.

Define the set $\mathfrak{S}\subset\mathbb{N}^{k-1}$ by declaring
$(n_{2},\ldots,n_{k})\in\mathfrak{S}$ if and only if (59) holds for
appropriate polynomials $P_{m}^{(s,N)}$ as described in the statement of
Theorem 1.9. Similarly, we define $\mathfrak{S}_{l,j}$ by declaring
$(n_{2},\ldots,n_{k})\in\mathfrak{S}_{l,j}$ if and only if

$\mathcal{L}_{N,l,j}^{(n_{2},n_{3},\ldots,n_{k})}(t_{1})=\frac{1}{t_{1}^{l+\sum_{q=2}^{k}qn_{q}-1}}\sum_{m=0}^{l+\sum_{q=2}^{k}(q-1)n_{q}}P_{l,j,m}^{(N,s)}(t_{1})\frac{\mathrm{d}^{m}}{\mathrm{d}t_{1}^{m}}\mathbf{\Psi}_{N}(t_{1},0,\ldots,0),$

where $P_{l,j,m}^{(N,s)}(t_{1})$ are polynomials of $t_{1}$ of degree at most
$l+\sum_{q=2}^{k}qn_{q}-1$, and the coefficients of these polynomials are
polynomials in $N$ and $s$ with degrees at most $l+\sum_{q=2}^{k}(q-1)n_{q}$
and $l+\sum_{q=2}^{k}(q-1)n_{q}-1$ respectively. Finally, we define the
intersection of all these sets:

$\widetilde{\mathfrak{S}}\overset{\textnormal{def}}{=}\bigcap_{1\leq m\leq
l}\mathfrak{S}_{l,m}\cap\mathfrak{S}.$ (60)

We can now begin the proof.

######  Lemma 4.17.

Suppose $(n_{3},\ldots,n_{k})\in\mathbb{N}^{k-2}$ is so that for any
$j=3,\ldots,k$,

$(m_{2},m_{3},\ldots,m_{j-1},m_{j},n_{j+1},\ldots,n_{k})\in\widetilde{\mathfrak{S}}$

whenever $m_{2},m_{3},\ldots,m_{j-1}\in\mathbb{N}$, $0\leq m_{j}\leq n_{j}-1$.
Then, we also have

$(n_{2},\ldots,n_{k})\in\widetilde{\mathfrak{S}},$ (61)

for all $n_{2}\in\mathbb{N}$.

###### Proof.

We will prove the lemma by inducting on $n_{2}$. We start by noting that
whenever $(n_{3},\ldots,n_{k})$ satisfies the assumptions of the lemma,
$\mathsf{D}_{j}^{(m_{2},m_{3},\ldots,m_{k})}$ is a polynomial in $N$ for all
$j\geq 0$, for any such $(m_{2},m_{3},\ldots,m_{k})$ as in the statement of
the lemma, by virtue of (4.3). In particular, whenever this is the case, we
have that

$\mathsf{deg}_{N}\left(\mathsf{D}_{j}^{(m_{2},m_{3}\ldots,m_{k})}\right)\leq
j+1+\sum_{q=2}^{k}(q-1)m_{q}$ (62)

where $\mathsf{deg}_{N}(\cdot)$ denotes the degree of a polynomial in $N$.
Next, we start our induction by showing that (61) holds in the base case
$n_{2}=0$. Indeed, in this case, by means of Proposition 4.15, and also (62),
we immediately get that under the assumptions of the lemma,

$(0,n_{3},\ldots,n_{k})\in\mathfrak{S}.$

Then, by (4.13), we see that

$(0,n_{3},\ldots,n_{k})\in\mathfrak{S}_{1,1}.$

Hence, combining these two observations with the assumption of the lemma, and
applying Proposition 4.15 once more, we see that

