# The Distribution of polynomials in monotone independent elements

Marwa Banna  New York University Abu Dhabi, Division of Science, Mathematics,
Abu Dhabi, UAE<EMAIL_ADDRESS>and  Pei-Lun Tseng  New York University
Abu Dhabi, Division of Science, Mathematics, Abu Dhabi, UAE<EMAIL_ADDRESS>
###### Abstract.

Building on the work of Arizmendi and Celestino (2021), we derive the
$*$-distributions of polynomials in monotone independent and infinitesimally
monotone independent elements. For non-zero complex numbers $\alpha$ and
$\beta$, we derive explicitly the $*$-distribution of $p_{\alpha,\beta}=\alpha
ab+\beta ba$ whenever $a$ and $b$ are monotone or infinitesimally monotone
independent elements. This encompasses both cases of the commutator and anti-
commutator. We further push this approach to investigate more general
polynomials. As applications, we derive the limiting distribution with respect
to the partial trace of polynomials in a certain class of random matrices.

###### Key words and phrases:

Monotone independence, Infinitesimal monotone independence, Distributions of
polynomials, Partial traces

###### 2000 Mathematics Subject Classification:

46L53, 60B20

##  1\. Introduction

In non-commutative probability theory, diverse notions of independence arise
through the examination of purely algebraic representations of fundamental
concepts in classical probability theory. Five notions of independence were
identified to satisfy natural axioms of probability: tensor (classical), free,
Boolean, monotone, and, its mirror image, anti-monotone. These notions of
independence were classified by Speicher [27] and Muraki [19, 20] and also by
Ben Ghorbal and Schürmann [4] in the framework of category theory.

Following Voiculescu’s introduction of the concept of free independence,
numerous extensions and generalizations of freeness have arisen within the
non-commutative setting. Among these, one notable generalization is
infinitesimal freeness, see [11] and [6]. This extension has found practical
applications in investigating the occurrence of spikes in diverse deformed
random matrix models, as shown in [2, 26, 7]. Consequently, extensions of
infinitesimal Boolean and monotone independence have also been introduced in
[23, 13].

Just like the classical counterpart, these notions of independence enable the
computation of probability distributions for sums and products of independent
random variables when we have knowledge of their individual distributions.
This is achieved through the corresponding operations of additive and
multiplicative convolutions. For a more details, we refer to [32, 31] for free
convolutions, [28, 12] for Boolean convolutions, and [18, 5] for monotone
convolutions. In a similar vein, analogous convolutions, known as
infinitesimal additive and multiplicative convolutions, have been developed in
[3, 15, 24, 30] for the free case and [23, 29] for the Boolean and monotone
cases.

Beyond sums and products, there is a keen interest in determining
distributions of general polynomials in independent variables. A natural
starting point involves deriving the distributions of two particular
polynomials: the commutator $i(x_{1}x_{2}-x_{2}x_{1})$, and the anti-
commutator, $x_{1}x_{2}+x_{2}x_{1}$, for independent variables $x_{1}$ and
$x_{2}$. While the distribution of the commutator in the free setting was
derived combinatorially using cumulants in prior works [21, 10], the
distribution of the anti-commutator was only obtained recently in [22]. This
is also the case for Boolean independent variables where the distributions of
both the commutator and anti-commutator were determined recently in [17].
Furthermore, the infinitesimal free and Boolean distributions of commutators
and anti-commutators were, under certain assumptions, derived recently in
[17].

On the other hand, distributions of commutators and anti-commutators for
monotone independent variables and infinitesimally monotone independent remain
unexplored. One of our primary objectives in this paper is to address this
gap. As polynomials, we study closely linear spans in Section 3.1 for which we
obtain explicit results and extend our approach to more general polynomials in
Section 3.2. In fact, let $(\mathcal{C},\varphi)$ be a $*$-probability space
and $a$, $b\in\mathcal{C}$ such that $\mathcal{A}_{a}\prec\mathcal{A}_{b}$
with $1_{\mathcal{C}}\in\mathcal{A}_{b}$. We determine explicitly in Theorem
3.1 the $*$-distribution of $p_{\alpha,\beta}=\alpha ab+\beta ba$, where
$\alpha$ and $\beta$ are non-zero complex numbers. Theorem 3.1 yields for the
particular case where $\varphi(b)=0$ that for any $k\geq 1$,

$\varphi(p_{\alpha,\beta}^{k})=\begin{cases}\sqrt{\alpha\beta\varphi(b^{2})}^{k}\varphi(a^{k})&\text{
if }k\text{ is even,}\\\ 0&\text{ if }k\text{ is odd.}\end{cases}$

The framework of Theorem 3.1 encompasses both cases of the commutator $q=i(ab-
ba)$ and anti-commutator $p=ab+ba$ and yields for each $k\geq 1,$

$\displaystyle\varphi(p^{k})$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\varphi(b^{2})}}\left([\varphi(b)+\sqrt{\varphi(b^{2})}]^{k+1}-[\varphi(b)-\sqrt{\varphi(b^{2})}]^{k+1}\right)\varphi(a^{k});$
$\displaystyle\varphi(q^{k})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1-(-1)^{k+1}\right)\left(\sqrt{\varphi(b^{2})-\varphi(b)^{2}}\right)^{k}\varphi(a^{k}).$

Moreover, using the upper triangular technique developed in [30, 23], we
extend our results to the infinitesimal monotone setting where we study linear
spans in Section 4.1 and general polynomials in Section 4.2.

In the work of Lenczewski [14], random matrix models for asymptotic monotone
independence with respect to the partial trace were given and were further
discussed in [9]. On the other hand, Mingo and Tseng [17] introduced recently
a constructive approach for generating random matrix models for which
asymptotic monotone independence holds. We extend in this paper the utility of
our results to derive the distribution of such random matrices with respect
the partial trace, as elaborated in Section 5.

Our approach builds up on the work of Arizmendi and Celestino [1] who study
distributions of polynomials with cyclic monotone elements. The notion of
cyclic monotone independence was introduced by Collins, Hasebe, and Sakuma [8]
with the aim of describing the joint limiting spectral distribution of some
pairs of matrices. They derived spectral formulas for certain polynomials of
degree $2$ and $3$ in cyclic monotone elements, including the commutator and
anti-commutator. In [1], Arizmendi and Celestino offered a novel approach for
finding spectral distributions for polynomials in cyclic monotone elements. A
pivotal aspect of their method lies in the observation that many polynomials
can be expressed as the $(1,1)$-entry of matrix products, allowing for the
derivation of the distribution by examining the corresponding product
matrices. Building on this idea, we extend this approach to study polynomials
in monotone independent elements. Moreover, we push this further to
investigate more general polynomials which leads to computing the precise
distributions of a wide range of polynomials as highlighted in Theorem 3.5.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Monotone Independence
    2. 2.2 Infinitesimally Monotone Independence
    3. 2.3 Random Matrices for Monotone independence
  3. 3 Distribution of polynomials in monotone independent elements
    1. 3.1 The Linear Span of $ab$ and $ba$
    2. 3.2 General Polynomials
  4. 4 Infinitesimal Monotone Cases
    1. 4.1 Linear spans in $ab$ and $ba$
    2. 4.2 General Polynomials
  5. 5 Applications to Random Matrices

##  2\. Preliminaries

###  2.1. Monotone Independence

A _non-commutative probability space_ (or ncps for short) is a pair
$(\mathcal{C},\varphi)$ such that $\mathcal{C}$ is a unital algebra and
$\varphi$ is a linear functional on $\mathcal{C}$ that $\varphi(1)=0$.
Moreover, a ncps $(\mathcal{C},\varphi)$ is called a _$*$ -probability space_
if we further assume $\mathcal{C}$ is a $*$-algebra and $\varphi(a^{*}a)\geq
0$ for all $a\in\mathcal{C}$.

For $a_{1},\dots,a_{k}\in\mathcal{C}$, the _$*$ -distribution_ of
$\\{a_{1},\dots,a_{k}\\}$ is the set of all possible joints moments

$\mu_{a_{1},\dots,a_{k}}=\\{\varphi(a_{i_{1}}^{m_{1}}\cdots
a_{i_{n}}^{m_{n}})\mid n\geq 1,\ 1\leq i_{1},\dots,i_{n}\leq k\text{ and
}m_{1},\dots,m_{n}\in\\{1,*\\}\\}.$

######  Definition 2.1.

Suppose that $(\mathcal{C},\varphi)$ is a $*$-probability space, and
$\mathcal{A}$ and $\mathcal{B}$ are $*$-subalgebras of $\mathcal{C}$ that are
not necessary unital. We say that $(\mathcal{A},\mathcal{B})$ is _monotone
independent_ and write $\mathcal{A}\prec\mathcal{B}$ if

(2.1) $\displaystyle\varphi(a_{1}\cdots a_{n})$ $\displaystyle=$
$\displaystyle\varphi(a_{1}\cdots a_{k-1}\varphi(a_{k})a_{k+1}\cdots a_{n}),$

whenever $a_{j}\in\mathcal{A}\cup\mathcal{B}$ for all $j$, and
$a_{k-1}\in\mathcal{A}$, $a_{k}\in\mathcal{B}$, and $a_{k+1}\in\mathcal{A}$
for some $k$. One of the inequalities is eliminated if $k=1$ or $k=n.$

For two sets $A_{1}$ and $A_{2}$ of elements in $\mathcal{C}$, we say
$(A_{1},A_{2})$ is monotone independent and write $A_{1}\prec A_{2}$ if
$\mathcal{A}_{1}\prec\mathcal{A}_{2}$ where $\mathcal{A}_{i}$ is the
$*$-algebra generated by $A_{i}$ for $i=1,2$.

######  Remark 2.2.

Note that under the notion of monotone independence of pair algebras
$(\mathcal{A}_{1},\mathcal{A}_{2})$, we usually don’t consider
$\mathcal{A}_{1}$ to be unital; otherwise, for all $a\in\mathcal{A}_{2},$ we
have

$\varphi(a^{n})=\varphi(1\cdot a\cdot 1\cdot a\cdots 1\cdot
a)=\varphi(a)^{n}\varphi(1\cdot 1\cdots 1\cdot 1)=\varphi(a)^{n},$

which is not the case that we are interested in.

###  2.2. Infinitesimally Monotone Independence

Let $(\mathcal{C},\varphi)$ be a $*$-probability space and
$\varphi^{\prime}:\mathcal{C}\rightarrow\mathbb{C}$ be a linear functional
such that $\varphi^{\prime}(1)=0$ and $\varphi(a^{*})=\overline{\varphi(a)}$
for all $a\in\mathcal{C}$. Then we call the triple
$(\mathcal{C},\varphi,\varphi^{\prime})$ an infinitesimal $*$-probability
space.

