# Husimi Q-functions attached to hyperbolic Landau levels

⋆Z. Mouayn, ♭H. Chhaiba, ‡H. Kassogue, ΥP. K. Kikodio

###### Abstract.

We are concerned with a phase-space probability distribution which is known as
Husimi $Q$-function of a density operator with respect to a set of coherent
states $|\widetilde{\kappa}_{z,B,R,m}\rangle$ attached to an $m$th hyperbolic
Landau level and labeled by points $z$ of an open disk of radius $R$, where
$B>0$ is proportional to a magnetic field strength. For a density operator
representing a projector on a Fock state $\left|j\right\rangle$ we obtain the
$Q_{j}$ distribution and discuss some of its basic properties such as its
characteristic function and its main statistical parameters. We achieve the
same program for the thermal density operator (mixed states) of the isotonic
oscillator for which we establish a lower bound for the associated
thermodynamical potential. We recover most of the results of the Euclidean
setting (flat case) as the parameter $R$ goes to infinity by making appeal to
asymptotic formulas involving orthogonal polynomials and special functions. As
a tool, we establish a summation formula for the special Kampé de Fériet
function $\digamma_{2:0:0}^{1:2:2}$.

Keywords. Coherent states; Isotonic oscillator; Husimi Q-Function; Hyperbolic
Landau Level; Thermodynamical potential.

⋆ Department of Mathematics, Faculty of Sciences and Technics (M’Ghila),
Sultan Moulay Slimane University, P.O. Box. 523, Béni Mellal, Morocco.
<EMAIL_ADDRESS>
♭ Residence num 408 Hassan 2 avenue appt 6, Diour Jamaa, Rabat, Morocco.
<EMAIL_ADDRESS>
‡ High Institute of Applied Technologies (TechnoLAB-ISTA), Bamako, Mali.
<EMAIL_ADDRESS>
Υ 13, Market avenue, Mongt-Ngafula, Kinshasa, DR Congo.
<EMAIL_ADDRESS>
##  1\. Introduction

Coherent states (CS) are an overcomplete family of normalized ket vectors
$\left|\zeta\right\rangle$, $\zeta\in X$, in a Hilbert space $\mathcal{H}$
corresponding to a specific quantum model and provide $\mathcal{H}$ with the
following resolution of the identity

$1_{\mathcal{H}}=\int\limits_{X}\left|\zeta\right\rangle\left\langle\zeta\right|d\mu\left(\zeta\right).$
(1.1)

Here, $X$ represents the phase-space domain and $d\mu\left(\zeta\right)$ the
associated integration measure on $X$. These CS have long been known for the
harmonic oscillator, whose properties have been used as a model for more
quantum systems [1]. In a such system of states the density operator
$\hat{\rho}$ for an arbitrary (pure or mixed) state can be represented by the
classical Husimi $Q$-function [2]:

$Q_{\hat{\rho}}(\zeta):=\left\langle\zeta\right|\hat{\rho}\left|\zeta\right\rangle.$
(1.2)

In particular, for a pure state of the form
$\hat{\rho}=\left|\phi\right\rangle\left\langle\phi\right|$ where
$\left|\phi\right\rangle\in\mathcal{H}$ is an arbitrary vector, the
$Q$-function can also be expressed as

$Q_{\left|\phi\right\rangle\left\langle\phi\right|}(\zeta)=\left(\mathcal{N}_{\zeta}\right)^{-1}\left|W\left[\phi\right]\left(\zeta\right)\right|^{2}$
(1.3)

in terms of the associated coherent state transform W defined from
$\mathcal{H}$ into a specific $L^{2}$ subspace of $X.$ Here,
$\mathcal{N}_{\zeta}$ is a factor ensuring the normalization condition
$\langle\zeta\left|\zeta\right\rangle=1.$ For a general density operator
$\hat{\rho},$ the function $Q_{\hat{\rho}}(\zeta)$ is normalized, since

$1=\mathrm{Tr}\left[\hat{\rho}\right]=\int\limits_{X}Q_{\hat{\rho}}(\zeta)d\mu(\zeta).$
(1.4)

As $Q_{\hat{\rho}}$ is introduced in a way that guarantees it to be non
negative, then it allows the representation of quantum states by a probability
distribution in the phase space $X$. It provides, equivalently to the Glauber-
Surdashan or Wigner representations, a basis for a formal equivalence between
the quantum and classical descriptions of optical coherence [3].

In this paper, we are concerned with the Husimi function of a density operator
with respect to a set of generalized CS (GCS)
$\left|\widetilde{\kappa}_{z,B,R,m}\right\rangle$ belonging to the Hilbert
space $L^{2}\left(\mathbb{R}_{+}\right),$ labeled by points
$z\in\mathbb{D}_{R}=\left\\{\zeta\in\mathbb{C},\left|\zeta\right|<R\right\\}$,
$R>0$ and attached to a higher $m$th hyperbolic Landau level in the presence
of a magnetic field whose strength is proportional to $B>0$. We first
introduce GCS via a group theoretical method by displacing an $m$th Laguerre
function by mean of a square integrable irreducible unitary representation of
the affine group. These GCS are then used to obtain the Husimi distribution,
$Q_{j}^{(B,R,m)}$, for a projector operator
$\hat{\rho}_{j}=\left|j\right\rangle\left\langle j\right|$ on a Fock state
$\left|j\right\rangle$, which is known as a pure state. For this distribution
we write down the characteristic function from which we derive the main value
and the variance. As in the Euclidean setting [4] this distribution may be
useful for tackling the hyperbolic version of Ginibre-type processes [5].
Next, from $Q_{j}^{(B,R,m)}$ we deduce the Husimi function
$Q_{\beta}^{(B,R,m)}$ for the thermal density operator (heat semi-group)
$\hat{\rho}_{\beta}$ of the Hamiltonian of the isotonic oscillator. Since
$Q_{j}^{(B,R,m)}$ may be also viewed as a lower symbol for
$\hat{\rho}_{\beta}$ we apply a Berezin-Lieb inequality [6] (with a specific
choice of a convex function) to obtain a lower bound for the thermodynamical
potential associated with this Hamiltonian. We also establish a formula for
the characteristic function of $Q_{\beta}^{(B,R,m)}$, from which we derive the
mean and the variance parameters. Finally, we recover most of the results [4]
of the flat case as the radius parameter $R$ goes to infinity. As a tool we
establish a summation formula for the special Kampé de Fériet function
$F^{1:2:2}_{2:0:0}$.

The paper is organized as follows. In Section 2, we recall some notations and
facts about orthogonal polynomials and special functions we will be using. In
section 3, we discuss the construction of GCS labeled by elements of the
affine group as well as their relationship to hyperbolic Landau levels.
Section 4 deals with the Q-representation for a pure state as well as the
corresponding characteristic function and the main statistical parameters. The
same task is acheived in Section 5 for the the thermal density operator of the
isotonic oscillator. For the latter one, we also give a lower bound for the
thermodynamical potential. In Section 6, we establish a summation formula for
the Kampé de Fériet function $\digamma_{2:0:0}^{1:2:2}$. Section 7 is devoted
to some concluding remarks. Proofs of our results are detailed in appendices.

##  2\. Some notations and definitions

This section collects the basic notations and definitions of special functions
and orthogonal polynomials used in the rest of the paper. The reader may
proceed to Section 3 and refer back here as necessary. For more details on the
theory of these functions we refer to [8, 11, 9, 10, 7].

1. For $a\in\mathbb{C}$, the shifted factorial or Pochhammer symbol is defined by

$(a)_{k}=a(a+1)\dots(a+k-1),\qquad k\in\mathbb{N}$ (2.1)

where by convention $(a)_{0}=1$. When $a=-n$ with
$n\in\mathbb{N}^{*}=\mathbb{N}-\\{0\\}$,

$(-n)_{k}=\left\\{\begin{array}[]{cl}\frac{(-1)^{k}n!}{(n-k)!},&0\leq k\leq
n\\\ 0,&k>n\end{array}\right..$ (2.2)

2. For $a\in\mathbb{C}$ and $k\in\mathbb{N}$, the binomial coefficient is defined by

$\binom{a}{k}=\frac{a(a-1)\dots(a-k+1)}{k!}=\frac{(-1)^{k}(-a)_{k}}{k!}$ (2.3)

3. For $z\in\mathbb{C}$, the gamma function is defined by

$\Gamma(z)=\int\limits_{0}^{\infty}t^{z-1}e^{-t}dt,\qquad{\rm Re}\,z>0.$ (2.4)

Note that $\Gamma(n+1)=n!$ if $n\in\mathbb{N}$, and

$(a)_{k}=\frac{\Gamma(a+k)}{\Gamma(a)}$ (2.5)

if $a\in\mathbb{C}\setminus\mathbb{Z}_{-}$.
4. For $a,b\in\mathbb{C}$ such that $\mbox{Re}\,a,\mbox{Re}\,b>0$, the beta function is defined by

$\mathcal{B}(a,b)=\int\limits_{0}^{1}t^{a-1}(1-t)^{b-1}dt=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}.$
(2.6)