$(1,n_{3},\ldots,n_{k})\in\mathfrak{S}.$

where, to obtain the claim about the degree of $N$, we note that for any
$h_{2},\dots,h_{k}\in\mathbb{N}$ with $h_{2}+\cdots+h_{k}\leq
n_{3}+\cdots+n_{k}$, and $h_{j}^{\prime}$ such that $h_{j}^{\prime}\leq
h_{j}$, we have

$\mathsf{deg}_{N}\left(\mathsf{D}_{\sum_{n=1}^{k-2}nh_{n+1}+(k-2)h_{k}-\sum_{n=3}^{k-1}h_{n}^{\prime}+(k-1)j}^{(0,n_{3}-h_{3},\ldots,n_{k-1}-h_{k-1},n_{k}-j-h_{k})}\right)\leq
1+\sum_{q=3}^{k}(q-1)n_{q}.$

From this, it follows, again by means of (4.13) that

$(1,n_{3},\ldots,n_{k})\in\mathfrak{S}_{1,1}$

and also

$(0,n_{3},\ldots,n_{k})\in\mathfrak{S}_{2,1}\cap\mathfrak{S}_{2,2}.$

Furthermore, applying the recursive relation in (51) iteratively, while using
the assumption of the lemma, we see that

$(0,n_{3},\ldots,n_{k})\in\widetilde{\mathfrak{S}},$

where we crucially use the fact that out of the matrices that appear in
equation (51), only $\mathbf{Q}_{2}^{(l)}$ has entries that depend on $N$,
whose entries are $O(N)$, and that it only appears as a multiplicant of
$\mathbf{L}_{l-2}^{(n_{2},\ldots,n_{k})}$. This completes the proof of the
base case. For the inductive step, assume the lemma holds for all $n_{2}\leq
a_{2}-1$. Then, observe that by means of Proposition 4.15 and (4.13), we
immediately get that

$(a_{2},n_{3},\ldots,n_{k})\in\mathfrak{S}\cap\mathfrak{S}_{1,1}.$

Then, we claim that applying Proposition 4.15 and (4.13) yet another time
gives that,

$(a_{2}+1,n_{3},\ldots,n_{k})\in\mathfrak{S}\cap\mathfrak{S}_{1,1}.$ (63)

Indeed, the integer tuples that appear on the right-hand side of (58) can be
classified as:

  * •

$(n_{2},n_{3},\ldots,n_{k})$ with $n_{2}\leq a_{2}$,

  * •

$(m_{2},m_{3},\ldots,m_{k})$ such that there exists $j\in\\{3,\ldots,k\\}$
with $m_{j}\leq n_{j}-1$, $m_{l}=n_{l}$ for all $l=j+1,\ldots,k$.

But then observe that $(a_{2},n_{3},\ldots,n_{k})$ does not appear in the
second sum on the right-hand side of (58) since this is only possible for the
terms corresponding to $j=0$, in which case we must have

$h_{2}+h_{3}+\cdots+h_{k}=i>0,$

and so either $n_{2}\leq a_{2}-1$ or the second bullet point above applies.
Hence, first combining all these observations with (53), and then using (4.13)
we get (63). Thus, using (4.13) again, one can then see,

$(a_{2},n_{3},\ldots,n_{k})\in\mathfrak{S}\cap\mathfrak{S}_{1,1}\cap\mathfrak{S}_{2,1}\cap\mathfrak{S}_{2,2}.$

Now, using the recursive relation (51) iteratively as before, while using the
inductive assumption, one sees that

$(a_{2},n_{3},\ldots,n_{k})\in\bigcap_{1\leq m\leq l}\mathfrak{S}_{l,m},$

which completes the inductive step and hence the proof of the result. ∎

###### Proof of Theorem 1.9.

To prove the theorem, we will show that

$\widetilde{\mathfrak{S}}=\mathbb{N}^{k-1}.$

This will be done by inducting on $k$. First, note that an application of
Lemma 4.17 with $n_{3}=\ldots=n_{k}=0$ immediately gives the result for $k=2$.
Thus, for the sake of induction, assume that