For $a_{1},\dots,a_{k}\in\mathcal{C}$, the _infinitesimal $*$-distribution_ of
$\\{a_{1},\dots,a_{k}\\}$ is the pair
$(\mu_{a_{1},\dots,a_{k}},\mu^{\prime}_{a_{1},\dots,a_{k}})$ where
$\mu_{a_{1},\dots,a_{k}}$ is the $*$-distribution of $\\{a_{1},\dots,a_{k}\\}$
and

$\mu^{\prime}_{a_{1},\dots,a_{k}}=\\{\varphi^{\prime}(a_{i_{1}}^{m_{1}}\cdots
a_{i_{n}}^{m_{n}})\mid n\geq 1,\ 1\leq i_{1},\dots,i_{n}\leq k\text{ and
}m_{1},\dots,m_{n}\in\\{1,*\\}\\}.$

######  Definition 2.3.

Let $(\mathcal{C},\varphi,\varphi^{\prime})$ be an infinitesimal
$*$-probability space and let $\mathcal{A}$ and $\mathcal{B}$ be
$*$-subalgebras of $\mathcal{C}$ that are not necessary unital. We say that
$(\mathcal{A},\mathcal{B})$ is _infinitesimally monotone independent_ and
write
$\mathcal{A}\mathrel{\scalebox{0.9}[1.0]{$\prec$}\mkern-5.0mu\scalebox{0.4}[1.0]{$\prec$}\mkern-5.5mu\scalebox{0.4}[1.0]{$\prec$}}\mathcal{B}$
if

$\displaystyle\varphi(a_{1}\cdots a_{n})$ $\displaystyle=$
$\displaystyle\varphi(a_{1}\cdots a_{k-1}\varphi(a_{k})a_{k+1}\cdots a_{n})$
$\displaystyle\varphi^{\prime}(a_{1}\cdots a_{n})$ $\displaystyle=$
$\displaystyle\varphi^{\prime}(a_{1}\cdots a_{k-1}\varphi(a_{k})a_{k+1}\cdots
a_{n})+\varphi(a_{1}\cdots a_{k-1}\varphi^{\prime}(a_{k})a_{k+1}\cdots
a_{n}),$

whenever $a_{j}\in\mathcal{A}\cup\mathcal{B}$ for all $j$, and
$a_{k-1}\in\mathcal{A}$, $a_{k}\in\mathcal{B}$, and $a_{k+1}\in\mathcal{A}$
for some $k$. One of the inequalities is eliminated if $k=1$ or $k=n.$

For two sets $A_{1}$ and $A_{2}$ in $\mathcal{C}$, we say that $(A_{1},A_{2})$
is infinitesimally monotone independent and write
$A_{1}\mathrel{\scalebox{0.9}[1.0]{$\prec$}\mkern-5.0mu\scalebox{0.4}[1.0]{$\prec$}\mkern-5.5mu\scalebox{0.4}[1.0]{$\prec$}}A_{2}$
if
$\mathcal{A}_{1}\mathrel{\scalebox{0.9}[1.0]{$\prec$}\mkern-5.0mu\scalebox{0.4}[1.0]{$\prec$}\mkern-5.5mu\scalebox{0.4}[1.0]{$\prec$}}\mathcal{A}_{2}$
with $\mathcal{A}_{i}$ being the $*$-algebra generated by $A_{i}$ for each
$i=1,2$.

Note that the infinitesimal monotone independence can be understood as
monotone independence with amalgamation. To be precise, let
$(\mathcal{C},\varphi,\varphi^{\prime})$ be an infinitesimal $*$-probability
space and define

$\widetilde{\mathcal{C}}=\Big{\\{}\begin{bmatrix}a&a^{\prime}\\\
0&a\end{bmatrix}\Big{|}a,a^{\prime}\in\mathcal{C}\Big{\\}},\
\widetilde{\mathbb{C}}=\Big{\\{}\begin{bmatrix}c&c^{\prime}\\\
0&c\end{bmatrix}\Big{|}c,c^{\prime}\in\mathbb{C}\Big{\\}}.$

Finally, let
$\widetilde{\varphi}:\widetilde{\mathcal{C}}\to\widetilde{\mathbb{C}}$ be the
linear functional defined by

$\widetilde{\varphi}\Big{(}\begin{bmatrix}a&a^{\prime}\\\
0&a\end{bmatrix}\Big{)}=\begin{bmatrix}\varphi(a)&\varphi^{\prime}(a)+\varphi(a^{\prime})\\\
0&\varphi(a)\end{bmatrix}.$

Then the triple
$(\widetilde{\mathcal{C}},\widetilde{\mathbb{C}},\widetilde{\varphi})$ forms a
$\widetilde{\mathbb{C}}$-valued probability space (see [30]). The
correspondence to independence with amalgamation is illustrated by the
following theorem by Perales and Tseng [23].

######  Theorem 2.4.

Let $(\mathcal{C},\varphi,\varphi^{\prime})$ be an infinitesimal
$*$-probability space and let $\mathcal{A}$ and $\mathcal{B}$ be
$*$-subalgebras of $\mathcal{C}$ that are not necessarily unital. Then
$(\mathcal{A},\mathcal{B})$ is infinitesimally monotone independent if and
only if $(\widetilde{\mathcal{A}},\widetilde{\mathcal{B}})$ is monotone
independent with respect to $\widetilde{\varphi}$ where

$\widetilde{\mathcal{A}}=\Big{\\{}\begin{bmatrix}a&a^{\prime}\\\
0&a\end{bmatrix}\Big{|}a,a^{\prime}\in\mathcal{A}\Big{\\}}\text{ and
}\widetilde{\mathcal{B}}=\Big{\\{}\begin{bmatrix}a&a^{\prime}\\\
0&a\end{bmatrix}\Big{|}a,a^{\prime}\in\mathcal{B}\Big{\\}}.$

###  2.3. Random Matrices for Monotone independence

######  Definition 2.5.

Let $(\mathcal{C}_{n},\varphi_{n})_{n}$ be a sequence of ncps and
$(a_{n},b_{n})$ be a pair of elements in $\mathcal{C}_{n}$ for each $n$. We
say that the sequence $(a_{n},b_{n})_{n}$ is _asymptotically monotone
independent_ with respect to $\varphi_{n}$ if there exist a ncps
$(\mathcal{C},\varphi)$ and elements $a,b\in\mathcal{C}$ with $a\prec b$ such
that $(a_{n},b_{n})$ converges to $(a,b)$ in distribution; that is,

$\lim_{n\to\infty}\varphi_{n}(p(a_{n},b_{n}))=\varphi(p(a,b))\text{ for any
polynomial }p$

and $(a,b)$ is monotone independent with respect to $\varphi$.

We present the random matrix model given by Lenczewski [14] where asymptotic
monotone independence holds with respect to the partial trace. Let $A_{N}$ be
an $N\times N$ random matrix and $N_{0}\leq N$ be a fixed natural number.
$A_{N}$ can be represented as a composition of four block matrices in the
following manner:

$A_{N}=\begin{bmatrix}A^{(1,1)}&A^{(1,2)}\\\
A^{(2,1)}&A^{(2,2)}\end{bmatrix},$

where $A^{(1,1)}$ corresponds to a square submatrix of size $N_{0}\times
N_{0}$, $A^{(2,2)}$ to a square submatrix of size $N\times N$, $A^{(1,2)}$ to
a submatrix of size $N_{0}\times(N-N_{0})$, and $A^{(2,1)}$ to a submatrix of
size $(N-N_{0})\times N_{0}$. The partial trace $\psi_{N}$ of $A_{N}$ is then
defined by

(2.2) $\psi_{N}(A_{N}):=\frac{1}{N_{0}}(E\circ Tr_{N_{0}})(A_{N}^{(1,1)})$

where $Tr_{N_{0}}$ is the non-normalized trace. Here we note that the partial
trace $\psi_{N}$ depends on $N_{0}$. Lenczewski showed that if $A_{N}$ and
$B_{N}$ are independent GUE matrices, then ($T_{A_{N}},B_{N})$ is
asymptotically monotone independent with respect to $\psi_{N}$ where

$T_{A_{N}}=\begin{bmatrix}0&A^{(1,2)}\\\ A^{(2,1)}&0\end{bmatrix}.$

In [17], Mingo and Tseng considered the notion of infinitesimal freeness and
showed a method for utilizing infinitesimal idempotents to construct monotone
independent random matrix models. By this construction, they recover the
random matrix models in [14] as particular cases. We illustrate this
construction for the reader’s convenience and start by recalling some
notation.

Let $(\mathcal{C},\varphi,\varphi^{\prime})$ be an infinitesimal probability
space, and let $j\in\mathcal{C}$ be an _infinitesimal idempotent_ element,
that is, $\varphi(j^{k})=0$ for all $k\geq 1$ and $j=j^{2}$. Assume
furthermore that $\varphi^{\prime}(j)\neq 0$ and let $\mathcal{A}$ be a unital
subalgebra of $\mathcal{C}$ that is infinitesimally free from $\\{j\\}$.
Finally, define

$\psi(a)=\frac{1}{\varphi^{\prime}(j)}\varphi^{\prime}(aj)\text{ for all
}a\in\mathcal{C},$

and note that $\psi(a)=\varphi(a)$ for all $a\in\mathcal{A}.$ It is clear that
$\psi$ is a linear functional on $\mathcal{C}$ with $\psi(1)=1$, and hence,
$(\mathcal{C},\psi)$ is a ncps. Finally, set $j^{(-1)}:=j$ and $j^{(1)}:=1-j$,
and define $\mathcal{J}_{a}(\mathcal{A})$ to be the algebra generated by

$\\{j^{(\varepsilon_{1})}a_{1}j^{(\varepsilon_{2})}a_{2}\cdots
j^{(\varepsilon_{k-1})}a_{k-1}j^{(\varepsilon_{k})}\mid k\geq
0,\varepsilon_{1},\dots,\varepsilon_{k}\in\\{\pm 1\\}\text{ with
}\varepsilon_{1}\neq\cdots\neq\varepsilon_{k},a_{1},\dots,a_{k}\in\mathcal{A}\\}.$

We refer to [11, 15] for more details on infinitesimal free independence and
recall now the following result in [17, Theorem 5.5].

######  Theorem 2.6.

Suppose $\mathcal{A}$ and $\mathcal{B}$ are unital subalgebras of
$\mathcal{C}$ that $\\{\mathcal{A},\mathcal{B}\\}$ are infinitesimally free
from $j$. If $\mathcal{A}$ and $\mathcal{B}$ are free, then
$(\mathcal{J}_{a}(\mathcal{A}),\mathcal{B})$ is monotone independent.

We illustrate now the following result by Au [2], which when combined with
Theorem 2.6, yields more matrix models where monotone independence holds
asymptotically with respect to the partial trace.

######  Theorem 2.7.

Let $W_{N}$ be an $N\times N$ Wigner matrix, $E_{N}^{(j,k)}$ be the unit
matrix in the $(j,k)$-th coordinate, and $J_{N}$ be the matrix with all-one
entries. For any fixed $N_{0}$, $W_{N}$, $(E_{N}^{(j,k)})_{j,k\in N_{0}}$, and
$\frac{1}{N}J_{N}$ are asymptotically infinitesimally free.

It is noteworthy that the limit operators of $(E_{N}^{(j,j)})_{j\in N_{0}}$
and $\frac{1}{N}J_{N}$ are infinitesimal idempotents. Consequently, we can
construct many asymptotically monotone independent random matrix models by
combining Theorem 2.6 and Theorem 2.7. In particular, choose $A_{N}$ and
$B_{N}$ to be asymptotically free and asymptotically infinitesimally free from
$j_{N}=\sum_{j=1}^{N_{0}}E_{N}^{j,j}$. It is easy to see that
$\psi_{N}(\cdot)=\frac{1}{\varphi_{N}^{\prime}(j_{N})}\varphi_{N}^{\prime}(\cdot
j_{N})$ is the partial trace in (2.2) and that
$T_{A_{N}}=j_{N}A_{N}(1-J_{N})+(1-J_{N})A_{N}J_{N}.$ This yields that
$(T_{A_{N}},B_{N})$ is asymptotically monotone independent with respect to the
partial trace. Moreover, the limiting distribution of $T_{A_{N}}$ with respect
to $\psi_{N}$ can then be computed following [17, Proposition 5.3] which,
whenever $A_{N}$ and $J_{N}$ are asymptotically infinitesimally free, yields
that

(2.3) $\displaystyle\lim_{N\to\infty}\psi_{N}(T_{A_{N}}^{k})$ $\displaystyle=$
$\displaystyle\begin{cases}\Big{[}\lim\limits_{N\to\infty}\big{(}(E\circ
tr_{N})(A_{N}^{2})-(E\circ tr_{N})(A_{N})^{2}\big{)}\Big{]}^{k/2}&\text{ if
}k\text{ is even,}\\\ 0&\text{ if }k\text{ is odd,}\end{cases}$

where $tr_{N}$ denotes the normalized trace.