5. For $a_{1},\dots,a_{p}\in\mathbb{C}$ and $c_{1},\dots,c_{q}\in\mathbb{C}\setminus\mathbb{Z}_{-}$, the generalized hypergeometric function is defined by the series

${}_{p}\digamma_{q}\left({\genfrac{}{}{0.0pt}{}{a_{1},...,a_{p}}{c_{1},...,c_{q}}}\big{|}z\right)=\sum_{k=0}^{+\infty}\frac{(a_{1})_{k}\cdots(a_{p})_{k}}{(c_{1})_{k}\cdots(c_{q})_{k}}\frac{z^{k}}{k!},$
(2.7)

which terminates whenever at least one of the $p$ parameters $a_{i}$ equals
$-1,-2,-3,\dots$. It converges for $|z|<\infty$ if $p\leq q$ or for $|z|<1$ if
$p=q+1$ and it diverges for all $z\neq 0$ if $p>q+1$. Special cases of this
function are the Gauss hypergeometric function
${}_{2}\digamma_{1}({\genfrac{}{}{0.0pt}{}{a,b}{c}}|z)$, the confluent
hypergeometric function ${}_{1}\digamma_{1}({\genfrac{}{}{0.0pt}{}{a}{c}}|z)$
and the binomial series ${}_{1}\digamma_{0}({\genfrac{}{}{0.0pt}{}{a}{-}}|z)$.
The later one reduces to $(1-z)^{-a}$ if $|z|<1$.
6. For $a\in\mathbb{R}$, the series

$I_{a}(z)=\sum_{k=0}^{+\infty}\frac{\left(\frac{1}{2}z\right)^{a+2k}}{k!\Gamma(a+k+1)}\text{,
}\qquad z\in\mathbb{C}$ (2.8)

is called the modified Bessel function of the first kind and order $a$.
7. The Jacobi polynomial of parameters $a$ and $b$ is defined by

$P_{n}^{(a,b)}(x)=2^{-n}\sum_{k=0}^{n}\binom{n+a}{k}\binom{n+b}{n-k}(x+1)^{k}(x-1)^{n-k},\qquad
a,b>-1$ (2.9)

and can be expressed in terms of the terminating ${}_{2}\digamma_{1}$-sum as

$P_{n}^{(a,b)}(x)=\frac{\Gamma(b+n+1)}{n!\Gamma(b+1)}\left(\frac{x-1}{2}\right)^{n}{}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{-n,-(a+n)}{b+1}}\bigg{|}\frac{x+1}{x-1}\right).$
(2.10)

8. The Laguerre polynomial of parameter $a$ is defined by

$L_{n}^{\left(a\right)}\left(x\right)=\sum_{k=0}^{n}(-1)^{k}\binom{n+a}{n-k}\frac{x^{k}}{k!},\qquad
a>-1$ (2.11)

and can be expressed in terms of the terminating ${}_{1}\digamma_{1}$-sum as

${}_{1}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{-n}{a+1}}\Big{|}x\right)=\frac{n!}{(a+1)_{n}}L_{n}^{(a)}(x).$
(2.12)

9. For $a_{1},\dots,a_{p},b_{1},\dots,b_{q},c_{1},\dots,c_{k}\in\mathbb{C}$ and $\alpha_{1},\dots,\alpha_{l},\beta_{1},\dots,\beta_{m},\gamma_{1},\dots,\gamma_{n}\in\mathbb{C}\setminus\mathbb{Z}_{-}$, the Kampé de Fériet function in two variables is defined by the series

$\digamma_{l:m:n}^{p:q:k}\left({\genfrac{}{}{0.0pt}{}{a_{1}...a_{p};b_{1}...b_{q};c_{1}...c_{k}}{\alpha_{1}...\alpha_{l};\beta_{1}...\beta_{m};\gamma_{1}...\gamma_{n}}}\bigg{|}x,y\right)=\sum_{r,s=0}^{+\infty}\frac{\displaystyle\prod_{i=1}^{p}(a_{i})_{r+s}\prod_{i=1}^{q}(b_{i})_{r}\prod_{i=1}^{k}(c_{i})_{s}}{\displaystyle\prod_{i=1}^{l}(\alpha_{i})_{r+s}\prod_{i=1}^{m}(\beta_{i})_{r}\prod_{i=1}^{n}(\gamma_{i})_{s}}\frac{x^{r}}{r!}\frac{y^{s}}{s!}$
(2.13)

where for convergence, $p+q<l+m+1,\quad p+k<l+n+1,\quad|x|<\infty$ and
$|y|<\infty$. A special case is the Humbert series

$\Phi_{1}\left({\genfrac{}{}{0.0pt}{}{a,b}{c}}\Big{|}w,z\right)=\digamma_{1:0:0}^{1:1:0}\left({\genfrac{}{}{0.0pt}{}{a;b;-}{c;-;-}}\bigg{|}w,z\right)=\sum_{k,l=0}^{+\infty}\frac{(a)_{k+l}(b)_{k}}{(c)_{k+l}}\frac{w^{k}z^{l}}{k!l!};\qquad|w|<1,\,|z|<\infty$
(2.14)

wich reduces to

$\Phi_{1}\left({\genfrac{}{}{0.0pt}{}{a,b}{c}}\Big{|}w,0\right)={}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{a,b}{c}}\Big{|}w\right)$
(2.15)

for $z=0$.

##  3\. Generalized Coherent states (GCS)

###  3.1. GCS labeled by elements of the affine group

We recall that the affine group is the set
$\mathbf{G}=\mathbb{R}\times\mathbb{R}^{+}$, endowed with group law
$\left(x,y\right).\left(x^{\prime},y^{\prime}\right)=\left(x+yx^{\prime},yy^{\prime}\right)$.
$\mathbf{G}$ is a locally compact group with the left Haar measure
$d\mu\left(x,y\right)=y^{-2}dxdy$. We shall consider one of the two
inequivalent infinite dimensional irreducible unitary representations of the
affine group $\mathbf{G}$, denoted $\pi_{+}$, realized on the Hilbert space
$\mathcal{H}:=$ $L^{2}\left(\mathbb{R}^{+},\xi^{-1}d\xi\right)$ as

$\pi_{+}\left(x,y\right)\left[\varphi\right]\left(\xi\right):=e^{\frac{i}{2}x\xi}\varphi\left(y\xi\right),\qquad\varphi\in\mathcal{\
H},\quad\xi\in\mathbb{R}^{+}\text{.}$ (3.1)

This representation is square integrable since it is easy to find a vector
$\phi_{0}\in\mathcal{H}$ such that the function
$\left(x,y\right)\mapsto\left\langle\pi_{+}\left(x,y\right)\left[\phi_{0}\right],\phi_{0}\right\rangle_{\mathcal{H}}$
belongs to $L^{2}\left(\mathbf{G},d\mu\right)$. This condition can also be
expressed by saying that the self-adjoint operator
$\delta:\mathcal{H\rightarrow H}$ defined as
$\delta\left[\varphi\right](\xi)=\xi^{-\frac{1}{2}}\varphi\left(\xi\right)$
gives

$\int\limits_{\mathbf{G}}\left\langle\varphi_{1},\pi_{+}\left(x,y\right)\left[\psi_{1}\right]\right\rangle\left\langle\pi_{+}\left(x,y\right)\left[\varphi_{2}\right],\psi_{2}\right\rangle
d\mu\left(x,y\right)=\left\langle\varphi_{1},\varphi_{2}\right\rangle\left\langle\delta^{\frac{1}{2}}\left[\varphi_{1}\right],\delta^{\frac{1}{2}}\left[\varphi_{2}\right]\right\rangle$
(3.2)

for all $\psi_{1},\psi_{2},\varphi_{1},\varphi_{2}\in\mathcal{H}$. The
operator $\delta$ is unbounded because $\mathbf{G}$ is not unimodular [12].

Now, as in [13], for $B>0$ and $m=0,1,\dots,\lfloor B-1/2\rfloor$ where
$\lfloor a\rfloor$ denotes the greatest integer not exceeding $a$, we consider
a set of CS labeled by elements $(x,y)\in\mathbf{G}$, which were obtained by
acting, via the representation operator $\pi_{+}\left(x,y\right)$, on the
admissible vector

$\phi_{B,m}\left(\xi\right):=\left(\frac{\Gamma\left(2B-m\right)}{m!}\right)^{-\frac{1}{2}}\xi^{B-m}e^{-\frac{1}{2}\xi}L_{m}^{\left(2(B-m)-1\right)}\left(\xi\right)$
(3.3)

where $L_{n}^{\alpha}(.)$ denotes the Laguerre polynomial defined in Eq.
(2.11). Precisely,

$\left|\tau_{(x,y),B,m}\right\rangle:=\pi_{+}\left(x,y\right)\left[\phi_{B,m}\right]$
(3.4)

and satisfy the resolution of the identity operator

$\mathbf{1}_{\mathcal{H}}=c_{B,m}\int\limits_{\mathbf{G}}d\mu\left(x,y\right)\left|\tau_{(x,y),B,m}\right\rangle\left\langle\tau_{(x,y),B,m}\right|$
(3.5)

where $c_{B,m}:=2\left(B-m\right)-1$ and the Dirac’s bra-ket notation
$|\Phi\rangle\langle\Phi|$ means the rank-one operator
$\phi\longmapsto\langle\Phi,\phi\rangle_{\mathcal{H}}.\Phi$ with
$\Phi,\phi\in\mathcal{H}$. In the $\xi$-coordinate, wavefunctions of the
states in Eq. (3.4) read

$\left\langle\xi\right|\tau_{(x,y),B,m}\rangle=\left(\frac{\Gamma\left(2B-m\right)}{m!}\right)^{-\frac{1}{2}}\left(\xi
y\right)^{B-m}e^{-\frac{1}{2}\xi\left(y-ix\right)}L_{m}^{\left(c_{B,m}\right)}\left(\xi
y\right)\text{, }\qquad\xi>0.$ (3.6)

For $m=0$, the states $\tau_{(x,y),B,0}$ coincide with the well known affine
coherent states [14]. These affine states are closely related to analytic
wavelets [15].