$(n_{2},\ldots,n_{k})\in\widetilde{\mathfrak{S}}$ (64)

for any $n_{2},\ldots,n_{k-1}\in\mathbb{N}$ and $n_{k}\leq a_{k}-1$. Note that
combining this assumption with Lemma 4.17, we get that

$(n_{2},0,\ldots,0,a_{k})\in\widetilde{\mathfrak{S}}$

for all $n_{2}\in\mathbb{N}$. Next, observe that by applying iteratively Lemma
4.17 on the $n_{3}$-variable, one gets,

$(n_{2},n_{3},0,\ldots,0,a_{k})\in\widetilde{\mathfrak{S}}$ (65)

for all $n_{2},n_{3},\in\mathbb{N}$. This, in turn, implies, by virtue of
Lemma 4.17 and (64), that

$(n_{2},0,1,0,\ldots,0,a_{k})\in\widetilde{\mathfrak{S}}.$

Hence, again applying Lemma 4.17 iteratively on the $n_{3}$-variable, we see
that

$(n_{2},n_{3},1,0,\ldots,0,a_{k})\in\widetilde{\mathfrak{S}}$

for any $n_{2},n_{3}\in\mathbb{N}$, which then implies

$(n_{2},0,2,0,\ldots,0,a_{k})\in\widetilde{\mathfrak{S}}.$

Continuing this, where we increase the $n_{4}$-variable by $1$ at each step,
we conclude that

$(n_{2},n_{3},n_{4},0,\ldots,0,a_{k})\in\widetilde{\mathfrak{S}}$

for any $n_{2},n_{3},n_{4}\in\mathbb{N}$. This process can then be continued
to conclude the inductive step that was set up in (64), which is best
described via the following figure:

${(n_{2},n_{3},n_{4},0,\ldots,0,a_{k})}$${(n_{2},0,0,1,0,\ldots,0,a_{k})}$${(n_{2},n_{3},0,1,0,\ldots,0,a_{k})}$${(n_{2},0,1,1,0,\ldots,0,a_{k})}$${(n_{2},n_{3},1,1,0,\ldots,0,a_{k})}$${(n_{2},0,2,1,\ldots,0,a_{k})}$${(n_{2},n_{3},n_{4},1,0,\ldots,0,a_{k})}$${(n_{2},0,0,2,0,\ldots,0,a_{k})}$${(n_{2},n_{3},n_{4},n_{5},0,\ldots,0,a_{k})}$

Figure 1: In the diagram above, any number of integer tuples pointing at a
fixed tuple implies that the latter tuple can be shown to be in
$\widetilde{\mathfrak{S}}$ via a (potentially iterative) application of Lemma
4.17, where we assume that all tuples of the form described by the bases of
the arrows are in $\widetilde{\mathfrak{S}}$. The dashed arrows further mean
that the pointed tuple is obtained not by a mere iterative application of
Lemma 4.17, but rather by an iterative application of the process described in
the figure up to that point.

Continuing this process, we reach at its conclusion that

$(n_{2},n_{3},n_{4},n_{5},\ldots,n_{k-1},a_{k})\in\widetilde{\mathfrak{S}},$

which completes the inductive step set up in (64) and hence the proof of the
theorem. ∎

As mentioned in the introduction, the procedure described above not only
proves the existence of a relation as in Theorem 1.11, but provides a
recursive way to compute the polynomials $P_{m}^{(s,N)}(t)$ (and hence
$\mathcal{A}_{m}^{(s)}(t)$).