##  3\. Distribution of polynomials in monotone independent elements

In this section, we extend the approach in [1] to the monotone setting and
study a larger class of polynomials in monotone independent elements. For two
monotone independent elements $a$ and $b$, we start by studying in Section 3.1
distributions of linear spans of $ab$ and $ba$ for which we derive explicit
expressions. As particular cases, we determine the distributions of the
commutator and anti-commutator. Then, in Section 3.2, we extend this machinery
furthermore to study to more general polynomials in monotone independent
elements.

###  3.1. The Linear Span of $ab$ and $ba$

Let $(\mathcal{C},\varphi)$ be a $*$-probability space. For any $a$ and $b$ in
$\mathcal{C}$, we denote by $p_{\alpha,\beta}:=p_{\alpha,\beta}(a,b)$, the
linear span of $ab$ and $ba$, given by $p_{\alpha,\beta}=\alpha ab+\beta ba$
for some non-zero complex numbers $\alpha$ and $\beta$.

######  Theorem 3.1.

Assume that $\mathcal{A}_{a}\prec\mathcal{A}_{b}$ with
$1_{\mathcal{C}}\in\mathcal{A}_{b}$, then for each $k\geq 1$, we have for any
$k\in\mathbb{N}$,

$\displaystyle\varphi(p_{\alpha,\beta}^{k})=\frac{\varphi(a^{k})}{2^{k+1}\gamma}\left[\left((\alpha+\beta)x+\gamma\right)^{k+1}-\left((\alpha+\beta)x-\gamma\right)^{k+1}\right]$

where $x=\varphi(b)$ and
$\gamma=\sqrt{(\alpha-\beta)^{2}\varphi(b)^{2}+4\alpha\beta\varphi(b^{2})}$.

######  Remark 3.2.

Note that by monotone independence, the $*$-distribution of $p_{\alpha,\beta}$
depends on the distribution of $b$ only through its first and second moments
$\varphi(b)$ and $\varphi(b^{2})$. Now, if we assume that $\varphi(b)=0$, then
for any $k\geq 1$,

$\varphi(p_{\alpha,\beta}^{k})=\frac{1}{2}\sqrt{\alpha\beta\varphi(b^{2})}^{k}\varphi(a^{k})+\frac{(-1)^{k}}{2}\sqrt{\alpha\beta\varphi(b^{2})}^{k}\varphi(a^{k})=\begin{cases}\sqrt{\alpha\beta\varphi(b^{2})}^{k}\varphi(a^{k})&\text{
if }k\text{ is even,}\\\ 0&\text{ if }k\text{ is odd.}\end{cases}$

Furthermore, if $a$ is an even operator, then we have the $*$-distribution of
$p_{\alpha,\beta}$ is a mere scaling of the distribution of $a$.

Note that, as a direct consequence, we obtain the $*$-distribution of anti-
commutators and commutators in monotone independent variables by taking
$(\alpha,\beta)=(1,1)$ or $(\alpha,\beta)=(i,-i)$ respectively. More
precisely, we obtain the following moment expressions.

######  Corollary 3.3.

Consider $a$, $b\in\mathcal{C}$ such that
$\mathcal{A}_{a}\prec\mathcal{A}_{b}$ with
$1_{\mathcal{C}}\in\mathcal{A}_{b}$. Let $p=ab+ba$ and $q=i(ab-ba)$, then for
each $k\geq 1,$

$\displaystyle\varphi(p^{k})$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\varphi(b^{2})}}\left([\varphi(b)+\sqrt{\varphi(b^{2})}]^{k+1}-[\varphi(b)-\sqrt{\varphi(b^{2})}]^{k+1}\right)\varphi(a^{k});$
$\displaystyle\varphi(q^{k})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1-(-1)^{k+1}\right)\left(\sqrt{\varphi(b^{2})-\varphi(b)^{2}}\right)^{k}\varphi(a^{k}).$

In order to derive explicit expressions for the distributions of spans of $ab$
and $ba$, we prove first the following proposition that plays a crucial role
in bridging the gap between matrices with monotone entries and the
distribution of polynomials.

######  Proposition 3.4.

Let $(\mathcal{C},\varphi)$ be a $*$-probability space. Suppose that
$\mathcal{A}$ and $\mathcal{B}$ are $*$-subalgebras of $\mathcal{C}$ such that
$\mathcal{A}\prec\mathcal{B}$. Consider $A_{k}=[a_{i,j}^{(k)}]_{i,j\in[n]}\in
M_{n}(\mathcal{A})$ for each $k=1,\dots,m$ and
$B_{s}=[b_{i,j}^{(s)}]_{i,j\in[n]}\in M_{n}(\mathcal{B})$ for each
$s=0,1,\dots,m$. Then we have

$(Tr_{n}\otimes\varphi)(D_{0}A_{1}B_{1}A_{2}B_{2}\cdots
B_{m-1}A_{m}D_{m})=(Tr_{n}\otimes\varphi)(D_{0}^{\prime}A_{1}B_{1}^{\prime}A_{2}B_{2}^{\prime}\cdots
B_{m-1}^{\prime}A_{m}D_{m}^{\prime})$

where $D_{0}=I_{n}$ or $B_{0}$, $D_{m}=I_{n}$ or $B_{m}$, and for each $k$,

$B_{k}^{\prime}=\left(\varphi(b_{i,j}^{(k)})_{i,j\in[n]}\right)\in
M_{n}(\mathbb{C}).$

###### Proof.

Without loss of generality, we may assume $D_{0}=I_{n}$ and $D_{m}=B_{m}$ (the
proof of all other cases are similar). Note that by monotone independence with
respect to $\varphi$, we obtain

$\displaystyle(Tr_{n}\otimes\varphi)$
$\displaystyle(A_{1}B_{1}A_{2}B_{2}\cdots A_{m}B_{m})$
$\displaystyle=\sum_{i_{1},\dots,i_{2m}\in[n]}\varphi\left(a_{i_{1},i_{2}}^{(1)}b_{i_{2},i_{3}}^{(1)}a_{i_{3},i_{4}}^{(2)}b_{i_{4},i_{5}}^{(2)}\cdots
a_{i_{2m-1},i_{2m}}^{(m)}b_{i_{2m},i_{1}}^{(m)}\right)$
$\displaystyle=\sum_{i_{1},\dots,i_{2m}\in[n]}\varphi\left(a_{i_{1},i_{2}}^{(1)}\varphi(b_{i_{2},i_{3}}^{(1)})a_{i_{3},i_{4}}^{(2)}\varphi(b_{i_{4},i_{5}}^{(2)})\cdots
a_{i_{2m-1},i_{2m}}^{(m)}\varphi(b_{i_{2m},i_{1}}^{(m)})\right)$
$\displaystyle=(Tr_{n}\otimes\varphi)(A_{1}B_{1}^{\prime}A_{2}B_{2}^{\prime}\cdots
A_{m}B_{m}^{\prime}).$

∎

Having this in hand, we are ready to prove the above results.

###### Proof of Theorem 3.1.

Let

$B_{0}=\begin{bmatrix}\alpha&b\\\ 0&0\end{bmatrix},\ A=\begin{bmatrix}a&0\\\
0&a\end{bmatrix},\text{ and }B_{1}=\begin{bmatrix}b&0\\\
\beta&0\end{bmatrix}.$

Then it is obvious that $B_{0}AB_{1}=\mathrm{Diag}(\alpha ab+\beta ba,0)$.
Note that following Proposition 3.4, for any $k\geq 1$, we have

$\displaystyle\varphi(p_{\alpha,\beta}^{k})$ $\displaystyle=$
$\displaystyle(Tr_{2}\otimes\varphi)\left((B_{0}AB_{1})^{k}\right)$
$\displaystyle=$
$\displaystyle(Tr_{2}\otimes\varphi)\left(B_{0}(AB_{1}B_{0})^{k-1}AB_{1}\right)$
$\displaystyle=$
$\displaystyle(Tr_{2}\otimes\varphi)\left(\begin{bmatrix}\alpha&\varphi(b)\\\
0&0\end{bmatrix}\left(\begin{bmatrix}a&0\\\
0&a\end{bmatrix}\begin{bmatrix}\alpha\varphi(b)&\varphi(b^{2})\\\
\alpha\beta&\beta\varphi(b)\end{bmatrix}\right)^{k-1}\begin{bmatrix}a&0\\\
0&a\end{bmatrix}\begin{bmatrix}\varphi(b)&0\\\ \beta&0\end{bmatrix}\right)$
$\displaystyle=$
$\displaystyle(Tr_{2}\otimes\varphi)\left(\begin{bmatrix}a^{k}&0\\\
0&a^{k}\end{bmatrix}\begin{bmatrix}\alpha&\varphi(b)\\\
0&0\end{bmatrix}\left(\begin{bmatrix}\alpha\varphi(b)&\varphi(b^{2})\\\
\alpha\beta&\beta\varphi(b)\end{bmatrix}\right)^{k-1}\begin{bmatrix}\varphi(b)&0\\\
\beta&0\end{bmatrix}\right).$

We observe that for all $x,y\in\mathbb{C}$, $\begin{bmatrix}\alpha x&y\\\
\alpha\beta&\beta x\end{bmatrix}=SDS^{-1}$ where

$D=\begin{bmatrix}\frac{(\alpha+\beta)x-\gamma}{2}&0\\\
0&\frac{(\alpha+\beta)x+\gamma}{2}\end{bmatrix},S=\begin{bmatrix}\frac{(\alpha-\beta)x-\gamma}{2\alpha\beta}&\frac{(\alpha-\beta)x+\gamma}{2\alpha\beta}\\\
1&1\end{bmatrix},\
S^{-1}=\begin{bmatrix}\frac{-\alpha\beta}{\gamma}&\frac{\gamma+(\alpha-\beta)x}{2\gamma}\\\
\frac{\alpha\beta}{\gamma}&\frac{\gamma-(\alpha-\beta)x}{2\gamma}\end{bmatrix}$

and
$\gamma=\sqrt{\varphi(b)^{2}(\alpha-\beta)^{2}+4\alpha\beta\varphi(b^{2})}$.
Then

$\displaystyle\begin{bmatrix}\alpha&x\\\ 0&0\end{bmatrix}\begin{bmatrix}\alpha
x&y\\\ \alpha\beta&\beta x\end{bmatrix}^{k-1}\begin{bmatrix}x&0\\\
\beta&0\end{bmatrix}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\frac{(\alpha-\beta)x-\gamma}{2\beta}+x&\frac{(\alpha-\beta)x+\gamma}{2\beta}+x\\\
0&0\end{bmatrix}\begin{bmatrix}\frac{\left((\alpha+\beta)x-\gamma\right)^{k-1}}{2^{k-1}}&0\\\
0&\frac{\left((\alpha+\beta)x+\gamma\right)^{k-1}}{2^{k-1}}\end{bmatrix}\begin{bmatrix}\frac{-(\beta^{2}+\alpha\beta)x+\beta\gamma}{2\gamma}&0\\\
\frac{(\beta^{2}+\alpha\beta)x+\beta\gamma}{2\gamma}&0\end{bmatrix}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{k+1}\gamma}\left[\left((\alpha+\beta)x-\gamma\right)^{k}\left(-(\alpha+\beta)x+\gamma\right)+\left((\alpha+\beta)x+\gamma\right)^{k}\left((\alpha+\beta)x+\gamma\right)\right]\begin{bmatrix}1&0\\\
0&0\end{bmatrix}.$

Thus, we complete the proof by concluding that

$\varphi(p_{\alpha,\beta}^{k})=\frac{\varphi(a^{k})}{2^{k+1}\gamma}\left[\left((\alpha+\beta)x+\gamma\right)^{k+1}-\left((\alpha+\beta)x-\gamma\right)^{k+1}\right].$

∎

###  3.2. General Polynomials

Let $(\mathcal{C},\varphi)$ be a $*$-probability space. Suppose $\mathcal{A}$
and $\mathcal{B}$ are $*$-subalgebras that $1_{\mathcal{C}}\in\mathcal{B}$. In
this section, we will demonstrate the method for computing the
$*$-distribution of a polynomial of the form:

$p=\sum_{j=1}^{n}b_{j}y_{j}b_{j}^{\prime}$

where
$b_{1},b_{1}^{\prime},\ldots,b_{n},b_{n}^{\prime}\in\mathcal{B}$,$y_{1},\ldots,y_{n}\in\mathcal{Q}$
and

(3.1) $\mathcal{Q}=\\{a_{0}b_{1}a_{1}\cdots b_{k}a_{k}\mid k\geq
0,a_{0},\dots,a_{k}\in\mathcal{A}\text{ and
}b_{1},\dots,b_{k}\in\mathcal{B}\\}.$

We write $\tilde{y}=a_{0}\varphi(b_{1})a_{1}\cdots\varphi(b_{k})a_{k}$
whenever $y=a_{0}b_{1}a_{1}\cdots b_{k}a_{k}$ for some
$a_{1},\dots,a_{k}\in\mathcal{A}$ and $b_{1},\dots,b_{k}\in\mathcal{B}$.