To describe the connection of these GCS with hyperbolic Landau levels we may
first identify the affine group with the Poincaré upper half-plane
$\mathbb{H}^{2}\equiv$ $\mathbf{G}$ . Then, with vectors in Eq. (3.4) one can
associate, as usual ([16, p.188]), the CS transform
$\mathcal{B}_{m}:\mathcal{H}\rightarrow
L^{2}\left(\mathbb{H}^{2},d\mu_{B}\right)$ defined by [13]:

$\mathcal{B}_{m}[\phi]\left(x,y\right)=\left(c_{B,m}\right)^{\frac{1}{2}}\int\limits_{0}^{+\infty}\overline{\left\langle\xi\right|\tau_{(x,y),B,m}\rangle}\phi(\xi)\xi^{-1}d\xi$
(3.7)

whose range is the eigenspace of the Schrödinger operator (in suitable units
and up to an additive constant) describing the dynamics of a charged particle
moving on $\mathbb{H}^{2}$ under the action of a magnetic field of strength
proportional to $B$

$\Delta_{B}=y^{2}\left(\partial_{x}^{2}+\partial_{y}^{2}\right)-2iBy\partial_{x},$
(3.8)

associated with the eigenvalue $\epsilon_{m}^{B}=(B-m)\left(1-B+m\right),$
$m=0,1,...,\left\lfloor B-\frac{1}{2}\right\rfloor$. That is,
$\mathcal{B}_{m}[\mathcal{H}]\equiv\left\\{\Phi\in
L^{2}\left(\mathbb{H}^{2},d\mu\right),\quad\Delta_{B}f=\epsilon_{m}^{B}f\right\\}$.
The operator $\Delta_{B}$ is also known as $B$-weight Maass operator [18, 19,
17, 20] which is an elliptic densely defined operator on the Hilbert space
$L^{2}(\mathbb{H}^{2},d\mu)$, with a unique self-adjoint realization also
denoted by $\Delta_{B}$. Its spectrum consists of two parts: a continuous part
$\left[1/4,+\infty\right[$, corresponding to scattering states and the finite
number of eigenvalues $\epsilon_{m}^{B}$ each one with infinite degeneracy
called hyperbolic Landau levels [13]. Finally, the reproducing kernel of the
Hilbert space $\mathcal{B}_{m}[\mathcal{H}]$ can be obtained from the
overlapping function
$\langle\tau_{w,B,m},\tau_{\zeta,B,m}\rangle_{\mathcal{H}}$ between two CS as

$K_{m}^{B}\left(w,\zeta\right)=\alpha_{B,m}\left(\frac{\left|w-\bar{\zeta}\right|^{2}}{4\Im
w\Im\zeta}\right)^{-B+m}\left(\frac{\zeta-\bar{w}}{w-\bar{\zeta}}\right)^{B}{}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{-m,-m-2B}{2(B-m)}}\bigg{|}\frac{4\Im
w\Im\zeta}{\left|w-\bar{\zeta}\right|^{2}}\right),$ (3.9)

$w,\zeta\in\mathbb{H}^{2}$ where
$\alpha_{B,m}:=\frac{\left(-1\right)^{m}\Gamma\left(2B-m\right)}{m!\Gamma\left(2B-2m\right)}$
and ${}_{2}\digamma_{1}$ being the Gauss hypergeometric function.

The finite number of the levels of infinite degeneracy increases with the
strength of the magnetic field, a property. In [21] the first bounds on the
size of the fundamental region, which correspond to a ‘cell’ (Pauli exclusion
principle allows only one electron per cell), have been obtained. Note that
[22] improves this estimate, so that, given a bounded region
$\Delta\subset\mathbb{H}^{2}$ in a hyperbolic Landau level, assuming the Pauli
exclusion principle, it is possible to use these discrete states to count the
number of particles distributed on such $\Delta$.

###  3.2. GCS labeled by points of the disk

We can also write a version of the GCS in Eq. (3.4) as states labeled by
points $z$ of the unit disk $\mathbb{D}=\left\\{z\in\mathbb{C},\text{
}\left|z\right|<1\right\\}$ by using the inverse Cayley transform
$\mathcal{C}^{-1}:\mathbb{D}\rightarrow\mathbf{G}$ given by

$\mathcal{C}^{-1}\left(z\right)=\left(-2\frac{\Im
z}{\left|1-z\right|^{2}},\frac{1-\left|z\right|^{2}}{\left|1-z\right|^{2}}\right).$
(3.10)

We precisely define GCS in $L^{2}(\mathbb{R}_{+},\xi^{-1}d\xi)$ by setting

$\kappa_{z,B,m}:=\left(\frac{1-\bar{z}}{1-z}\right)^{B}\pi_{+}\left(\mathcal{C}^{-1}\left(z\right)\right)\left[\phi_{B,m}\right].$
(3.11)

Direct calculations lead to their wave functions in $\xi$-coordinates as

$\left\langle\xi\right|\kappa_{z,B,m}\rangle=\sqrt{\frac{m!}{\Gamma\left(2B-m\right)}}\frac{\left|1-z\right|^{2m}}{\left(1-z\right)^{2B}}\left(\left(1-z\bar{z}\right)\xi\right)^{B-m}\exp\left(-\frac{\xi}{2}\frac{1+z}{1-z}\right)L_{m}^{\left(c_{B,m}\right)}\left(\xi\frac{1-z\bar{z}}{\left|1-z\right|^{2}}\right).$
(3.12)

By Eq. (3.12) we recover the CS in ([23, Eq. (4.4)]) (up to
$\left(-1\right)^{m}$) where $\nu$ should be taken as our $B$ and we may view
Eq. (3.11) as the analog of “displacing the vacuum”
$\left(D(z)\left|0\right\rangle\right)$ way in the construction of canonical
CS for the harmonic oscillator. When $m=0,$ the wavefunction of GCS in Eq.
(3.12) reduces to

$\left\langle\xi\right|\kappa_{z,B,0}\rangle=\frac{1}{\sqrt{\Gamma\left(2B\right)}}\left(\frac{\left(1-z\bar{z}\right)\xi}{\left(1-z\right)^{2}}\right)^{B}\exp\left(-\frac{\xi}{2}\frac{1+z}{1-z}\right),\qquad\xi>0$
(3.13)

and coincides with those constructed by Molanar et al ([24, Eq.(2.2)]) for the
Morse potential by an algebraic way based on supersymetry and shape invariance
properties where the shape parameter may be taken as our $B>0$. The GCS in Eq.
(3.13) where first introduced by Nieto et al [25] as generalized minimal
uncertainty states for the Hamiltonian of Morse potential [26].

Moreover, the GCS in Eq. (3.12) may slightly be modified in order to perform
them as vectors of $L^{2}(\mathbb{R}_{+},d\xi)$ labeled by points of the disk
$\mathbb{D}_{R}:=\left\\{z\in\mathbb{C},\text{ }\left|z\right|<R\right\\}$ as

$\left\langle\xi\right|\widetilde{\kappa}_{z,B,R,m}\rangle:=\sqrt{\frac{2}{\xi}}\,\langle\xi^{2}|\kappa_{\frac{z}{R},BR^{2},m}\rangle$
(3.14)

for every fixed $m=0,1,2,\dots,\lfloor BR^{2}-\frac{1}{2}\rfloor$ provided
that $2BR^{2}>1$. These new states obey the normalization condition
$\langle\widetilde{\kappa}_{z,B,R,m},\widetilde{\kappa}_{z,B,R,m}\rangle_{L^{2}(\mathbb{R}_{+},d\xi)}=1$
and satisfy the resolution of the identity operator as

$1_{L^{2}(\mathbb{R}_{+},d\xi)}=\int\limits_{\mathbb{D}_{R}}\left|\widetilde{\kappa}_{z,B,R,m}\right\rangle\left\langle\widetilde{\kappa}_{z,B,R,m}\right|d\mu_{B,R,m}(z),$
(3.15)

with respect to the measure

$d\mu_{B,R,m}(z):=\frac{(2(BR^{2}-m)-1)}{\pi
R^{2}\left(1-\frac{z\bar{z}}{R^{2}}\right)^{2}}d\mu(z),$ (3.16)