###### Proof of Proposition 1.10 and Corollary 1.12..

Both results are mere consequences of the relations:

$\mathbb{E}_{N}^{(s)}\left[e^{-\textnormal{i}t\sum_{j=1}^{N}\frac{\mathsf{x}_{j}^{(N)}}{N}}\sum_{j=1}^{N}\Big{(}\mathsf{x}_{j}^{(N)}-\textnormal{i}\Big{)}^{2}\right]=-\frac{4N(Ns-
t_{1})}{2t_{1}}\frac{\mathrm{d}\mathbb{E}_{N}^{(s)}\left[e^{-\textnormal{i}\frac{t_{1}}{N}\sum_{j=1}^{N}\mathsf{x}_{j}^{(N)}}\right]}{\mathrm{d}t_{1}}-4N\frac{\mathbb{E}_{N}^{(s)}\left[e^{-\textnormal{i}\frac{t_{1}}{N}\sum_{j=1}^{N}\mathsf{x}_{j}^{(N)}}\right]}{2}$

and

$\displaystyle\frac{1}{N^{4}}\mathbb{E}_{N}^{(s)}\Big{[}e^{-\textnormal{i}\frac{t_{1}}{N}\sum_{j=1}^{N}\mathsf{x}_{j}^{(N)}}\mathrm{p}_{2}^{2}\Big{(}\mathsf{x}_{1}^{(N)},\ldots,\mathsf{x}_{N}^{(N)}\Big{)}\Big{]}=\frac{4s^{2}+2}{t_{1}^{2}}\frac{\mathrm{d}^{2}}{\mathrm{d}t_{1}^{2}}$
$\displaystyle\mathbb{E}_{N}^{(s)}\left[e^{-\textnormal{i}\frac{t_{1}}{N}\sum_{j=1}^{N}\mathsf{x}_{j}^{(N)}}\right]$
$\displaystyle+\left(\frac{4s}{Nt_{1}}-\frac{12s^{2}}{t_{1}^{3}}\right)\frac{\mathrm{d}}{\mathrm{d}t_{1}}\mathbb{E}_{N}^{(s)}\left[e^{-\textnormal{i}\frac{t_{1}}{N}\sum_{j=1}^{N}\mathsf{x}_{j}^{(N)}}\right]$
$\displaystyle+\left(\frac{1}{N^{2}}-\frac{2}{t_{1}^{2}}-\frac{4s}{N^{2}t_{1}^{2}}\right)\mathbb{E}_{N}^{(s)}\left[e^{-\textnormal{i}\frac{t_{1}}{N}\sum_{j=1}^{N}\mathsf{x}_{j}^{(N)}}\right].$

Indeed, Corollary 1.12 follows immediately after taking the $N\to\infty$ limit
of these relations, whereas Proposition 1.10 follows by evaluating these
relations at $t=0$, and then using the expressions for the first few moments
of the statistic
$\mathrm{p}_{1}\Big{(}\mathsf{x}_{1}^{(N)},\ldots,\mathsf{x}_{N}^{(N)}\Big{)}$,
which were obtained in [20]. ∎

## Appendix

As mentioned before, here we give explicit formulae for the matrices
$\\{\mathbf{Q}_{m}^{(l)}\\}_{m=0}^{k+1}$ that were used in the proof of
Theorem 1.9. Through the rest of this section, we fix $l\in\mathbb{N}$. Then,
we define, for $i,j=1,\ldots,l$,

$\displaystyle\left(\mathbf{B}^{(l)}\right)_{ij}=\begin{cases}(-1)^{i+j-1}/j(j+1)&j\geq
i;\\\ -1/i&j=i-1;\\\ 0&j<i-1;\\\ (-1)^{i-1}/l&j=l.\end{cases}$

For $i=1,\ldots,l$, $j=1,\ldots,l-1$, we define

$\displaystyle\left(\mathbf{Q}_{0}^{(l)}\right)_{ij}$
$\displaystyle=\begin{cases}(-1)^{i+j}\frac{j(l-j-2)+1-2s}{j(j+1)}&\text{if
}i\leq j\leq l-1;\\\ \frac{j(j+2s)-l+1}{j+1}&\text{if }j=i-1;\\\ 0&\text{if
}j<i-1.\end{cases}$ $\displaystyle\left(\mathbf{Q}_{1}^{(l)}\right)_{ij}$
$\displaystyle=\begin{cases}(-1)^{i+j}\frac{1}{j(j+1)}&\text{if }i\leq j\leq
l-1;\\\ -\frac{j}{j+1}&\text{if }j=i-1;\\\ 0&\text{if }j<i-1.\end{cases}$