######  Theorem 3.5.

Let $p$ be a polynomial defined as above and assume
$\mathcal{A}\prec\mathcal{B}$ with $1_{\mathcal{C}}\in\mathcal{B}$. Then for
any $m\in\mathbb{N}$,

$\varphi(p^{m})=\sum_{j_{1},\dots,j_{m}\in[n]}\varphi(\tilde{y}_{j_{1}}\cdots\tilde{y}_{j_{m}})\varphi(b_{j_{1}})\varphi(b_{j_{1}}^{\prime}b_{j_{2}})\varphi(b_{j_{2}}^{\prime}b_{j_{3}})\cdots\varphi(b^{\prime}_{j_{m-1}}b_{j_{m}})\varphi(b_{j_{m}}^{\prime}).$

In order to prove the above result, we start by computing $*$-distributions of
polynomials in monotone independent variables of the following form:

######  Proposition 3.6.

Suppose
$a_{1},\dots,a_{n},b_{1},\dots,b_{n},c_{1},\dots,c_{n},d_{1},\dots,d_{n}$ are
elements from a $*$-probability space $(\mathcal{C},\varphi)$ that
$\\{a_{1},\dots,a_{n},c_{1},\dots,c_{n}\\}\prec\\{b_{1},\dots,b_{n},d_{1},\dots,d_{n}\\}$.
Then for any $m\in\mathbb{N}$,

$\varphi\left(\left[\sum_{k=1}^{n}a_{k}b_{k}c_{k}\right]^{m}\right)=\varphi\left(\left[\sum_{k=1}^{n}a_{k}\varphi(b_{k})c_{k}\right]^{m}\right)$

and

$\varphi\left(\left[\sum_{k=1}^{n}b_{k}a_{k}d_{k}\right]^{m}\right)=\sum_{j_{1},\dots,j_{m}\in[n]}\varphi(a_{j_{1}}a_{j_{2}}\cdots
a_{j_{m}})\varphi(b_{j_{1}})\varphi(d_{j_{1}}b_{j_{2}})\varphi(d_{j_{2}}b_{j_{3}})\cdots\varphi(d_{j_{m-1}}b_{j_{m}})\varphi(d_{j_{m}}).$

###### Proof.

Setting $B:=\mathrm{Diag}(b_{1},\dots,b_{n})$,
$B^{\prime}:=\mathrm{Diag}(\varphi(b_{1}),\dots,\varphi(b_{n}))$,

$A:=\begin{bmatrix}a_{1}&\cdots&a_{n}\\\ \vdots&\ddots&\vdots\\\
0&\cdots&0\end{bmatrix},\text{ and }C:=\begin{bmatrix}c_{1}&\cdots&0\\\
\vdots&\ddots&\vdots\\\ c_{n}&\cdots&0\end{bmatrix},$

we get $ABC=\mathrm{Diag}(\sum_{k=1}^{n}a_{k}b_{k}c_{k},0,\dots,0)$. Then
Proposition 3.4 directly yields for each $m\geq 1$,

$\displaystyle\varphi\left(\left[\sum_{k=1}^{n}a_{k}b_{k}c_{k}\right]^{m}\right)$
$\displaystyle=$ $\displaystyle
Tr_{n}\otimes\varphi([ABC]^{m})=Tr_{n}\otimes\varphi([AB^{\prime}C]^{m})=\varphi\left(\left[\sum_{k=1}^{n}a_{k}\varphi(b_{k})c_{k}\right]^{m}\right).$

In order to prove the second expression, we define now the matrices
$A:=\mathrm{Diag}(a_{1},\dots,a_{n})$, $B^{\prime}=\varphi\otimes Id_{n}(B)$,
$D^{\prime}=\varphi\otimes Id_{n}(D)$, and $(DB)^{\prime}=\varphi\otimes
Id_{n}(DB)$ with

$B:=\begin{bmatrix}b_{1}&\cdots&b_{n}\\\ \vdots&\ddots&\vdots\\\
0&\cdots&0\end{bmatrix},\text{ and }D=\begin{bmatrix}d_{1}&\cdots&0\\\
\vdots&\ddots&\vdots\\\ d_{n}&\cdots&0\end{bmatrix}.$

One can easily check that
$BAD=\mathrm{Diag}(\sum_{k=1}^{n}b_{k}a_{k}d_{k},0,\dots,0)$ and hence we
write

$\displaystyle\varphi\left(\left[\sum_{k=1}^{n}b_{k}a_{k}d_{k}\right]^{m}\right)$
$\displaystyle=$ $\displaystyle
Tr_{n}\otimes\varphi([BAD]^{m})=Tr_{n}\otimes\varphi(B^{\prime}[ADB]^{m-1}AD^{\prime})$
$\displaystyle=$ $\displaystyle
Tr_{n}\otimes\varphi(B^{\prime}[A(DB)^{\prime}]^{m-1}AD^{\prime}),$

where the second equality holds by monotone independence while the last one
follows from Proposition 3.4. We proceed by noting that

$\displaystyle\left[(A(DB)^{\prime})^{m-1}\right]_{j,i}$ $\displaystyle=$
$\displaystyle\sum_{i_{1},\dots,i_{m-2}\in[n]}a_{j}\varphi(d_{j}b_{i_{1}})a_{i_{1}}\varphi(d_{i_{1}}b_{i_{2}})\cdots
a_{i_{m-2}}\varphi(d_{i_{m-2}}b_{i})$ $\displaystyle=$
$\displaystyle\sum_{i_{1},\dots,i_{m-2}\in[n]}a_{j}a_{i_{1}}\cdots
a_{i_{m-2}}\varphi(d_{j}b_{i_{1}})\varphi(d_{i_{1}}b_{i_{2}})\cdots\varphi(d_{i_{m-2}}b_{i}),$

from which we obtain the expression of the $(1,1)$-entry of
$B^{\prime}[A(DB)^{\prime}]^{m-1}AD^{\prime}$ is

$\displaystyle\sum_{i,j\in[n]}\varphi(b_{j})\left(\sum_{i_{1},\dots,i_{m-2}\in[n]}a_{j}a_{i_{1}}\cdots
a_{i_{k-2}}\varphi(d_{j}b_{i_{1}})\varphi(d_{i_{1}}b_{i_{2}})\cdots\varphi(d_{i_{m-2}}b_{i})\right)a_{i}\varphi(d_{i})$
$\displaystyle=$
$\displaystyle\sum_{i,j\in[n]}\sum_{i_{1},\dots,i_{m-2}\in[n]}a_{j}a_{i_{1}}\cdots
a_{i_{k-2}}a_{i}\varphi(b_{j})\varphi(d_{j}b_{i_{1}})\varphi(d_{i_{1}}b_{i_{2}})\cdots\varphi(d_{i_{m-2}}b_{i})\varphi(d_{i})$
$\displaystyle=$
$\displaystyle\sum_{j_{1},\dots,j_{m}\in[n]}a_{j_{1}}a_{j_{2}}\cdots
a_{j_{m}}\varphi(b_{j_{1}})\varphi(d_{j_{1}}b_{j_{2}})\varphi(d_{j_{2}}b_{j_{3}})\cdots\varphi(d_{j_{m-1}}b_{j_{m}})\varphi(d_{j_{m}}).$

Hence, we conclude that

$\varphi\left(\left[\sum_{k=1}^{n}b_{k}a_{k}d_{k}\right]^{m}\right)=\sum_{j_{1},\dots,j_{m}\in[n]}\varphi(a_{j_{1}}a_{j_{2}}\cdots
a_{j_{m}})\varphi(b_{j_{1}})\varphi(d_{j_{1}}b_{j_{2}})\varphi(d_{j_{2}}b_{j_{3}})\cdots\varphi(d_{j_{m-1}}b_{j_{m}})\varphi(d_{j_{m}}).$

∎

We still need to prove the following lemma before giving the proof of Theorem
3.5. We start by noting that since $\mathcal{B}$ is unital then
$\mathcal{A}\subset\mathcal{Q}$ with $\mathcal{Q}$ defined in (3.1). Recall
that for each $y\in\mathcal{Q}$, we write
$\tilde{y}=a_{0}\varphi(b_{1})a_{1}\cdots\varphi(b_{k})a_{k}$ whenever
$y=a_{0}b_{1}a_{1}\cdots b_{k}a_{k}$ for some
$a_{1},\dots,a_{k}\in\mathcal{A}$ and $b_{1},\dots,b_{k}\in\mathcal{B}$.

######  Lemma 3.7.

Assume $\mathcal{A}\prec\mathcal{B}$ with $1_{\mathcal{C}}\in\mathcal{B}$.
Suppose $y_{1},\dots,y_{n}\in\mathcal{Q}$ and
$b_{1},\dots,b_{n}\in\mathcal{B}$, then
$\\{b_{1},\dots,b_{n},y_{1},\dots,y_{n}\\}$ and
$\\{b_{1},\dots,b_{n},\tilde{y}_{1},\dots,\tilde{y}_{n}\\}$ have the same
$*$-distribution. Moreover,
$\\{y_{1},\dots,y_{n}\\}\prec\\{b_{1},\dots,b_{n}\\}$.