$d\mu(z)$ being the Lebesgue measure on the unit disk $\mathbb{D}$. Recalling
that the GCS $\left|\widetilde{\kappa}_{z,B,R,m}\right\rangle$ originate from
Eq. (3.11) according to a group theoretical approach [27], it becomes natural
to seek for a number states expansion [28] for them. Indeed, we can show that
they decompose as

$\left|\widetilde{\kappa}_{z,B,R,m}\right\rangle=\left(\mathcal{N}_{B,R,m}(z)\right)^{-\frac{1}{2}}\sum_{j=0}^{+\infty}\gamma_{j}^{B,R,m}(z)\left|j\right\rangle_{B,R,m}$
(3.17)

where

$\mathcal{N}_{B,R,m}\left(z\right):=\pi^{-1}(2(BR^{2}-m)-1)\left(1-z\bar{z}R^{-2}\right)^{-2BR^{2}},$
(3.18)

and the wavefunctions of the number states $\left|j\right\rangle_{B,R,m}$ are
given by

$\langle\xi|j\rangle_{B,R,m}:=\left(\frac{2j!}{\Gamma(2(BR^{2}-m)+j)}\right)^{\frac{1}{2}}\xi^{2(BR^{2}-m)-\frac{1}{2}}e^{-\frac{\xi^{2}}{2}}L_{j}^{(2(BR^{2}-m)-1)}(\xi^{2}),\qquad\xi\in\mathbb{R}_{+},\text{
}$ (3.19)

form an orthonormal basis in $L^{2}(\mathbb{R}_{+},d\xi)$. The coefficients

$\begin{split}z\mapsto\gamma_{j}^{B,R,m}\left(z\right):=&\,(-1)^{m}\left(\frac{(2(BR^{2}-m)-1)}{\pi}\frac{(m\land
j)!\Gamma(2(BR^{2}-m)+m\lor j)}{(m\lor j)!\Gamma(2(BR^{2}-m)+m\land
j)}\right)^{\frac{1}{2}}\\\ &\frac{(-1)^{m\land
j}}{\left(1-\frac{z\bar{z}}{R^{2}}\right)^{m}}\left(\frac{z\bar{z}}{R^{2}}\right)^{\frac{1}{2}|m-j|}e^{-i(m-j)\arg
z}P_{m\land
j}^{(|m-j|,2(BR^{2}-m)-1)}\left(1-\frac{2z\bar{z}}{R^{2}}\right)\end{split}$
(3.20)

are in fact orthonormalized functions of the Hilbert space
$L^{2}\left(\mathbb{D}_{R},R^{-2}\left(1-\frac{z\bar{z}}{R^{2}}\right)^{2BR^{2}-2}d\mu(z)\right)$.
Here $m\wedge j:=\min\\{m,j\\}$ and $m\vee j:=\max\\{m,j\\}$. The key
ingredient in proving Eq. (3.17) is the use of the generating formula ([10,
p.137]):

$\sum_{n=0}^{+\infty}\lambda^{n}{}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{-n,b}{1+a}}\Big{|}t\right)L_{n}^{(a)}(x)=\frac{(1-\lambda)^{b-1-a}}{(1-\lambda+t\lambda)^{b}}\exp\left(\frac{-x\lambda}{1-\lambda}\right){}_{1}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{b}{1+a}}\Big{|}\frac{\lambda
xt}{(1-\lambda)(1-\lambda+t\lambda)}\right)$ (3.21)

where the parameters $a\geq 0$ and $|\lambda|<\min\\{1,|1-t|^{-1}\\}$.

We may note in passing that for $R=1$, the GCS
$\left|\widetilde{\kappa}_{z,B,1,m}\right\rangle$ in Eq. (3.17) can be
considered as a class of generalized negative binomial states [29] and the
above functions $\gamma_{j}^{B,1,m}(z)$ constitute, in fact, an orthonormal
basis for the eigenspace

$\mathcal{E}_{B,m}\left(\mathbb{D}\right)=\left\\{f\in
L^{2}\left(\mathbb{D},\left(1-z\bar{z}\right)^{2B-2}d\mu(z)\right),\quad\widetilde{\Delta}_{B}f=\sigma_{B,m}f\right\\}$
(3.22)

of the $B$-weight Maass-Laplacian

$\widetilde{\Delta}_{B}=-4\left(1-z\overline{z}\right)\left(\left(1-z\overline{z}\right)\frac{\partial^{2}}{\partial
z\partial\overline{z}}-2B\overline{z}\frac{\partial}{\partial\overline{z}}\right)$
(3.23)

associated with eigenvalues $\sigma_{B,m}=4m\left(2B-m-1\right)$.
Additionally, the mapping

$W_{B,m}\left[\varphi\right]\left(z\right)=\left(\mathcal{N}_{B,1,m}\left(z\right)\right)^{\frac{1}{2}}\langle\varphi\left|\widetilde{\kappa}_{z,B,1,m}\right\rangle_{L^{2}(\mathbb{R}_{+},d\xi)}$
(3.24)

is a unitary isomorphism from $L^{2}(\mathbb{R}_{+},d\xi)$ onto the Hilbert
space $\mathcal{E}_{B,m}\left(\mathbb{D}\right)$ whose reproducing kernel can
be obtained from the overlapping function
$\langle\widetilde{\kappa}_{z,B,1,m},\,\widetilde{\kappa}_{w,B,1,m}\rangle_{L^{2}(\mathbb{R}_{+},d\xi)}$
between two GCS. Explicitly,

$\begin{split}K_{B,m}\left(z,w\right)=&\,\pi^{-1}(2\left(B-m\right)-1)\left(1-z\overline{w}\right)^{-2B}\left(\frac{\left|1-z\overline{w}\right|^{2}}{\left(1-z\overline{z}\right)\left(1-w\overline{w}\right)}\right)^{m}\\\
&\times
P_{m}^{\left(0,2\left(B-m\right)-1\right)}\left(2\frac{\left(1-z\overline{z}\right)\left(1-w\overline{w}\right)}{\left|1-z\overline{w}\right|^{2}}-1\right)\end{split}$
(3.25)

for $z,w\in\mathbb{D}$.

##  4\. Husimi’s Q-function for a pure state

By using Eq. (3.17) to Eq. (3.20), one can write according to Eq. (1.2), the
Husimi function of the pure state

$\widehat{\rho}_{j}:=|j\rangle_{B,R,m\text{ \ }B,R,m}\langle j|,$ (4.1)

as

$Q_{j}^{B,R,m}(z):=\langle\widetilde{\kappa}_{z,B,R,m}|\widehat{\rho}_{j}|\widetilde{\kappa}_{z,B,R,m}\rangle,\qquad
z\in\mathbb{D}_{R}.$ (4.2)

Explicitly,

$Q_{j}^{B,R,m}(z)=\tau_{j}^{B,R,m}\left(1-\frac{z\bar{z}}{R^{2}}\right)^{2(BR^{2}-m)}\left(\frac{z\bar{z}}{R^{2}}\right)^{|m-j|}\left(P_{m\wedge
j}^{(|m-j|,2(BR^{2}-m)-1)}\left(1-\frac{2z\bar{z}}{R^{2}}\right)\right)^{2}$
(4.3)

where

$\tau_{j}^{B,R,m}:=\frac{(m\wedge j)!\Gamma(2(BR^{2}-m)+m\vee j)}{(m\vee
j)!\Gamma(2(BR^{2}-m)+m\wedge j)}$ (4.4)

for every fixed $m=0,1,\dots,\lfloor BR^{2}-\frac{1}{2}\rfloor$ provided that
$2BR^{2}>1$. Doing so enables us, by letting $R\to\infty$, to recover the
(radial) Husimi’s Q-function for a pure state associated with an $m$th
Euclidean Landau level ([4, p.5]):

$Q_{j}^{(m),\text{ Euclid}}(\sqrt{2B}z)=\frac{(m\wedge j)!}{(m\vee
j)!}e^{-2Bz\bar{z}}\left(2Bz\bar{z}\right)^{|m-j|}\left(L_{m\wedge
j}^{(|m-j|)}(2Bz\bar{z})\right)^{2},\;\qquad m=0,1,2,\dots,\quad
z\in\mathbb{C},$ (4.5)

see Appendix A for the proof. We also note that the characteristic function
for the random variable $X$ having $Q_{j}^{(m),\text{ Euclid}}$ as its density
function reads ([4, p.6]):

$\displaystyle\phi_{X}(u)$ $\displaystyle:=$
$\displaystyle\left(1-iu\right)^{-(m+j+1)}\sum_{k=0}^{m\land
j}\binom{m}{k}\binom{j}{k}(iu)^{2k},\qquad u\in\mathbb{R}$ $\displaystyle=$
$\displaystyle\frac{(m+j)!}{m!j!}\left(1-iu\right)^{-(m+j+1)}{}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{-m,-j}{-m-j}}\Big{|}1+u^{2}\right)$

and was involved by Shirai while discussing Ginibre-type processes [5]. Like
the Euclidean setting, one can check that the Husimi function given by Eq.
(4.3) is a probability distribution on the phase space $\mathbb{D}_{R}$. That
is,