For $i=1,\ldots,l$, $j=1,\ldots,l-2$, we define

$\displaystyle\left(\mathbf{Q}_{2}^{(l)}\right)_{ij}=\begin{cases}(-1)^{i+j}\frac{(s-2)N+j(j+3)-(j+2)(l-2)+2(s-1)}{(j+1)(j+2)}&i-1\leq
j\leq l-2;\\\ \frac{(j+s)(j-N)}{j+2}&j=i-2;\\\ 0&1\leq j<i-2.\end{cases}$

For $i=1,\ldots,l$, $j=1,\ldots,l+m-2$, we define

$\left(\mathbf{Q}_{m}^{(l)}\right)_{ij}=\begin{cases}(-1)^{m-j-1}&i=1,j=1,\ldots,m-1;\\\
\frac{(-1)^{i+j+m}(2-m)}{(j-m+1)(j-m+2)}&j>i+m-2\\\ \frac{i+m-2}{i}&i\neq
1,j=i+m-2\\\ 0&i\neq 1,j\leq i+m-3.\end{cases}$

## References

  * [1] G. Akemann and G. Vernizzi, Characteristic polynomials of complex random matrix models, Nuclear Physics B, 660 (2003), p. 532–556.
  * [2] D. J. Aldous, Representations for partially exchangeable arrays of random variables, J. Multivariate Anal., 11 (1981), pp. 581–598.
  * [3] D. J. Aldous, Exchangeability and related topics,, École d’été de probabilités de Saint-Flour, XIII—1983, Lecture Notes in Math., 1117 (1985), pp. 1–198.
  * [4] D. J. Aldous, Exchangeability and continuum limits of discrete random structures, in Proceedings of the International Congress of Mathematicians. Volume I, Hindustan Book Agency, New Delhi, 2010, pp. 141–153.
  * [5] , More uses of exchangeability: representations of complex random structures, in Probability and mathematical genetics, vol. 378 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2010, pp. 35–63.
  * [6] S. A. Altuğ, S. Bettin, I. Petrow, Rishikesh, and I. Whitehead, A recursion formula for moments of derivatives of random matrix polynomials, Q. J. Math., 65 (2014), pp. 1111–1125.
  * [7] J. Andrade and C. Best, Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices, Journal of Physics A: Mathematical and Theoretical, (2024).
  * [8] G. Arfken, Mathematical methods for physicists, Academic Press, New York-London, 1966.
  * [9] L.-P. Arguin, D. Belius, P. Bourgade, M. Radziwiłł, and K. Soundararajan, Maximum of the riemann zeta function on a short interval of the critical line, Communications on Pure and Applied Mathematics, 72 (2019), pp. 500–535.
  * [10] L.-P. Arguin, P. Bourgade, and M. Radziwiłł, The fyodorov-hiary-keating conjecture. i, arXiv preprint arXiv:2007.00988, (2020).
  * [11] L.-P. Arguin, P. Bourgade, and M. Radziwiłł, The fyodorov-hiary-keating conjecture. ii, 2023.
  * [12] T. Assiotis, Random entire functions from random polynomials with real zeros, Adv. Math., 410 (2022), pp. Paper No. 108701, 28.
  * [13] T. Assiotis, B. Bedert, M. A. Gunes, and A. Soor, On a distinguished family of random variables and painlevé equations, Probability and Mathematical Physics, 2 (2021), p. 613–642.
  * [14] T. Assiotis, M. A. Gunes, and A. Soor, Convergence and an explicit formula for the joint moments of the circular jacobi $\beta$-ensemble characteristic polynomial, Mathematical Physics, Analysis and Geometry, 25 (2022), p. 15.
  * [15] T. Assiotis, J. P. Keating, and J. Warren, On the joint moments of the characteristic polynomials of random unitary matrices, International Mathematics Research Notices, 2022 (2022), pp. 14564–14603.
  * [16] T. Assiotis and J. Najnudel, The boundary of the orbital beta process, Moscow Mathematical Journal, 21 (2021), p. 659–694.
  * [17] T. Austin, On exchangeable random variables and the statistics of large graphs and hypergraphs, Probab. Surv., 5 (2008), pp. 80–145.