###### Proof.

Let $\mathcal{A}_{b}$ be the unital $*$-algebra generated by
$\\{b_{1},\dots,b_{n}\\}$ and $\mathcal{A}_{y}$ be the $*$-algebra generated
by $\\{y_{1},\dots,y_{n}\\}$. It suffices to show that

(3.2) $\varphi(c_{0}d_{1}c_{1}\cdots
d_{m}c_{m})=\varphi(c_{0}\tilde{d}_{1}c_{1}\cdots\tilde{d}_{m}c_{m})$

for any $c_{0},\dots,c_{m}\in\mathcal{A}_{b}$ and
$d_{1},\dots,d_{m}\in\mathcal{A}_{y}$ where for each $i\in[n]$,
$d_{i}=a_{0}^{(i)}b_{1}^{(i)}a_{1}^{(i)}\cdots b_{n_{i}}^{(i)}a_{n_{i}}^{(i)}$
for some $n_{i}\geq 1$,
$a_{0}^{(i)},a_{1}^{(i)},\dots,a_{n_{i}}^{(i)}\in\mathcal{A}$ and
$b_{1}^{(i)},\dots,b_{n_{i}}^{(i)}\in\mathcal{B}$. This follows directly by
monotone independence, which yields that both sides of (3.2) are equal to

$\varphi(a_{0}^{(1)}\dots a_{n_{1}}^{(1)}a_{0}^{(2)}\dots
a_{n_{2}}^{(2)}\cdots a_{0}^{(m)}\dots
a_{n_{m}}^{(m)})\prod_{j=1}^{m}\varphi(c_{m})\prod_{k=1}^{n_{j}}\varphi(b^{(j)}_{k}).$

∎

###### Proof of Theorem 3.5.

We start by noting that, by Lemma 3.7, we have that
$\\{y_{1},\dots,y_{n}\\}\prec\\{b_{1},b_{1}^{\prime},\dots,b_{n},b_{n}^{\prime}\\}$.
Then for any $m\in\mathbb{N}$,

$\displaystyle\varphi(p^{m})$ $\displaystyle=$
$\displaystyle\varphi([\sum_{j=1}^{n}b_{j}y_{j}b_{j}^{\prime}]^{m})=\varphi([\sum_{j=1}^{n}b_{j}\tilde{y}_{j}b_{j}^{\prime}]^{m})$
$\displaystyle=$
$\displaystyle\sum_{j_{1},\dots,j_{m}\in[n]}\varphi(\tilde{y}_{j_{1}}\cdots\tilde{y}_{j_{m}})\varphi(b_{j_{1}})\varphi(b_{j_{1}}^{\prime}b_{j_{2}})\varphi(b_{j_{2}}^{\prime}b_{j_{3}})\cdots\varphi(b^{\prime}_{j_{m-1}}b_{j_{m}})\varphi(b_{j_{m}})$

where the second equality holds by Lemma 3.7 while the last one follows from
Proposition 3.6. ∎

##  4\. Infinitesimal Monotone Cases

Let $(\mathcal{C},\varphi,\varphi^{\prime})$ be an infinitesimal
$*$-probability space and consider its associated
$\widetilde{\mathbb{C}}$-valued probability space
$(\widetilde{\mathcal{C}},\widetilde{\mathbb{C}},\widetilde{\varphi})$,
described in Subsection 2.2. Note that by Theorem 2.4 and the fact that
$\widetilde{\mathbb{C}}$ is commutative, we extend our approach to the
framework of
$(\widetilde{\mathcal{C}},\widetilde{\mathbb{C}},\widetilde{\varphi})$. This
allows passing directly many results on $*$-distributions of polynomials in
monotone independent variables to the infinitesimal setting.

For two infinitesimally monotone elements $a$ and $b$, we start by
illustrating, in Section 4.1, how this allows to derive an explicit expression
of the infinitesimal distribution of linear spans of $ab$ and $ba$, covering
the cases of the commutator and anti-commutator. Then, in Section 4.2, we
extend the results to more general polynomials in infinitesimal monotone
independent elements.

###  4.1. Linear spans in $ab$ and $ba$

The commutativity of $\widetilde{\mathbb{C}}$ allows deriving explicit
expressions of the infinitesimal $*$-distribution of linear spans. Let $a$ and
$b$ be two elements in $\mathcal{C}$ such that
$a\mathrel{\scalebox{0.9}[1.0]{$\prec$}\mkern-5.0mu\scalebox{0.4}[1.0]{$\prec$}\mkern-5.5mu\scalebox{0.4}[1.0]{$\prec$}}b$
and set $A=\mathrm{Diag}(a,a)$ and $B=~{}\mathrm{Diag}(b,b)$. Then, by Theorem
2.4, we have that $A\prec B$ with respect to $\widetilde{\varphi}$. We obtain,
following the same lines of proof as in Theorem 3.1, the explicit distribution
of the linear span $P_{\alpha,\beta}:=P_{\alpha,\beta}(A,B)=\alpha AB+\beta
BA$ for any non-zero complex numbers $\alpha$ and $\beta$. More precisely,
setting $X=\widetilde{\varphi}(B)$ and
$\Gamma=\sqrt{(\alpha-\beta)^{2}\widetilde{\varphi}(B)^{2}+4\alpha\beta\widetilde{\varphi}(B^{2})}$,
we obtain for each $k\geq 1$,

(4.1)
$\displaystyle\widetilde{\varphi}(P_{\alpha,\beta}^{k})=\frac{\widetilde{\varphi}(A^{k})\Gamma^{-1}}{2^{k+1}}\left[\left((\alpha+\beta)X+\Gamma\right)^{k+1}-\left((\alpha+\beta)X-\Gamma\right)^{k+1}\right].$

The key point that enables deriving such an expression following the same
lines as in the scalar case is the fact that $\widetilde{\mathbb{C}}$ is
commutative, and more precisely, $X\Gamma=\Gamma X$. Note that by comparing
the $(1,2)$-entry of both sides of (4.1), we obtain explicitly the $k$th
moment of $p_{\alpha,\beta}$ with respect to $\varphi^{\prime}$.

######  Theorem 4.1.

Let $a$ and $b$ be two elements in an infinitesimal $*$-probability space
$(\mathcal{C},\varphi,\varphi^{\prime})$ such that
$\mathcal{A}_{a}\mathrel{\scalebox{0.9}[1.0]{$\prec$}\mkern-5.0mu\scalebox{0.4}[1.0]{$\prec$}\mkern-5.5mu\scalebox{0.4}[1.0]{$\prec$}}\mathcal{A}_{b}$
with $1_{\mathcal{C}}\in\mathcal{A}_{b}$. For given non-zero complex numbers
$\alpha$ and $\beta$, let $p_{\alpha,\beta}=\alpha ab+\beta ba$. Then for each
$k\geq 0$,

$\varphi^{\prime}(p_{\alpha,\beta}^{k})=\frac{\varphi^{\prime}(a^{k})}{2^{k+1}\gamma}\Bigg{[}((\alpha+\beta)\varphi(b)+\gamma)^{k}\Big{(}(\alpha+\beta)\big{(}(k+1)\varphi^{\prime}(b)-\varphi(b)\omega/\gamma^{2}\big{)}+k\omega/\gamma\Big{)}\\\
-((\alpha+\beta)\varphi(b)-\gamma)^{k}\Big{(}(\alpha+\beta)\big{(}(k+1)\varphi^{\prime}(b)-\varphi(b)\omega/\gamma^{2}\big{)}-k\omega/\gamma\Big{)}\Bigg{]},$

where

(4.2) $\
\gamma=\sqrt{(\alpha-\beta)^{2}\varphi(b)^{2}+4\alpha\beta\varphi(b^{2})}\quad\text{and}\quad\omega=(\alpha-\beta)^{2}\varphi(b)\varphi^{\prime}(b)+2\alpha\beta\varphi^{\prime}(b^{2}).$

######  Remark 4.2.

We note that the infinitesimal $*$-distribution of $p_{\alpha,\beta}$ only
depends on that of $a$ and on $\varphi(b),\varphi(b^{2}),\varphi^{\prime}(b)$
and $\varphi^{\prime}(b^{2})$. In the particular case where
$\varphi(b)=\varphi^{\prime}(b)=0$, this reduces to

$\varphi^{\prime}(p_{\alpha,\beta}^{k})=\frac{k}{4}(\alpha\beta)^{k/2}\Big{(}1+(-1)^{k}\Big{)}\varphi^{\prime}(b^{2})\varphi(b^{2})^{k/2-1}\varphi^{\prime}(a^{k}).$

Again, as a direct consequence of Theorem 4.1, we obtain the infinitesimal
$*$-distribution of anti-commutators and commutators by taking
$(\alpha,\beta)=(1,1)$ or $(\alpha,\beta)=(i,-i)$ respectively.

######  Corollary 4.3.

Let $p=ab+ba$ and $q=i(ab-ba)$, then for each $k\geq 1$, we have

$\displaystyle\varphi^{\prime}(p^{k})$ $\displaystyle=$
$\displaystyle\frac{\varphi^{\prime}(a^{k})}{4\sqrt{\varphi(b^{2})}\varphi(b^{2})}\Bigg{[}(\varphi(b)+\sqrt{\varphi(b^{2})})^{k}\Big{(}2(k+1)\varphi^{\prime}(b)\varphi(b^{2})-\varphi^{\prime}(b^{2})\big{[}\varphi(b)-k\sqrt{\varphi(b^{2})}\big{]}\Big{)}$
$\displaystyle+(\varphi(b)-\sqrt{\varphi(b^{2})})^{k}\Big{(}2(k+1)\varphi^{\prime}(b)\varphi(b^{2})-\varphi^{\prime}(b^{2})\big{[}\varphi(b)+k\sqrt{\varphi(b^{2})}\big{]}\Big{)}\Bigg{]}$

and

$\varphi^{\prime}(q^{k})=\frac{\varphi^{\prime}(a^{k})\big{[}\varphi^{\prime}(b^{2})-2\varphi(b)\varphi^{\prime}(b)\big{]}}{8}\Big{(}\varphi(b^{2})-\varphi(b)^{2}\Big{)}^{\frac{k-3}{2}}\Bigg{[}4k\sqrt{\varphi(b^{2})-\varphi(b)^{2}}+\varphi(b)\big{[}(-1)+(-1)^{k}\big{]}\Bigg{]}.$

###### Proof of Theorem 4.1.

We start by noting that

$\widetilde{\varphi}(P_{\alpha,\beta}^{k})=\widetilde{\varphi}\left(\begin{bmatrix}p_{\alpha,\beta}^{k}&0\\\
0&p_{\alpha,\beta}^{k}\end{bmatrix}\right)=\begin{bmatrix}\varphi(p_{\alpha,\beta}^{k})&\varphi^{\prime}(p_{\alpha,\beta}^{k})\\\
0&\varphi(p_{\alpha,\beta}^{k})\end{bmatrix}.$

Thus in order to derive an expression for
$\varphi^{\prime}(p_{\alpha,\beta}^{k})$, we shall compute the $(1,2)$-entry
of the right hand side of (4.1). In order to do this, let us note for any
$z,z^{\prime}\in\mathbb{C}$ and matrix $M=\begin{bmatrix}z&z^{\prime}\\\
0&z\end{bmatrix}\in\widetilde{\mathbb{C}}$,

$M^{1/2}=\sqrt{M}=\begin{bmatrix}\sqrt{z}&\frac{z^{\prime}}{2\sqrt{z}}\\\
0&\sqrt{z}\end{bmatrix}\text{ and
}M^{-1}=\begin{bmatrix}z^{-1}&-z^{-2}z^{\prime}\\\ 0&z^{-1}\end{bmatrix}.$

Setting $x=\varphi(b)$ and $y=\varphi^{\prime}(b)$, we observe that

$X=\widetilde{\varphi}(B)=\begin{bmatrix}x&y\\\
0&x\end{bmatrix}\quad\text{and}\quad\Gamma=\sqrt{(\alpha-\beta)^{2}\widetilde{\varphi}(B)^{2}+4\alpha\beta\widetilde{\varphi}(B^{2})}\\\
=\begin{bmatrix}\gamma&\frac{\omega}{\gamma}\\\ 0&\gamma\end{bmatrix},$

where $\gamma$ and $\omega$ are defined in (4.2), and finally obtain that

$\frac{1}{2^{k+1}}\Gamma^{-1}=\frac{1}{2^{k+1}}\begin{bmatrix}\gamma^{-1}&-\gamma^{-2}\cdot\omega/\gamma\\\
0&\gamma^{-1}\end{bmatrix}=\frac{1}{2^{k+1}\gamma}\begin{bmatrix}1&-\omega/\gamma^{2}\\\
0&1\end{bmatrix}.$