$\int\limits_{\mathbb{D}_{R}}Q_{j}^{B,R,m}(z)d\mu_{B,R,m}(z)=1$ (4.7)

where $d\mu_{B,R,m}(z)$ is given by Eq. (3.16). Using polar coordinates
$z=re^{i\theta}$, $r\in[0,R)$, $\theta\in\left[0,2\pi\right)$ and setting

$\mathfrak{Q}_{j}^{B,R,m}(\lambda):=R^{-2}\left(2(BR^{2}-m)-1\right)\left(1-\lambda
R^{-2}\right)^{-2}{Q}_{j}^{B,R,m}(\lambda),$ (4.8)

with $\lambda=r^{2},$ then Eq. (4.7) ensures

$\int\limits_{0}^{R^{2}}\mathfrak{Q}_{j}^{B,R,m}(\lambda)d\lambda=1.$ (4.9)

For $m=0$, the density function in Eq. (4.8) reduces to

$\mathfrak{Q}_{j}^{B,R,0}(\lambda)=R^{-2}\frac{\Gamma(2BR^{2}-1+j+1)}{\Gamma(2BR^{2}-1)\Gamma(j+1)}\left(1-\frac{\lambda}{R^{2}}\right)^{(2BR^{2}-1)-1}\left(\frac{\lambda}{R^{2}}\right)^{(j+1)-1}$
(4.10)

which coincides (up to a scale factor $1/R^{2}$) with the Beta distribution
$\mathcal{B}e(j+1,2BR^{2}-1)$ whose characteristic function is known to be
given by the confluent hypergeometric series as

$\phi_{\mathcal{B}e}(u)={}_{1}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{{j+1}}{{2BR^{2}+j}}}\bigg{|}iuR^{2}\right).$
(4.11)

If for $R=1$ we denote by $\mathfrak{X}_{m}^{\left(1\right)}$ the random
variable having $\lambda\mapsto\mathfrak{Q}_{j}^{B,1,m}(\lambda)$ as its
density function then $\mathfrak{X}_{m}^{(1)}$ would play the role of the
hyperbolic analog of the continuous random variable denoted by $S_{k}^{n,b}$
in Shirai’s paper [5] and

$\Pr\left(\mathfrak{X}^{(1)}_{m}\leq
r^{2}\right)=\int\limits_{0}^{r^{2}}\mathfrak{Q}_{j}^{B,1,m}(\xi)d\xi\equiv\lambda_{j}^{m,B}(r^{2})$
(4.12)

would provide a probabilistic representation of eigenvalues
$\lambda_{j}^{m,B}(r^{2}):=||\gamma_{j}^{B,1,m}\mathbf{1}_{\mathbf{D}_{r}}||^{2}$
of the restricted operator $\left(K_{B,m}(z,w)\right)_{\mathbf{D}_{r}}$ to the
disk $\mathbf{D}_{r}\subset\mathbb{D}$ of radius $r\leq 1,$ where
$K_{B,m}\left(z,w\right)$ defined in Eq. (3.25) is the projection operator
onto the $m$th hyperbolic Landau eigenspace
$\mathcal{E}_{B,m}\left(\mathbb{D}\right)$ in Eq. (3.22). Furthermore, the
operator $\left(K_{B,m}\left(z,w\right)\right)_{\mathbf{D}_{r}}$ may be
unitarly intertwined via the Bargmann-type transform $W_{B,m}$ in Eq. (3.24)
to be acting on the quantum (or signals) Hilbert space
$L^{2}(\mathbb{R}_{+},d\xi)$ as a localization operator “à la Daubechies”[30]
of the indicator function $\mathbf{1}_{\mathbf{D}_{r}}$ of the disk
$\mathbf{D}_{r}$.

For these reasons, it would be useful to investigate properties of the random
variable $\mathfrak{X}_{m}^{(R)}$ having $\mathfrak{Q}_{j}^{B,R,m}(\lambda)$
as its density function by first writing down its characteristic function
which is defined, as usual, by the Fourier-type integral transform ([31,
p.22]):

$\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right):=\int\limits_{0}^{R^{2}}e^{iu\lambda}\mathfrak{Q}_{j}^{B,R,m}(\lambda)d\lambda,\quad
u\in\mathbb{R}.$ (4.13)

Straightforward calculations (see Appendix B) lead to the expression

$\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)=\tau_{j}^{B,R,m}(2(BR^{2}-m)-1)\sum_{k=0}^{2(m\wedge
j)}C_{j,k}^{B,R,m}\Gamma(2(BR^{2}-m)-1+k)\frac{\left(-iuR^{2}\right)^{k}}{k!}$
$\times\frac{\Gamma\left(\left|m-j\right|+k+1\right)}{\Gamma\left(2(BR^{2}-m)+\left|m-j\right|+2k\right)}._{1}\digamma_{1}\left(\begin{array}[]{c}\left|m-j\right|+k+1\\\
2(BR^{2}-m)+\left|m-j\right|+2k\end{array}\mid iuR^{2}\right)$ (4.14)

where

$C_{j,k}^{B,R,m}=\frac{(2(m\wedge j))!(|m-j|+1))_{2(m\wedge
j)}((2(BR^{2}-m)+|m-j|)_{2(m\wedge j)})^{2}}{((m\wedge
j)!)^{2}((2(BR^{2}-m)+|m-j|)_{m\wedge j})^{2}}$
$\times\frac{(-1)^{k}(2(BR^{2}-m)+|m-j|-1)_{k}(2k+2(BR^{2}-m)+|m-j|-1)}{(2(m\wedge
j)-k)!(|m-j|+1)_{k}(2(BR^{2}-m)+|m-j|-1)_{2(m\wedge j)+k+1}}$
$\times\digamma_{2:1:1}^{2:2:2}\left(\begin{array}[]{c}-2(m\wedge
j)+k,-2(BR^{2}-m)-m-j+1-k;-m,-j;-m,-j\\\ -2(m\wedge
j),-m-j;-2(BR^{2}-m)-m-j+1;-2(BR^{2}-m)-m-j+1\end{array}\bigg{|}1,1\right).$
(4.15)

Here, $\digamma_{2:2:2}^{2:1:1}$ denotes the Kampé de Fériet function of two
variables defined by Eq. (2.13). For $m=0$, Eq. (4.14) reduces to Eq. (4.11)
as expected.

Now, by having $\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)$, we
immediately derive the main statistical parameters which are the mean value
and the variance of $\mathfrak{X}_{m}^{\left(R\right)}$. Namely, we obtain the
following formulas

$\mathbb{E}(\mathfrak{X}_{m}^{\left(R\right)})=\tau_{j}^{B,R,m}R^{2}\frac{(|m-j|+1)!\Gamma(2(BR^{2}-m))}{\Gamma(2(BR^{2}-m)+|m-j|+1)}\left(C_{j,0}^{B,R,m}-\frac{(2(BR^{2}-m)-1)C_{j,1}^{B,R,m}}{(2(BR^{2}-m)+|m-j|+1)}\right)$
(4.16)

and

$\mathbb{V}ar\left(\mathfrak{X}_{m}^{\left(R\right)}\right)=R^{4}\tau_{j}^{B,R,m}\frac{(|m-j|+2)!\Gamma(2(BR^{2}-m))}{\Gamma(2(BR^{2}-m)+|m-j|+2)}$
(4.17)
$\bigg{(}C_{j,0}^{B,R,m}-\frac{2((2(BR^{2}-m)-1))C_{j,1}^{B,R,m}}{(2(BR^{2}-m)+|m-j|+2)}+\frac{(2(BR^{2}-m)-1)(2(BR^{2}-m))C_{j,2}^{B,R,m}}{(2(BR^{2}-m)+|m-j|+2)(2(BR^{2}-m)+|m-j|+3)}\bigg{)}$
$-\bigg{[}R^{2}\tau_{j}^{B,R,m}\frac{(|m-j|+1)!\Gamma(2(BR^{2}-m))}{\Gamma(2(BR^{2}-m)+|m-j|+1)}\left(C_{j,0}^{B,R,m}-\frac{(2(BR^{2}-m)-1)C_{j,1}^{B,R,m}}{(2(BR^{2}-m)+|m-j|+1)}\right)\bigg{]}^{2}$

For the proofs of Eq. (4.16) and Eq. (4.17) see Appendix D. In particular, for
$m=0$ the above formulas reduce to

$\mathbb{E}\left(\mathfrak{X}_{0}^{\left(R\right)}\right)=\frac{(j+1)R^{2}}{2BR^{2}+j},\quad\mathbb{V}ar(\mathfrak{X}_{0}^{\left(R\right)})=\frac{(j+1)(2BR^{2}-1)R^{4}}{(2BR^{2}+j)^{2}(2BR^{2}+j+1)},$
(4.18)

which are (up to a scale factor $1/R^{2}$) the mean value and variance of the
Beta distribution $\mathcal{B}e\left(j+1,2BR^{2}-1\right)$. Finally, we
establish (see Appendix C) the following limit

$\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)\rightarrow\left(1-\frac{iu}{2B}\right)^{-(m+j+1)}\sum_{k=0}^{m\land
j}\binom{m}{k}\binom{j}{k}\left(\frac{iu}{2B}\right)^{2k},\qquad u<2B$ (4.19)

as $R\rightarrow+\infty$, which turn out to be (up to a scale factor $1/2B$)
the characteristic function in the corresponding Euclidean setting in Eq. (4).