  * [18] E. C. Bailey, S. Bettin, G. Blower, J. B. Conrey, A. Prokhorov, M. O. Rubinstein, and N. C. Snaith, Mixed moments of characteristic polynomials of random unitary matrices, Journal of Mathematical Physics, 60 (2019).
  * [19] Y. Barhoumi-Andréani, A new approach to the characteristic polynomial of a random unitary matrix, arXiv preprint arXiv:2011.02465, (2020).
  * [20] E. Basor, P. Bleher, R. Buckingham, T. Grava, A. Its, E. Its, and J. P. Keating, A representation of joint moments of cue characteristic polynomials in terms of painlevé functions, Nonlinearity, 32 (2019), p. 4033–4078.
  * [21] H. Bateman, A. Erdélyi, W. Magnus, and F. Oberhettinger, Tables of integral transforms, vol. 1, McGraw-Hill New York, 1954.
  * [22] E. B. Bogomolny and J. P. Keating, Random matrix theory and the Riemann zeros. I. Three-and four-point correlations, Nonlinearity, 8 (1995), pp. 1115–1131.
  * [23] , Gutzwiller’s trace formula and spectral statistics: beyond the diagonal approximation, Phys.Rev.Lett., 77 (1996), pp. 1472–1475.
  * [24] , Random matrix theory and the Riemann zeros II: $n$-point correlations, Nonlinearity, 9 (1996), pp. 911–935.
  * [25] A. Borodin, Determinantal point processes, in The Oxford handbook of random matrix theory, Oxford Univ. Press, Oxford, 2011, pp. 231–249.
  * [26] A. Borodin and G. Olshanski, Infinite random matrices and ergodic measures, Communications in Mathematical Physics, 223 (2001), p. 87–123.
  * [27] S. Chatterjee, A generalization of the Lindeberg principle, Ann. Probab., 34 (2006), pp. 2061–2076.
  * [28] J. B. Conrey, The fourth moment of derivatives of the Riemann zeta-function, Quart. J. Math. Oxford Ser. (2), 39 (1988), pp. 21–36.
  * [29] J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Autocorrelation of random matrix polynomials, Comm. Math. Phys., 237 (2003), pp. 365–395.
  * [30] , Integral moments of $L$-functions, Proc. London Math. Soc. (3), 91 (2005), pp. 33–104.
  * [31] J. B. Conrey and A. Ghosh, On mean values of the zeta-function. II, Acta Arith., 52 (1989), pp. 367–371.
  * [32] , A conjecture for the sixth power moment of the Riemann zeta-function, Internat. Math. Res. Notices, (1998), pp. 775–780.
  * [33] J. B. Conrey and S. M. Gonek, High moments of the Riemann zeta-function, Duke Math. J., 107 (2001), pp. 577–604.
  * [34] J. B. Conrey and J. P. Keating, Moments of zeta and correlations of divisor-sums: I, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 373 (2015), p. 20140313.
  * [35] , Moments of zeta and correlations of divisor-sums: II, In Advances in the Theory of Numbers, 373 (2015), pp. 75–85.
  * [36] , Moments of zeta and correlations of divisor-sums: III, Indagationes Mathematicae, 26 (2015), pp. 736–747.
  * [37] , Moments of zeta and correlations of divisor-sums: IV, Research in Number Theory, 2 (2016), p. 24.
  * [38] , Moments of zeta and correlations of divisor-sums: V, Proceedings of the London Mathematical Society, 118 (2019), pp. 729–752.
  * [39] J. B. Conrey, M. O. Rubinstein, and N. C. Snaith, Moments of the derivative of characteristic polynomials with an application to the riemann zeta function, Communications in Mathematical Physics, 267 (2006), pp. 611–629.
  * [40] J. B. Conrey and N. C. Snaith, In support of $n$-correlation, Comm. Math. Phys., 330 (2014), pp. 639–653.
  * [41] P.-O. Dehaye, Joint moments of derivatives of characteristic polynomials, Algebra Number Theory 2, 31–68, (2008).