Note that

$\displaystyle((\alpha+\beta)X-\Gamma)^{k+1}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}((\alpha+\beta)x-\gamma)^{k+1}&(k+1)\big{(}(\alpha+\beta)x-\gamma\big{)}^{k}\big{(}(\alpha+\beta)y-\omega/\gamma\big{)}\\\
0&((\alpha+\beta)x-\gamma)^{k+1}\end{bmatrix}$ $\displaystyle=$
$\displaystyle((\alpha+\beta)x-\gamma)^{k}\begin{bmatrix}(\alpha+\beta)x-\gamma&(k+1)\big{(}(\alpha+\beta)y-\omega/\gamma\big{)}\\\
0&(\alpha+\beta)x-\gamma\end{bmatrix}.$

Therefore,

$\displaystyle\frac{1}{2^{k+1}}\Gamma^{-1}\left((\alpha+\beta)X-\Gamma\right)^{k+1}$
$\displaystyle=$
$\displaystyle\frac{((\alpha+\beta)x-\gamma)^{k}}{2^{k+1}\gamma}\begin{bmatrix}1&-\omega/\gamma^{2}\\\
0&1\end{bmatrix}\begin{bmatrix}(\alpha+\beta)x-\gamma&(k+1)((\alpha+\beta)y-\omega/\gamma)\\\
0&(\alpha+\beta)x-\gamma\end{bmatrix}$ $\displaystyle=$
$\displaystyle\frac{((\alpha+\beta)x-\gamma)^{k}}{2^{k+1}\gamma}\begin{bmatrix}(\alpha+\beta)x-\gamma&(k+1)((\alpha+\beta)y-\omega/\gamma)-\omega/\gamma^{2}((\alpha+\beta)x-\gamma)\\\
0&(\alpha+\beta)x-\gamma\end{bmatrix}$ $\displaystyle=$
$\displaystyle\frac{((\alpha+\beta)x-\gamma)^{k}}{2^{k+1}\gamma}\begin{bmatrix}(\alpha+\beta)x-\gamma&(\alpha+\beta)\Big{(}(k+1)y-x\omega/\gamma^{2}\Big{)}-k\omega/\gamma\\\
0&(\alpha+\beta)x-\gamma\end{bmatrix}$

Similarly, we also have

$\frac{1}{2^{k+1}}\Gamma^{-1}((\alpha+\beta)X+\Gamma)^{k+1}\\\
=\frac{((\alpha+\beta)x+\gamma)^{k}}{2^{k+1}\gamma}\begin{bmatrix}(\alpha+\beta)x+\gamma&(\alpha+\beta)\Big{(}(k+1)y-x\omega/\gamma^{2}\Big{)}+k\omega/\gamma\\\
0&(\alpha+\beta)x+\gamma\end{bmatrix}.$

Thus, the right hand side of Equation (4.1) is

$\frac{\varphi^{\prime}(a^{k})}{2^{k+1}\gamma}\Bigg{[}((\alpha+\beta)\varphi(b)+\gamma)^{k}\Big{(}(\alpha+\beta)\big{(}(k+1)\varphi^{\prime}(b)-\varphi(b)\omega/\gamma^{2}\big{)}+k\omega/\gamma\Big{)}\\\
-((\alpha+\beta)\varphi(b)-\gamma)^{k}\Big{(}(\alpha+\beta)\big{(}(k+1)\varphi^{\prime}(b)-\varphi(b)\omega/\gamma^{2}\big{)}-k\omega/\gamma\Big{)}\Bigg{]},$

which completes the proof. ∎

###  4.2. General Polynomials

The commutativity of $\widetilde{\mathbb{C}}$ allows again deriving
infinitesimal $*$-distributions of general polynomials. We start by
considering polynomials of degree $3$ as in Proposition 3.6. To see how this
works, let $a_{1},\dots,a_{n}$, $b_{1},\dots,b_{n}$, $c_{1},\dots,c_{n}$,
$d_{1},\dots,d_{n}$ be elements from a $*$-incps
$(\mathcal{C},\varphi,\varphi^{\prime})$ satisfying
$\\{a_{1},\dots,a_{n},c_{1},\dots,c_{n}\\}\mathrel{\scalebox{0.9}[1.0]{$\prec$}\mkern-5.0mu\scalebox{0.4}[1.0]{$\prec$}\mkern-5.5mu\scalebox{0.4}[1.0]{$\prec$}}\\{b_{1},\dots,b_{n},d_{1},\dots,d_{n}\\}$.
Then setting
$A_{i}=\mathrm{Diag}(a_{i},a_{i}),B_{i}=\mathrm{Diag}(b_{i},b_{i}),C_{i}=\mathrm{Diag}(c_{i},c_{i}),$
and $D_{i}=\mathrm{Diag}(d_{i},d_{i})$ for all $i\in[n]$, we get by Theorem
2.4 that
$\\{A_{1},\dots,A_{n},C_{1},\dots,C_{n}\\}\prec\\{B_{1},\dots,B_{n},D_{1},\dots,D_{n}\\}$
in $(\widetilde{\mathcal{C}},\widetilde{\mathbb{C}},\widetilde{\varphi})$. By
using the fact that $\widetilde{\mathbb{C}}$ is commutative and following the
lines of the proof of Proposition 3.6, we obtain for any $m\geq 1$,

(4.3)
$\widetilde{\varphi}\left(\left[\sum_{k=1}^{n}A_{k}B_{k}C_{k}\right]^{m}\right)=\widetilde{\varphi}\left(\left[\sum_{k=1}^{n}A_{k}\widetilde{\varphi}(B_{k})C_{k}\right]^{m}\right)$

and

(4.4)
$\widetilde{\varphi}\left(\left[\sum_{k=1}^{n}B_{k}A_{k}D_{k}\right]^{m}\right)\\\
=\sum_{j_{1},\dots,j_{m}\in[n]}\widetilde{\varphi}(A_{j_{1}}A_{j_{2}}\cdots
A_{j_{m}})\widetilde{\varphi}(B_{j_{1}})\widetilde{\varphi}(D_{j_{1}}B_{j_{2}})\widetilde{\varphi}(D_{j_{2}}B_{j_{3}})\cdots\widetilde{\varphi}(D_{j_{m-1}}B_{j_{m}})\widetilde{\varphi}(D_{j_{m}}).$

Comparing the $(1,2)$-entries of the both sides of (4.3) and (4.4), we obtain
the expressions for
$\varphi^{\prime}\left([\sum_{k=1}^{n}a_{k}b_{k}c_{k}]^{m}\right)$ and
$\varphi^{\prime}\left([\sum_{k=1}^{n}b_{k}a_{k}d_{k}]^{m}\right)$
respectively.

This applies to more general polynomials for which we recall the setting: for
a given incps $(\mathcal{C},\varphi,\varphi^{\prime})$ and $\mathcal{A}$ and
$\mathcal{B}$ $*$-subalgebras of $\mathcal{C}$ with
$1_{\mathcal{C}}\in\mathcal{B}$, we set

$\mathcal{Q}=\\{a_{0}b_{1}a_{1}\cdots b_{k}a_{k}\mid k\geq
0,a_{0},\dots,a_{k}\in\mathcal{A}\text{ and
}b_{1},\dots,b_{k}\in\mathcal{B}\\},$

and denote by

$\widetilde{\mathcal{Q}}=\\{A_{0}B_{1}A_{1}\cdots B_{k}A_{k}\mid k\geq
0,a_{0},\dots,a_{k}\in\mathcal{A}\text{ and
}b_{1},\dots,b_{k}\in\mathcal{B}\\}$

where $A_{i}=\mathrm{Diag}(a_{i},a_{i})$ and
$B_{j}=\mathrm{Diag}(b_{j},b_{j})$ for each $0\leq i\leq n,1\leq j\leq n$. For
each element $Y=A_{0}B_{1}A_{1}\cdots B_{k}A_{k}$ in
$\widetilde{\mathcal{Q}}$, we set
$\widetilde{Y}=A_{0}\widetilde{\varphi}({B}_{1})A_{1}\cdots\widetilde{\varphi}({B}_{k})A_{k}$.

Assume
$\mathcal{A}\mathrel{\scalebox{0.9}[1.0]{$\prec$}\mkern-5.0mu\scalebox{0.4}[1.0]{$\prec$}\mkern-5.5mu\scalebox{0.4}[1.0]{$\prec$}}\mathcal{B}$
with $1_{\mathcal{C}}\in\mathcal{B}$, we illustrate how to derive
infinitesimal $*$-distributions of polynomials $p$ of the form
$p=\sum_{j=1}^{n}b_{j}y_{j}b_{j}^{\prime}$ with
$b_{1},b_{1}^{\prime},\dots,b_{n},b_{n}^{\prime}\in\mathcal{B}$ and
$y_{1},\dots,y_{n}\in\mathcal{Q}$. Setting
$P=\sum_{j=1}^{n}B_{j}Y_{j}B_{j}^{\prime}$, we have for any $m\in\mathbb{N}$,

(4.5) $\displaystyle\widetilde{\varphi}(P^{m})$ $\displaystyle=$
$\displaystyle\widetilde{\varphi}(\big{[}\sum_{j=1}^{n}B_{j}\widetilde{Y_{j}}B_{j}^{\prime}\big{]}^{m})$
$\displaystyle=$
$\displaystyle\sum_{j_{1},\dots,j_{m}\in[n]}\widetilde{\varphi}(\widetilde{Y}_{j_{1}}\cdots\widetilde{Y}_{j_{m}})\widetilde{\varphi}(B_{j_{1}})\widetilde{\varphi}(B_{j_{1}}^{\prime}B_{j_{2}})\widetilde{\varphi}(B_{j_{2}}^{\prime}B_{j_{3}})\cdots\widetilde{\varphi}(B^{\prime}_{j_{m-1}}B_{j_{m}})\widetilde{\varphi}(B_{j_{m}}),$

where the second equality is an application of (4.4) while the first follows
by an analogous argument as in Lemma 3.7. Hence, we infer that
$\\{B_{1},\dots,B_{n},B_{1}^{\prime},$ $\dots,B_{n}^{\prime},Y_{1},$
$\dots,Y_{n}\\}$ and $\\{B_{1},\dots,B_{n},$
$B_{1}^{\prime},\dots,B_{n}^{\prime},$
$\widetilde{Y}_{1},\dots,\widetilde{Y}_{n}\\}$ have the same $*$-distribution
with respect to $\widetilde{\varphi}$. It remains to track the $(1,2)$-entry
of right hand side of (4.5) to obtain $\varphi^{\prime}(p^{m})$.