Remark 1. Note that ${Q}_{j}^{B,R,m}(z)$ vanishes at the origin $(0,0)$, the
points of the circle of radius $R$ and the points $z$ such that that
$R^{2}-2\left|z\right|^{2}$ are zeros of the Jacobi polynomials $P_{m\wedge
j}^{(|m-j|,2(B-m)-1)}(\cdot)$. Denoting by $x_{i}^{(m\wedge j)},\quad 1\leq
i\leq m\wedge j$ the ordered consecutive zeros of this polynomial, the zeros
of ${Q}_{j}^{B,R,m}(z)$ are then located on the concentric circles of radius
$r_{i}=\sqrt{\left(R^{2}-x_{i}^{(m\wedge j)}\right)/2}$. A discussion on the
number of zeros of the Husimi density and their location in connection with
properties of harmonic and anaharmonic oscillators can be found in [32],
particularly [4] in the Euclidean setting.

##  5\. Husimi’s Q-function for mixed states

###  5.1. A Hamiltonian operator for the GCS

Recall that the functions $\left\langle\xi|j\right\rangle_{B,R,m}$ in the
expansion (3.17) of the GCS $\left|\widetilde{\kappa}_{z,B,R,m}\right\rangle$
were given by the normalized Laguerre functions (3.19) in
$L^{2}(\mathbb{R}_{+},d\xi)$. Moreover, one may use the Rodriguez formula for
the Laguerre polynomial to show that these functions can also be obtained by a
$j$-fold application of the first order differential operator

$A_{\tau,m}=\frac{1}{\sqrt{2}}\left(-\frac{\tau-m-1/2}{\xi}+\xi+\frac{d}{d\xi}\right),\qquad\tau=BR^{2},$
(5.1)

on the ground state

$\langle\xi|0\rangle_{B,R,m}:=\left(\frac{2}{\Gamma(2(BR^{2}-m))}\right)^{\frac{1}{2}}\xi^{2(BR^{2}-m)-\frac{1}{2}}e^{-\frac{\xi^{2}}{2}},\qquad\xi\in\mathbb{R}_{+},$
(5.2)

as follows

$|j\rangle_{B,R,m}=\left(A_{\tau,m}\right)^{j}|0\rangle_{B,R,m}.$ (5.3)

The formal adjoint operator of $A_{\tau,m}$ in $L^{2}(\mathbb{R}_{+},d\xi)$ is
given by

$A_{\tau,m}^{\ast}=\frac{1}{\sqrt{2}}\left(-\frac{\tau-m-1/2}{\xi}+\xi-\frac{d}{d\xi}\right).$
(5.4)

Therefore we may associate to GCS
$\left|\widetilde{\kappa}_{z,B,R,m}\right\rangle$ the Hamiltonian operator

$H_{\tau,m}=A_{\tau,m}^{\ast}A_{\tau,m}+2\left(\tau-m\right).$ (5.5)

Explicitly,

$H_{\tau,m}:=-\frac{1}{2}\frac{d^{2}}{d\xi^{2}}+\frac{\left(2(\tau-m)-1\right)^{2}-1/4}{2\xi^{2}}+\frac{1}{2}\xi^{2}$
(5.6)

which appears in the literature under many names such as isotonic oscillator
[33, 34], Gol’dman-Krivchenkov Hamiltonian [35], pseudoharmonic oscillator
[36] or Laguerre operator [37], see also ([38, p.171]). The spectrum of
$H_{\tau,m}$ in $L^{2}(\mathbb{R}_{+},d\xi)$ is purely discrete and given by
the eigenvalues

$\eta_{j}=j+4(\tau-m)-1.$ (5.7)

The latter ones, together with their associated eigenfunctions
$|j\rangle_{B,R,m}$ can be used to define the heat semigroup associated with
$H_{\tau,m}$ through the discret spectral resolution

$e^{-\beta
H_{\tau,m}}\left[f\right]:=\sum_{j=0}^{+\infty}e^{-\beta\eta_{j}}\left\langle
f\right|j\rangle_{B,R,m}.\left|j\right\rangle_{B,R,m},\;\;f\in
L^{2}(\mathbb{R}_{+}).$ (5.8)

It is also well known that by using the Hille-Hardy formula ([39, p.242]),
this semigroup has the integral representation

$e^{-\beta
H_{\tau,m}}\left[f\right]\left(r\right)=\int\limits_{0}^{+\infty}W_{\beta}^{B,m}\left(r,\rho\right)f\left(\rho\right)d\rho$
(5.9)

where

$W_{\beta}^{B,m}\left(r,\rho\right)=\frac{2\sqrt{r\rho}e^{-\beta}}{(1-e^{-2\beta})}\exp\left(-\frac{1}{2}\left(r^{2}+\rho^{2}\right)\frac{1+e^{-2\beta}}{1-e^{-2\beta}}\right)I_{\left(2(BR^{2}-m)-1\right)}\left(\frac{2r\rho
e^{-\beta}}{1-e^{-2\beta}}\right)\text{.}$ (5.10)

Here $I_{a}\left(\cdot\right)$ denotes the modified Bessel function defined in
Eq. (2.8).

###  5.2. Husimi’s function

The standard statistical mechanics starts using the Gibb’s canonical
distribution, whose thermal density operator is represented by

$\hat{\rho}_{\beta}=\frac{1}{Z}e^{-\beta\hat{H}}$ (5.11)

where $Z=\mathrm{Tr}\left(e^{-\beta\hat{H}}\right)$ is the partition function,
$\hat{H}$ is the Hamiltonian of the system, $\beta=1/(kT)$ the inverse
temperature $T$ and $k$ the Boltzmann constant ($k=1$). In our context,
$\hat{H}\equiv H_{\tau,m}$ the isotonic oscillator and $\hat{\rho}_{\beta}$ is
the associated normalized heat operator. The Husimi distribution is defined as
the expectation value of the density operator $\hat{\rho}_{\beta}$ in the set
of states $|\widetilde{\kappa}_{z,B,R,m}\rangle$ as

$Q_{\beta}^{B,R,m}(z):=\langle\widetilde{\kappa}_{z,B,R,m}|\widehat{\rho}_{\beta}|\widetilde{\kappa}_{z,B,R,m}\rangle,\qquad
z\in\mathbb{D}_{R}.$ (5.12)

By using Eq. (5.8), the R.H.S of Eq. (5.12) takes the form

$Q_{\beta}^{B,R,m}(z)=\frac{1}{Z}\sum_{j=0}^{+\infty}e^{-\beta\eta_{j}}\left[Q_{j}^{B,R,m}(z)\right]$
(5.13)

where $Q_{j}^{B,R,m}$ is the Husimi function for the pure state
$|j\rangle\langle j|$ given by Eq. (4.3), where
$|j\rangle\equiv|j\rangle_{B,R,m}$. Replacing the $\eta_{j}$ by their
expressions $j+4(BR^{2}-m)-1$ and using direct calculations, we obtain the
partition function as

$Z=\frac{1}{1-e^{-\beta}}e^{-(4(BR^{2}-m)-1)\beta}$ (5.14)

such that

$\int\limits_{\mathbb{D}_{R}}Q_{\beta}^{B,R,m}(z)d\mu_{B,R,m}\left(z\right)=1.$
(5.15)

Therefore,

$Q_{\beta}^{B,R,m}(z)=\left(1-e^{-\beta}\right)\sum_{j=0}^{+\infty}\left(e^{-\beta}\right)^{j}\left[Q_{j}^{B,R,m}(z)\right].$
(5.16)

Next, if we make the last sum look like the moment generating function of the
generalized negative binomial distribution ([40, p.6]), then we get

$\begin{split}Q_{\beta}^{B,R,m}(z)=&\left(1-e^{-\beta}\right)\left(\frac{1-\frac{z\bar{z}}{R^{2}}}{1-\frac{z\bar{z}}{R^{2}}e^{-\beta}}\right)^{2BR^{2}}\left(\frac{\left(\frac{z\bar{z}}{R^{2}}-e^{-\beta}\right)\left(1-\frac{z\bar{z}}{R^{2}}e^{-\beta}\right)}{\left(1-\frac{z\bar{z}}{R^{2}}\right)^{2}}\right)^{m}\\\
&\times
P_{m}^{(2(BR^{2}-m)-1,0)}\left(1+\frac{2e^{-\beta}\left(1-\frac{z\bar{z}}{R^{2}}\right)^{2}}{\left(\frac{z\bar{z}}{R^{2}}-e^{-\beta}\right)\left(1-\frac{z\bar{z}}{R^{2}}e^{-\beta}\right)}\right).\end{split}$
(5.17)