  * [42] P.-O. Dehaye, A note on moments of derivatives of characteristic polynomials, 22nd International Conference on Formal Power Series and Algebraic Combinatorics, 681-692, Discrete Math. Theor. Comput. Sci. Proc. AN, (2010).
  * [43] P. Diaconis and D. Freedman, de Finetti’s theorem for Markov chains, Ann. Probab., 8 (1980), pp. 115–130.
  * [44] , Finite exchangeable sequences, Ann. Probab., 8 (1980), pp. 745–764.
  * [45] P. Forrester, Log-Gases and Random Matrices (LMS-34), Princeton University Press, 2010.
  * [46] P. Forrester and N. Witte, Application of the $\tau$-function theory of Painlevé equations to random matrices: PV, PIII, the LUE, JUE, and CUE, Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 55 (2002), pp. 679–727.
  * [47] P. J. Forrester, Log-gases and random matrices, vol. 34 of London Mathematical Society Monographs Series, Princeton University Press, Princeton, NJ, 2010.
  * [48] P. J. Forrester, Joint moments of a characteristic polynomial and its derivative for the circular $\beta$-ensemble, Probability and Mathematical Physics, 3 (2022), pp. 145–170.
  * [49] P. J. Forrester and N. S. Witte, Application of the $\tau$-function theory of Painlevé equations to random matrices: PIV, PII and the GUE, Comm. Math. Phys., 219 (2001), pp. 357–398.
  * [50] , Application of the $\tau$-function theory of Painlevé equations to random matrices: $\rm P_{V}$, $\rm P_{III}$, the LUE, JUE, and CUE, Comm. Pure Appl. Math., 55 (2002), pp. 679–727.
  * [51] Y. V. Fyodorov, G. A. Hiary, and J. P. Keating, Freezing transition, characteristic polynomials of random matrices, and the riemann zeta function, Phys. Rev. Lett., 108 (2012), p. 170601.
  * [52] Y. V. Fyodorov and J. P. Keating, Freezing transitions and extreme values: random matrix theory, and disordered landscapes, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), pp. 20120503, 32.
  * [53] M. A. Gunes, Characteristic polynomials of orthogonal and symplectic random matrices, jacobi ensembles & l-functions, arXiv preprint arXiv:2208.08827, (2022).
  * [54] C. Hammer, Higher transcendental functions, volume i: by harry bateman (compiled by the staff of the bateman manuscript project). 302 pages, new york, mcgraw-hill book co., inc., 1953., 1953.
  * [55] G. H. Hardy and J. E. Littlewood, Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes, Acta Math., 41 (1916), pp. 119–196.
  * [56] D. N. Hoover, Relations on probability spaces and arrays of random variables, Preprint, (1979).
  * [57] C. Hughes, On the characteristic polynomial of a random unitary matrix and the Riemann zeta function, PhD Thesis, University of Bristol, (2001).
  * [58] A. E. Ingham, Mean-Value Theorems in the Theory of the Riemann Zeta-Function, Proc. London Math. Soc. (2), 27 (1927), pp. 273–300.
  * [59] H. Iwaniec, W. Luo, and P. Sarnak, Low lying zeros of families of $L$-functions, Inst. Hautes Études Sci. Publ. Math., (2000), pp. 55–131.
  * [60] K. Johansson, Random matrices and determinantal processes, in Mathematical statistical physics, Elsevier B. V., Amsterdam, 2006, pp. 1–55.
  * [61] O. Kallenberg, On the representation theorem for exchangeable arrays, J. Multivariate Anal., 30 (1989), pp. 137–154.
  * [62] , Symmetries on random arrays and set-indexed processes, J. Theoret. Probab., 5 (1992), pp. 727–765.
  * [63] , Foundations of modern probability, Probability and its Applications (New York), Springer-Verlag, New York, second ed., 2002.