In order to clarify how to do so, we consider in the following proposition the
polynomials in (4.3) and (4.4). In order to soothe the notation, let us set
$\varphi^{(0)}=\varphi$ and $\varphi^{(1)}=\varphi^{\prime}$ and define, for
each $m\geq 0$,

$\mathcal{W}_{m}=\\{(1,0,0,\dots,0),(0,1,0,\cdots,0),\cdots,(0,0,\cdots,0,1)\\}\subset\mathbb{R}^{m+1}.$

######  Proposition 4.4.

Suppose that
$a_{1},\dots,a_{n},b_{1},\dots,b_{n},c_{1},\dots,c_{n},d_{1},\dots,d_{n}$ are
elements in $\mathcal{C}$ that
$\\{a_{1},\dots,a_{n},c_{1},\dots,c_{n}\\}\mathrel{\scalebox{0.9}[1.0]{$\prec$}\mkern-5.0mu\scalebox{0.4}[1.0]{$\prec$}\mkern-5.5mu\scalebox{0.4}[1.0]{$\prec$}}\\{b_{1},\dots,b_{n},d_{1},\dots,d_{n}\\}$
from a $*$-incps $(\mathcal{C},\varphi,\varphi^{\prime})$, then we have

$\displaystyle\varphi^{\prime}\left(\left[\sum_{k=1}^{n}a_{k}b_{k}c_{k}\right]^{m}\right)$
$\displaystyle=$
$\displaystyle\varphi^{\prime}\left(\left[\sum_{k=1}^{n}a_{k}\varphi(b_{k})c_{k}\right]^{m}\right)+\sum_{j=1}^{m}\varphi\left(\left[\sum_{k=1}^{n}a_{k}\varphi(b_{k})c_{k}\right]^{j-1}\left[\sum_{k=1}^{n}a_{k}\varphi^{\prime}(b_{k})c_{k}\right]\left[\sum_{k=1}^{n}a_{k}\varphi(b_{k})c_{k}\right]^{m-j}\right)$

and

$\displaystyle\varphi^{\prime}\left(\left[\sum_{k=1}^{n}b_{k}a_{k}d_{k}\right]^{m}\right)$
$\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}j_{1},\dots,j_{m}\in[n]\\\
(u_{1},\dots,u_{m+1})\in\mathcal{W}_{m}\end{subarray}}\varphi^{(u_{1})}(a_{j_{1}}a_{j_{2}}\cdots
a_{j_{m}})\varphi^{(u_{2})}(b_{j_{1}})\varphi^{(u_{3})}(d_{j_{1}}b_{j_{2}})\cdots\varphi^{(u_{m})}(d_{j_{m-1}}b_{j_{m}})\varphi^{(u_{m+1})}(d_{j_{m}}).$

###### Proof.

Note that the $(1.2)$-entry of
$\widetilde{\varphi}\left(\Big{[}\sum_{k=1}^{n}A_{k}B_{k}C_{k}\Big{]}^{m}\right)$
is
$\varphi^{\prime}\left(\Big{[}\sum_{k=1}^{n}a_{k}b_{k}c_{k}\Big{]}^{m}\right)$.
In addition, we observe that

$\displaystyle\left[\sum_{k=1}^{n}A_{k}\widetilde{\varphi}(B_{k})C_{k}\right]^{m}=\begin{bmatrix}\sum_{k=1}^{n}a_{k}\varphi(b_{k})c_{k}&\sum_{k=1}^{n}a_{k}\varphi^{\prime}(b_{k})c_{k}\\\
0&\sum_{k=1}^{n}a_{k}\varphi(b_{k})c_{k}\end{bmatrix}^{m}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\Big{[}\sum_{k=1}^{n}a_{k}\varphi(b_{k})c_{k}\Big{]}^{m}&\sum_{j=1}^{m}\Big{[}\sum_{k=1}^{n}a_{k}\varphi(b_{k})c_{k}\Big{]}^{j-1}\Big{[}\sum_{k=1}^{n}a_{k}\varphi^{\prime}(b_{k})c_{k}\Big{]}\Big{[}\sum_{k=1}^{n}a_{k}\varphi(b_{k})c_{k}\Big{]}^{m-j}\\\
0&\Big{[}\sum_{k=1}^{n}a_{k}\varphi(b_{k})c_{k}\Big{]}^{m}\end{bmatrix}.$

By computing
$\widetilde{\varphi}\left(\left[\sum_{k=1}^{n}A_{k}\widetilde{\varphi}(B_{k})C_{k}\right]^{m}\right)$
and comparing the $(1,2)$-entry of both sides of (4.3), we obtain the formula
for $\varphi^{\prime}([\sum_{k=1}^{n}a_{k}b_{k}c_{k}]^{m})$.

Similarly, note that the $(1,2)$-entry of the left hand side of (4.4) is
$\varphi^{\prime}([\sum_{k=1}^{n}b_{k}a_{k}d_{k}]^{m})$. Thus, we shall
compute the $(1,2)$-entry of the right hand side of (4.4). Observe that for
any $(j_{1},\dots,j_{m})\in[2]^{m}$,

$\widetilde{\varphi}(A_{j_{1}}A_{j_{2}}\cdots
A_{j_{m}})\widetilde{\varphi}(B_{j_{1}})\widetilde{\varphi}(D_{j_{1}}B_{j_{2}})\widetilde{\varphi}(D_{j_{2}}B_{j_{3}})\cdots\widetilde{\varphi}(D_{j_{m-1}}B_{j_{m}})\widetilde{\varphi}(D_{j_{m}})\\\
=\begin{bmatrix}\varphi(a_{j_{1}}\cdots
a_{j_{m}})&\varphi^{\prime}(a_{j_{1}}\cdots a_{j_{m}})\\\
0&\varphi(a_{j_{1}}\cdots
a_{j_{m}})\end{bmatrix}\begin{bmatrix}\varphi(b_{j_{1}})&\varphi^{\prime}(b_{j_{1}})\\\
0&\varphi(b_{j_{1}})\end{bmatrix}\cdots\begin{bmatrix}\varphi(d_{j_{m}})&\varphi^{\prime}(d_{j_{m}})\\\
0&\varphi(d_{j_{m}})\end{bmatrix}.$

Then it is easy to see that the $(1,2)$-entry of the right hand side of (4.4)
is

$\sum_{\begin{subarray}{c}j_{1},\dots,j_{m}\in[n]\\\
(u_{1},\dots,u_{m+1})\in\mathcal{W}_{m}\end{subarray}}\varphi^{(u_{1})}(a_{j_{1}}a_{j_{2}}\cdots
a_{j_{m}})\varphi^{(u_{2})}(b_{j_{1}})\varphi^{(u_{3})}(d_{j_{1}}b_{j_{2}})\cdots\varphi^{(u_{m})}(d_{j_{m-1}}b_{j_{m}})\varphi^{(u_{m+1})}(d_{j_{m}}).$

∎

##  5\. Applications to Random Matrices

Fix $N_{0}\in\mathbb{N}$ and let for each $N\times N$ random matrix $A_{N}$,
$T_{A_{N}}$ be the matrix defined by

$T_{A_{N}}=\begin{bmatrix}0&A^{(1,2)}\\\ A^{(2,1)}&0\end{bmatrix}\text{
whenever }A_{N}=\begin{bmatrix}A^{(1,1)}&A^{(1,2)}\\\
A^{(2,1)}&A^{(2,2)}\end{bmatrix}$

with $A^{(1,1)}$ (respectively $A^{(2,2)}$) be the corresponding square
$N_{0}\times N_{0}$ (respectively $N\times N$) submatrix, $A^{(1,2)}$ be the
corresponding $N_{0}\times N-N_{0}$ submatrix, $A^{(2,1)}$ be the
corresponding $N-N_{0}\times N_{0}$ submatrix. We also recall the definition
of the partial trace that is defined for a given $A_{N}$ by

$\psi_{N}(A_{N}):=\frac{1}{N_{0}}(E\circ Tr_{N})(A_{N}^{(1,1)}).$

where $Tr_{N}$ is the non-normalized trace. Here we note that the partial
trace $\psi_{N}$ depends on $N_{0}$.

######  Proposition 5.1.

Let $A_{N}$ and $B_{N}$ be independent $N\times N$ Wigner matrices and let for
some $m\geq 0$ and non-zero $\alpha\in\mathbb{C}$, $P_{N}$ be the polynomial
defined by

(5.1) $P_{N}=\alpha T_{A_{N}^{m}}B_{N}+\widebar{\alpha}B_{N}T_{A_{N}^{m}}.$

Then $(T_{A_{N}^{m}},B_{N})_{N}$ is asymptotically monotone independent with
respect to $\psi_{N}$ and the limiting distribution of $P_{N}$ with respect to
$\psi$ is $\mu=\frac{1}{2}(\delta_{\omega}+\delta_{-\omega})$ where $\delta$
is the dirac measure and $\omega=\sqrt{|\alpha|D_{m}}$ with

$D_{m}=\begin{cases}C_{m}-C^{2}_{m/2}&\text{ if }m\text{ is even,}\\\
C_{m}&\text{ if }m\text{ is odd,}\end{cases}$

and $C_{k}$ being the $k$-th Catalan number.

###### Proof.

We start by noting that $A_{N}$ and $B_{N}$ are asymptotically free with
respect to $E\circ tr_{N}$ where we recall that $tr_{N}$ is the normalized
trace. Also note that
$T_{A_{N}^{m}}=j_{N}A_{N}^{m}(1-J_{N})+(1-J_{N})A_{N}^{m}J_{N}$ where
$j_{N}=\sum_{j=1}^{N_{0}}E_{N}^{j,j}$ and that by Theorem 2.7 $A_{N}^{m}$ and
$B_{N}$ are asymptotically infinitesimally free from $J_{N}$ . Then it follows
by Theorem 2.6 that $(T_{A_{N}^{m}},B_{N})_{N}$ is asymptotically monotone
independent with respect to $\psi_{N}$. Moreover, the limiting distribution of
$T_{A_{N}^{m}}$ with respect to $\psi_{N}$ can then be easily computed
following [17, Proposition 5.3]. Indeed, we have for any $k\geq 1$,

$\displaystyle\lim_{N\to\infty}\psi_{N}(T_{A_{N}^{m}}^{k})$ $\displaystyle=$
$\displaystyle\begin{cases}\Big{[}\lim\limits_{N\to\infty}\big{(}(E\circ
tr_{N})(A_{N}^{2m})-(E\circ tr_{N})(A_{N}^{m})^{2}\big{)}\Big{]}^{k/2}&\text{
if }k\text{ is even,}\\\ 0&\text{ if }k\text{ is odd.}\end{cases}$

By the Wigner semicircular law, we infer that

$\displaystyle\lim_{N\to\infty}\psi_{N}(T_{A_{N}^{m}}^{k})$ $\displaystyle=$
$\displaystyle\begin{cases}D_{m}^{k/2}&\text{ if }k\text{ is even,}\\\
0&\text{ if }k\text{ is odd.}\end{cases}$

Finally, observing that $\lim_{N\to\infty}(E\circ tr_{N})(B_{N})=0$ and
$\lim_{N\to\infty}(E\circ tr_{N})(B_{N}^{2})=1$, we get by applying Theorem
3.1 and taking into account Remark 3.2 that for any $k\geq 1$,

$\displaystyle\lim_{N\to\infty}\psi_{N}(P_{N}^{k})=\begin{cases}\big{(}|\alpha|D_{m}\big{)}^{k/2}&\text{
if }k\text{ is even}\\\ 0&\text{ if }k\text{ is odd.}\end{cases}$

∎

######  Remark 5.2.

The above result also holds for $A_{N}$ being an $N\times N$ GUE matrix, and
$B_{N}$ an entry permutation of $A_{N}$ that is self-adjoint. In particular,
$B_{N}$ can be chosen to be the transpose of $A_{N}$. Indeed, by [16] and
[25], we have that $(A_{N},B_{N})_{N}$ is asymptotically free and by combining
Theorems 2.6 and 2.7, we obtain that $(T_{A_{N}},B_{N})_{N}$ is asymptotically
monotone independent with respect to $\psi_{N}$. Then by analogous arguments,
we obtain the same limiting $*$-distribution of $P_{N}$ with respect to
$\psi_{N}$ as in Proposition 5.1.