By letting $R\rightarrow\infty$ we prove (see Appendix E) that
$Q_{\beta}^{B,R,m}(z)\rightarrow q_{m}(|\sqrt{2B}z|^{2},\beta)$ where

$q_{m}(|\sqrt{2B}z|^{2},\beta)=\left(1-e^{-\beta}\right)\exp\left(-2|\sqrt{2B}z|^{2}(1-e^{-\beta})\right)e^{-\beta
m}L_{m}^{(0)}\left(-4|\sqrt{2B}z|^{2}\sinh^{2}\frac{\beta}{2}\right)$ (5.18)

is the Husimi function for mixed states of the flat case ([4, p.7]) as
expected. Now, setting

$\mathfrak{Q}_{\beta}^{B,R,m}(\lambda):=R^{-2}\left(2(BR^{2}-m)-1\right)\left(1-\lambda
R^{-2}\right)^{-2}{Q}_{\beta}^{B,R,m}(\lambda),$ (5.19)

with $\lambda=z\bar{z}$, Eq. (5.15) gives

$\int\limits_{0}^{R^{2}}\mathfrak{Q}_{\beta}^{B,R,m}(\lambda)d\lambda=1.$
(5.20)

Then, we denote by $\mathfrak{Y}_{m}^{\left(R\right)}$ the random variable
having $\lambda\mapsto\mathfrak{Q}_{\beta}^{B,R,m}(\lambda)$ as its density
function for which it would be useful to investigate some properties by
starting by its characteristic function which is defined, as usual, by

$\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u):=\int\limits_{0}^{R^{2}}e^{iu\lambda}\mathfrak{Q}_{\beta}^{B,R,m}(\lambda)d\lambda,\qquad
u\in\mathbb{R}.$ (5.21)

Straightforward calculations (see Appendix F) lead to the expression

$\begin{split}\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)=&(2(BR^{2}-m)-1)(1-e^{-\beta})e^{-m\beta}\sum_{k=0}^{m}\binom{2BR^{2}-m-1}{k}\binom{m}{k}\left(\frac{(1-e^{-\beta})^{2}}{e^{-\beta}}\right)^{k}\\\
&\mathcal{B}(k+1,2BR^{2}-2k-1)\,\Phi_{1}\left({\genfrac{}{}{0.0pt}{}{k+1,2BR^{2}}{2BR^{2}-k}}\Big{|}e^{-\beta},-iuR^{2}\right)\end{split}$
(5.22)

where $\mathcal{B}(\cdot,\cdot)$ is the beta function and $\Phi_{1}$ is the
Humbert series defined in Eq. (2.14).
Now, by having $\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}\left(u\right)$, we
immediately derive (see Appendix G for the proof) the mean value and the
variance of $\mathfrak{Y}_{m}^{\left(R\right)}$ that are given respectively by

$\begin{split}\mathbb{E}\left(\mathfrak{Y}_{m}^{\left(R\right)}\right)=&\;R^{2}(2(BR^{2}-m)-1)(e^{-\beta}-1)e^{-m\beta}\sum_{k=0}^{m}\binom{2BR^{2}-m-1}{k}\binom{m}{k}\left(\frac{(1-e^{-\beta})^{2}}{e^{-\beta}}\right)^{k}\\\
&\mathcal{B}(k+2,2BR^{2}-2k-1)\,{}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{k+2,2BR^{2}}{2BR^{2}-k+1}}\Big{|}e^{-\beta}\right)\end{split}$
(5.23)

and

$\mathbb{V}ar(\mathfrak{Y}_{m}^{\left(R\right)})=R^{4}(2(BR^{2}-m)-1)(1-e^{-\beta})e^{-m\beta}$
$\begin{split}\times&\sum_{k=0}^{m}\binom{2BR^{2}-m-1}{k}\binom{m}{k}\left(\frac{(1-e^{-\beta})^{2}}{e^{-\beta}}\right)^{k}\mathcal{B}(k+3,2BR^{2}-2k-1){}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{k+3,2BR^{2}}{2BR^{2}-k+2}}\Big{|}e^{-\beta}\right)\\\
-&\Bigg{[}R^{2}(2(BR^{2}-m)-1)(1-e^{-\beta})e^{-m\beta}\sum_{k=0}^{m}\binom{2BR^{2}-m-1}{k}\binom{m}{k}\left(\frac{(1-e^{-\beta})^{2}}{e^{-\beta}}\right)^{k}\\\
\times&\mathcal{B}(k+2,2BR^{2}-2k-1){}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{k+2,2BR^{2}}{2BR^{2}-k+1}}\Big{|}e^{-\beta}\right)\Bigg{]}^{2}.\end{split}$
(5.24)

We also establish (see Appendix H) the following limit

$\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)\to\frac{(1-e^{-\beta})}{1-e^{-\beta}-\frac{iu}{2B}}\left(\frac{1-e^{-\beta}-\frac{iu}{2B}e^{-\beta}}{1-e^{-\beta}-\frac{iu}{2B}}\right)^{m},\qquad
u<2B$ (5.25)

as $R\to+\infty$, which turn out to be (up to a scale factor $\frac{1}{2B}$)
the characteristic function in the Euclidean setting [4, p.8] as expected.

Remark 2. For $q_{m}(|\sqrt{2B}z|^{2},\beta)$ in Eq. (5.18), we may set
$\lambda=2Bz\overline{z}$ and use a duality to define a non negative integer-
valued random variable $\mathcal{Y}_{\lambda,\beta}$ by

$\Pr\left(\mathcal{Y}_{\lambda,\beta}\right)=q_{m}\left(\lambda,\beta\right),\qquad
m=0,1,2,...,$ (5.26)

to recover the Laguerre probability distribution with parameters
$\left(\lambda,\beta\right)$ which can be presented as follows. Let
$T=1/k\beta$ represents in suitable units $(k=1)$, the absolute temperature
and $N_{T}:=\left(e^{\frac{1}{T}}-1\right)^{-1}$ the average number of photons
associated with the thermal noisy state, then the photon count probability
distribution for a mixed light reads ([41]):

$q_{m}\left(\lambda,\beta\right)=\frac{(N_{T})^{m}}{(1+N_{T})^{m+1}}\exp\left(-\frac{{\lambda}}{1+N_{T}}\right)L_{m}^{(0)}\left(-\frac{\lambda}{N_{T}(1+N_{T})}\right).$
(5.27)

Finally, we can use the obtained Q-function $Q_{\beta}^{B,R,m}$ to get a lower
bound for the thermodynamical potential associated with the Hamiltonian
$H_{\tau,m}$, $\tau=BR^{2}$. This potential reads

$\Theta_{\beta}^{\tau,\eta}:=\frac{-1}{\beta}\mathrm{Tr}\left(\log\left(1+e^{-\beta(H_{\tau,m}-\eta)}\right)\right)$
(5.28)

where $\eta$ is the chemical potential. Putting $\epsilon=e^{\beta\eta}$, the
form of the potential in Eq. (5.28) suggests us to consider the function

$\zeta\mapsto\phi_{\epsilon}(\zeta)=-\log(1+\epsilon\zeta).$ (5.29)

So that we can rewrite Eq. (5.28) as

$\beta\Theta_{\beta}^{\tau,\eta}=\mathrm{Tr}\left(\phi_{\epsilon}\left(e^{-\beta
H_{\tau,m}}\right)\right).$ (5.30)

Applying the Berezin-Lieb inequality [6] for the lower symbol
$Q_{\beta}^{B,R,m}$ of the operator $e^{-\beta H_{\tau,m}}$ as defined by Eq.
(5.12), we obtain that

$\int\limits_{\mathbb{D}_{R}}[\phi_{\epsilon}\circ
Q_{\beta}^{B,R,m}](z)d\mu_{B,R,m}(z)\leq\mathrm{Tr}(\phi_{\epsilon}(e^{-\beta
H_{\tau,m}})).$ (5.31)

Making use of Eq. (5.29) and replacing the R.H.S of Eq. (5.31) by
$\beta\Theta_{\beta}^{\tau,\eta}$ as in Eq. (5.30), we get an inequality that
holds for every $m=0,1,\dots,\left\lfloor BR^{2}-1/2\right\rfloor$. Therefore,
we consider the maximum with respect to the integer $m$ of the quantity in the
L.H.S of Eq. (5.31) as

$\max_{m\in\mathbb{Z}_{+}\cap[0,BR^{2}-1/2]}\left[\frac{1}{\beta}\int\limits_{\mathbb{D}_{R}}\log\left(\frac{1}{1+\epsilon
Q_{\beta}^{B,R,m}(z)}\right)d\mu_{B,R,m}(z)\right]\leq\Theta_{\beta}^{BR^{2},\eta}$
(5.32)

for every $\beta>0$.

##  6\. A formula for a special Kampé de Fériet function

In this section we establish summation formula for the Kampé de Fériet
function $\digamma_{2:0:0}^{1:2:2}$, wich will be limit in Eq. (4.5) (see
appendix C).