  * [64] , Probabilistic symmetries and invariance principles, Probability and its Applications (New York), Springer, New York, 2005.
  * [65] N. M. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, vol. 45 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 1999.
  * [66] J. P. Keating and N. C. Snaith, Random matrix theory and $L$-functions at $s=1/2$, Comm. Math. Phys., 214 (2000), pp. 91–110.
  * [67] , Random matrix theory and $\zeta$(1/2+it), Communications in Mathematical Physics, 214 (2000), pp. 57–89.
  * [68] J. P. Keating and F. Wei, Joint moments of higher order derivatives of CUE characteristic polynomials I: asymptotic formulae, International Mathematics Research Notices, 2024 (2024), pp. 9607–9632.
  * [69] J. P. Keating and F. Wei, Joint moments of higher order derivatives of CUE characteristic polynomials II: Structures, recursive relations, and applications, Nonlinearity, 37 (2024), p. 085009.
  * [70] A. Lenard, Correlation functions and the uniqueness of the state in classical statistical mechanics, Communications in Mathematical Physics, 30 (1973), pp. 35–44.
  * [71] C. Male, Traffic distributions and independence: permutation invariant random matrices and the three notions of independence, Mem. Amer. Math. Soc., 267 (2020), pp. v+88.
  * [72] H. L. Montgomery, The pair correlation of zeros of the zeta function, in Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), vol. Vol. XXIV of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1973, pp. 181–193.
  * [73] J. Najnudel, On the extreme values of the riemann zeta function on random intervals of the critical line, Probability Theory and Related Fields, 172 (2018), pp. 387–452.
  * [74] K. Okamoto, Studies on the Painlevé equations. II. Fifth Painlevé equation $P_{\rm V}$, Japan. J. Math. (N.S.), 13 (1987), pp. 47–76.
  * [75] , Studies on the Painlevé equations. IV. Third Painlevé equation $P_{{\rm III}}$, Funkcial. Ekvac., 30 (1987), pp. 305–332.
  * [76] G. Olshanski, The problem of harmonic analysis on the infinite-dimensional unitary group, J. Funct. Anal., 205 (2003), pp. 464–524.
  * [77] G. Olshanski and A. Vershik, Ergodic unitary invariant measures on the space of infinite hermitian matrices,, American Mathematical Society Translations, Ser. 2, 175 (1996), pp. 137–175.
  * [78] D. Pickrell, Mackey analysis of infinite classical motion groups, Pacific J. Math., 150 (1991), pp. 139–166.
  * [79] Y. Qiu, Infinite random matrices & ergodic decomposition of finite and infinite Hua-Pickrell measures, Adv. Math., 308 (2017), pp. 1209–1268.
  * [80] M. Rubinstein, Low-lying zeros of $L$-functions and random matrix theory, Duke Math. J., 109 (2001), pp. 147–181.
  * [81] Z. Rudnick and P. Sarnak, Zeros of principal $L$-functions and random matrix theory, Duke Math. J., 81 (1996), pp. 269–322.
  * [82] H. Umemura, Second proof of the irreducibility of the first differential equation of Painlevé, Nagoya Math. J., 117 (1990), pp. 125–171.
  * [83] H. Umemura and H. Watanabe, Solutions of the third Painlevé equation. I, Nagoya Math. J., 151 (1998), pp. 1–24.
  * [84] B. Winn, Derivative moments for characteristic polynomials from the cue, Communications in Mathematical Physics, 315 (2012), p. 531–562.
  * [85] N. S. Witte and P. J. Forrester, Gap probabilities in the finite and scaled Cauchy random matrix ensembles, Nonlinearity, 13 (2000), pp. 1965–1986.

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