######  Proposition 5.3.

Let $A_{N}$ and $B_{N}$ be $N\times N$ independent Wigner matrices and
consider the polynomial given by

$P_{N}=B_{N}^{2n}T_{A_{N}}B_{N}^{2h}T_{A_{N}}B_{N}^{2n}+B_{N}^{2m}T_{A_{N}}B_{N}^{2s}T_{A_{N}}B_{N}^{2m}$

for some $n,m,h,s\geq 1$. Then $(T_{A_{N}},B_{N})_{N}$ is asymptotically
monotone independent with respect to $\psi_{N}$ and $P_{N}$ converges in
distribution with respect to $\psi$ to
$\mu=p\delta_{\vartheta+\zeta}+(1-p)\delta_{\vartheta-\zeta}$ where $\delta$
denotes the dirac measure,
$p=\frac{1}{2\zeta}(\zeta-\vartheta+C_{h}C_{n}^{2}+C_{s}C_{m}^{2})$ with

$\vartheta=\frac{1}{2}(C_{2m}C_{s}+C_{2n}C_{h}),\qquad\zeta=\frac{1}{2}\sqrt{(C_{2m}C_{s}-C_{2n}C_{h})^{2}+4C_{n+m}^{2}C_{s}C_{h}},$

and $C_{k}$ being the $k$-th Catalan number.

###### Proof.

We first note that $(T_{A_{N}},B_{N})_{N}$ is asymptotically monotone
independent with respect to $\psi_{N}$. Then there is a $*$-probability space
$(\mathcal{C},\psi)$ and elements $a$ and $b$ in $\mathcal{C}$ such that
$a\prec b$ with $1_{\mathcal{C}}\in\mathcal{A}_{b}$ and $(T_{A_{N}},B_{N})$
converges to $(a,b)$ in $*$-distribution. In other words, for any $k\geq 1$,

$\lim_{N\to\infty}\psi_{N}(P_{N}^{k})=\psi(p^{k})\text{ where
}p=b^{2n}ab^{2h}ab^{2n}+b^{2m}ab^{2s}ab^{2m}.$

Lemma 3.7 together with Theorem 3.5 yield that

$\displaystyle\psi(p^{k})$ $\displaystyle=$
$\displaystyle\psi\Big{(}[b^{2n}\cdot\psi(b^{2h})a^{2}\cdot
b^{2n}+b^{2m}\cdot\psi(b^{2s})a^{2}\cdot b^{2m}]^{k}\Big{)}$ $\displaystyle=$
$\displaystyle\sum_{j_{1},\dots,j_{k}=1}^{2}\psi(a_{j_{1}}\cdots
a_{j_{k}})\psi(b_{j_{1}})\psi(b_{j_{1}}b_{j_{2}})\psi(b_{j_{2}}b_{j_{3}})\cdots\psi(b_{j_{k-1}}b_{j_{k}})\psi(b_{j_{k}})$

where $a_{1}=\varphi(b^{2h})a^{2},a_{2}=\psi(b^{2s})a^{2},b_{1}=b^{2n}$, and
$b_{2}=b^{2m}.$ Since $A_{N}$ is an $N\times N$ Wigner matrix, then (2.3)
yields $\psi(a^{2k})=\lim_{N\to\infty}\psi_{N}(T_{A_{N}}^{2k})=1$ and hence
for all $(j_{1},\dots,j_{k})\in~{}[2]^{k}$,

$\psi(a_{j_{1}}\cdots
a_{j_{k}})=\psi(a^{2k})\psi(\tilde{b}_{j_{1}})\psi(\tilde{b}_{j_{2}})\cdots\psi(\tilde{b}_{j_{k}})=\psi(\tilde{b}_{j_{1}})\psi(\tilde{b}_{j_{2}})\cdots\psi(\tilde{b}_{j_{k}}),$

where $\tilde{b}_{1}=b^{2h}$ and $\tilde{b}_{2}=b^{2s}$. Thus, if for each
$i\in[2]$, we set $\alpha_{1}^{(i)}=\varphi(b_{i})$ and

$\alpha_{k}^{(i)}=\sum_{j_{1},\dots,j_{k-1}\in[2]}\psi(b_{i}\cdot
b_{j_{1}})\psi(b_{j_{1}}b_{j_{2}})\cdots\psi(b_{j_{k-2}}b_{j_{k-1}})\psi(b_{j_{k-1}})\psi(\tilde{b}_{j_{1}})\psi(\tilde{b}_{j_{2}})\cdots\psi(\tilde{b}_{j_{k-1}})\text{
for }k\geq 2$

then we obtain

$\displaystyle\psi(p^{k})$ $\displaystyle=$
$\displaystyle\psi(b_{1})\psi(\tilde{b}_{1})\alpha_{k}^{(1)}+\psi(b_{2})\psi(\tilde{b}_{2})\alpha_{k}^{(2)}.$

Now, observe that

$\psi(b_{1})=C_{n},\psi(b_{2})=C_{m},\psi(\tilde{b}_{1})=C_{h},\text{ and
}\psi(\tilde{b}_{2})=C_{s},$

and for all $u\in[k-1]$,

(5.2) $\psi(b_{j_{u}}b_{j_{u+1}})=\begin{cases}C_{2n}&\text{ if
}(j_{u},j_{u+1})=(1,1),\\\ C_{2m}&\text{ if }(j_{u},j_{u+1})=(2,2),\\\
C_{n+m}&\text{ if }(j_{u},j_{u+1})=(1,2)\text{ or }(2,1).\end{cases}$

We observe that for each $l\geq 2$,

$\alpha_{l}^{(2)}=\psi(b_{2}b_{1})\psi(\tilde{b_{1}})\alpha_{l-1}^{(1)}+\psi(b_{2}b_{2})\psi(\tilde{b_{2}})\alpha_{l-1}^{(2)}=C_{n+m}C_{h}\alpha_{l-1}^{(1)}+C_{2m}C_{s}\alpha_{l-1}^{(2)},$

which when iterated we obtain for each $l\geq 3$,

$\displaystyle\alpha_{l}^{(2)}$ $\displaystyle=$ $\displaystyle
C_{n+m}C_{h}\alpha_{l-1}^{(1)}+C_{n+m}C_{h}C_{2m}C_{s}\alpha_{l-2}^{(1)}+C_{2m}^{2}C_{s}^{2}\alpha_{l-2}^{(2)}$
$\displaystyle=$ $\displaystyle
C_{n+m}C_{h}\Big{(}\alpha_{l-1}^{(1)}+C_{2m}C_{s}\alpha_{l-2}^{(1)}+C_{2m}^{2}C_{s}^{2}\alpha_{l-3}^{(1)}\Big{)}+C_{2m}^{3}C_{s}^{3}\alpha_{l-3}^{(2)}$
$\displaystyle=$ $\displaystyle
C_{n+m}C_{h}\Big{(}\alpha_{l-1}^{(1)}+\sum_{j=1}^{l-3}C_{2m}^{j}C_{s}^{j}\alpha_{l-(j+1)}^{(1)}\Big{)}+C_{2m}^{l-2}C_{s}^{l-2}(C_{n+m}C_{n}C_{h}+C_{2m}C_{m}C_{s}).$

Using the expression obtained for $\alpha_{k-1}^{(2)}$, we have get for any
$k\geq 2$,

$\displaystyle\alpha_{k}^{(1)}$
$\displaystyle=\psi(b_{1}b_{1})\psi(\tilde{b_{1}})\alpha_{k-1}^{(1)}+\psi(b_{1}b_{2})\psi(\tilde{b_{2}})\alpha_{k-1}^{(2)}$
$\displaystyle=C_{2n}C_{h}\alpha_{k-1}^{(1)}+C_{n+m}C_{s}\alpha_{k-1}^{(2)}$
$\displaystyle=C_{2n}C_{h}\alpha_{k-1}^{(1)}+C_{n+m}^{2}C_{h}\Big{(}C_{s}\alpha_{k-2}^{(1)}+\sum_{j=1}^{k-4}C_{2m}^{j}C_{s}^{j+1}\alpha_{k-j-2}^{(1)}+C_{2m}^{k-3}C_{s}^{k-2}C_{n}\Big{)}$
$\displaystyle\quad+C_{n+m}C_{2m}^{k-2}C_{s}^{k-1}C_{m}.$

Then we observe

$\displaystyle\alpha_{k}^{(1)}-C_{2m}C_{s}\alpha_{k-1}^{(1)}=C_{2n}C_{h}\alpha_{k-1}^{(1)}+(C_{n+m}^{2}-C_{2m}C_{2n})C_{s}C_{h}\alpha_{k-2}^{(1)},$

and hence we obtain the following linear homogeneous recurrences

$\alpha_{k}^{(1)}-(C_{2m}C_{s}+C_{2n}C_{h})\alpha_{k-1}^{(1)}+(C_{2m}C_{2n}-C_{n+m}^{2})C_{s}C_{h}\alpha_{k-2}^{(1)}=0.$

Thus by solving the characteristic equation
$x^{2}-(C_{2m}C_{s}+C_{2n}C_{h})x+(C_{2m}C_{2n}-C_{n+m}^{2})C_{s}C_{h}=~{}0,$
we have $x=\vartheta\pm\zeta$ as a solution with

$\vartheta=\frac{1}{2}(C_{2m}C_{s}+C_{2n}C_{h})\text{ and
}\zeta=\frac{1}{2}\sqrt{(C_{2m}C_{s}-C_{2n}C_{h})^{2}+4C_{n+m}^{2}C_{s}C_{h}}.$

This yields that

$\alpha_{k}^{(1)}=s_{1}^{(1)}(\vartheta+\zeta)^{k}+s_{2}^{(1)}(\vartheta-\zeta)^{k}$

for some constants $s_{1}^{(1)}$ and $s_{2}^{(1)}$ that determined by the
initial conditions, and similarly

$\alpha_{k}^{(2)}=s_{1}^{(2)}(\vartheta+\zeta)^{k}+s_{2}^{(2)}(\vartheta-\zeta)^{k}$

for some constants $s_{1}^{(2)}$ and $s_{2}^{(2)}$. Hence, putting the above
terms together, we infer that for any $k\geq 1$,

$\psi(p^{k})=(C_{n}C_{h}s_{1}^{(1)}+C_{m}C_{s}s_{1}^{(2)})(\vartheta+\zeta)^{k}+(C_{n}C_{h}s_{2}^{(1)}+C_{m}C_{s}s_{2}^{(2)})(\vartheta-\zeta)^{k}.$

Now we observe that $\psi(p^{0})=1$ and
$\psi(p)=C_{h}C_{n}^{2}+C_{s}C_{m}^{2}$. Thus,

$\begin{cases}C_{n}C_{h}(s_{1}^{(1)}+s_{2}^{(1)})+C_{m}C_{s}(s_{1}^{(2)}+s_{2}^{(2)})=1\\\
C_{n}C_{h}[s_{1}^{(1)}(\vartheta+\zeta)+s_{2}^{(1)}(\vartheta-\zeta)]+C_{m}C_{s}[s_{1}^{(2)}(\vartheta+\zeta)+s_{2}^{(2)}(\vartheta-\zeta)]=C_{h}C_{n}^{2}+C_{s}C_{m}^{2}.\end{cases}$

As a result, we obtain

$\displaystyle C_{n}C_{h}s_{1}^{(1)}+C_{m}C_{s}s_{1}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2\zeta}(\zeta-\vartheta+C_{h}C_{n}^{2}+C_{s}C_{m}^{2})$
$\displaystyle C_{n}C_{h}s_{2}^{(1)}+C_{m}C_{s}s_{2}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2\zeta}(\zeta+\vartheta-C_{h}C_{n}^{2}-C_{s}C_{m}^{2})$

by solving a system of equations which concludes the proof by obtaining for
any $k\geq 1$,

$\psi(p^{k})=\frac{1}{2\zeta}(\zeta-\vartheta+C_{h}C_{n}^{2}+C_{s}C_{m}^{2})(\vartheta+\zeta)^{k}+\frac{1}{2\zeta}(\zeta+\vartheta-
C_{h}C_{n}^{2}-C_{s}C_{m}^{2})(\vartheta-\zeta)^{k}$

∎

## Acknowledgement

The authors would like to express their gratitude to Octavio Arizmendi and
Adrián Celestino for the valuable discussions that led to this project.

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