Proposition 1. Let $n\in\mathbb{N}^{*}$. For all real $x>0$ or $x<-2\sqrt{2}$,
the equality

$\sum_{k=0}^{n}\binom{n}{k}\digamma_{2:0:0}^{1:2:2}\left({\genfrac{}{}{0.0pt}{}{-n+k;a,b;c-a,c-b}{-n,c;-;-}}\bigg{|}1,1\right)x^{k}=(1+x)^{n}\left(\frac{x}{1+x}\right)^{a+b-c}{}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{a,b}{c}}\bigg{|}\frac{1+2x}{(1+x)^{2}}\right)$
(6.1)

holds true. It also holds true for $x\in\mathbb{C}\setminus\\{-1,0\\}$ if
$\max\\{a,b\\}\in\mathbb{Z}_{-}$ and $\max\\{c-a,c-b\\}\in\mathbb{Z}_{-}$.
Proof. We denote

$S_{n}:=\sum_{k=0}^{n}\binom{n}{k}\digamma_{2:0:0}^{1:2:2}\left({\genfrac{}{}{0.0pt}{}{-n+k;a,b;c-a,c-b}{-n,c;-;-}}\bigg{|}1,1\right)x^{k}.$
(6.2)

for $n\in\mathbb{N}^{*}$. Using the definition of $\digamma_{l:m:n}^{p:q:k}$
in Eq. (2.13) we obtain

$S_{n}=\sum_{s,t\geq
0}\frac{(a)_{s}(b)_{s}(c-a)_{t}(c-b)_{t}}{(-n)_{s+t}(c)_{s+t}}\frac{1}{s!t!}\sum_{k=0}^{n}\binom{n}{k}(-n+k)_{s+t}x^{k}.$
(6.3)

Since $(-n+k)_{s+t}=0$ if $s+t>n-k$, the second sum in the R.H.S of Eq. (6.3)
terminates as

$\sum_{k=0}^{n}\binom{n}{k}(-n+k)_{s+t}x^{k}=(-n)_{s+t}\sum_{k=0}^{n-s-t}\binom{n-s-t}{k}x^{k}=(-n)_{s+t}(1+x)^{n-s-t}$
(6.4)

when $(-n)_{k}=(-1)^{k}k!\binom{n}{k}$ for $0\leq k\leq n$,
$n\in\mathbb{N}^{*}$. Therefore, Eq. (6.3) takes the form

$S_{n}=(1+x)^{n}\sum_{s,t\geq
0}\frac{(a)_{s}(b)_{s}(c-a)_{t}(c-b)_{t}}{(c)_{s+t}}\frac{1}{s!t!}\left(\frac{1}{1+x}\right)^{s+t}$
(6.5)

wich also reads

$S_{n}=(1+x)^{n}\sum_{s\geq
0}\frac{(c-a)_{s}(c-b)_{s}}{(c)_{s}}{}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{a,b}{c+s}}\bigg{|}\frac{1}{1+x}\right)\frac{1}{s!}\left(\frac{1}{1+x}\right)^{s}.$
(6.6)

Using the Euler transformation ([10, p.33]):

${}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{a,b}{c}}\big{|}z\right)=(1-z)^{c-a-b}{}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{c-a,c-b}{c}}\big{|}z\right);\qquad|\arg(1-z)|<\pi,$
(6.7)

Eq. (6.6) becomes

$\begin{split}S_{n}=&\,(1+x)^{n}\left(\frac{x}{1+x}\right)^{c-a-b}\\\
&\times\sum_{s\geq
0}\frac{(c-a)_{s}(c-b)_{s}}{s!(c)_{s}}\left(\frac{x}{(1+x)^{2}}\right)^{s}{}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{c-a+s,c-b+s}{c+s}}\bigg{|}\frac{1}{1+x}\right).\end{split}$
(6.8)

Next, by applying the generating formula ([42, p.348]):

$\sum_{k\geq
0}\frac{(\bar{a})_{k}(\bar{b})_{k}}{k!(c)_{k}}t^{k}{}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{\bar{a}+k,\bar{b}+k}{c+k}}\big{|}y\right)={}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{\bar{a},\bar{b}}{c}}\big{|}t+y\right),\qquad|t|,\quad|y|,\quad|t+y|<1,$
(6.9)

for $\bar{a}=c-a$, $\bar{b}=c-b$, $t=\frac{x}{(1+x)^{2}}$ and
$y=\frac{1}{1+x}$, the sum $S_{n}$ can be written in a closed form as

$S_{n}=(1+x)^{n}\left(\frac{x}{1+x}\right)^{c-a-b}{}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{c-a,c-b}{c}}\bigg{|}\frac{1+2x}{(1+x)^{2}}\right).$
(6.10)

where the conditions $|t|,\quad|y|,\quad|t+y|<1$ translate into
$x\in]-\infty,-2\sqrt{2}[\cup]0,+\infty[$. Note that if
$\max\\{a,b\\}\in\mathbb{Z}_{-}$ and $\max\\{c-a,c-b\\}\in\mathbb{Z}_{-}$, the
convergence conditions $|t|,\quad|y|,\quad|t+y|<1$ are not necessary for
applying the formula in Eq. (6.9) since both series in L.H.S and R.H.S of Eq.
(6.8) will be finite sums. Then the resulting Eq. (6.10) holds true for all
$x\in\mathbb{C}$ in this case. This completes the proof. $\square$

##  7\. Concluding remarks

We have been dealing with the Husimi Q-function of a density operator with
respect to a set of coherent states attached to higher hyperbolic Landau
levels and labeled by points of an open disk of radius $R$ (phase space). For
both a pure state representing a projector on a Fock state and the thermal
density operator (mixed states) for the Hamiltonian of the isotonic
oscillator, we explicitly have established formulas for the Q-function and we
have written down the corresponding characteristic function from which the
mean statistical parameters have been derived. We also have obtained a lower
bound for a thermodynamical potential associated with the Hamiltonian under
consideration. Most of the results of the Euclidean setting (flat case) as
$R\to\infty$ have been recovered, by making use of asymptotic formulas
involving special functions and orthogonal polynomials. In particular, and as
tool, we obtained a summation formula for the Kampé de Fériet function
$\digamma_{2:0:0}^{1:2:2}$. The obtained results may be useful for tackling
the hyperbolic version of Ginibre-type processes [5]. They also can be
exploited in the quantization procedure or localization operators ”À la
Daubechies” [30]. Finally, these Q-functions will be used to determine Wehrl
entropies and discuss some related questions in our forthcoming paper.

### Acknowledgements

The authors would like to thank the Moroccan Association of Harmonic Analysis
& Spectral Geometry.

##  Appendix A Proof of Eq. (4.5)

By the asymptotic expansion for the ratio of the Gamma functions ([10, p.24]):

$\frac{\Gamma(z+\alpha)}{\Gamma(z+\beta)}=z^{\alpha-\beta}\left(1+\frac{(\alpha-\beta)(\alpha+\beta-1)}{2z}+\mathcal{O}\left(z^{-2}\right)\right)$
(A.1)

$|z|\rightarrow\infty,\ |arg(z)|\leq\pi-\varepsilon,\ 0<\varepsilon<\pi$, one
obtains that

$\frac{\Gamma(2(BR^{2}-m)+m\vee j)}{\Gamma(2(BR^{2}-m)+m\wedge
j)}\simeq(2BR^{2})^{|m-j|}\text{ as\ }R\rightarrow\infty.$ (A.2)

Next, using the well known approximation ([43, p.196]):

$\left(1+\frac{x}{n}\right)^{n}\;\simeq\;e^{x}\text{ as }n\rightarrow\infty$
(A.3)

we find that

$\left(1-\frac{|z|^{2}}{R^{2}}\right)^{2(BR^{2}-m)}=\left(1-\frac{2B|z|^{2}}{2BR^{2}}\right)^{2BR^{2}}\left(1-\frac{|z|^{2}}{R^{2}}\right)^{-2m}\simeq
e^{-2B|z|^{2}}\text{ as }R\rightarrow\infty.$ (A.4)

By another side, we use the limiting formula ([39, p.247]):

$L_{n}^{(a)}(x)=\lim_{b\rightarrow\infty}P_{n}^{(a,b)}\left(1-\frac{2x}{b}\right)$
(A.5)

to show that

$P_{m\wedge
j}^{(|m-j|,\,2(BR^{2}-m)-1)}\left(1-\frac{2|z|^{2}}{R^{2}}\right)\simeq
L_{m\wedge j}^{(|m-j|)}(2B|z|^{2})\text{ as }R\rightarrow\infty.$ (A.6)

Finally by Eq. (A.2), Eq. (A.4) and Eq. (A.6) we arrive at

$Q_{j}^{B,R,m}(z)\simeq\frac{(m\wedge j)!}{(m\vee
j)!}\;e^{-2B|z|^{2}}\;\left(2B|z|^{2}\right)^{|m-j|}\left(L_{m\wedge
j}^{(|m-j|)}(2B|z|^{2})\right)^{2}\text{ as }R\rightarrow\infty.$ (A.7)

The expression in the R.H.S of Eq. (A.7) is $Q_{j}^{\left(m\right),\text{
Euclid}}(\sqrt{2B}z)$ as denoted in Eq. (4.5). $\square$

##  Appendix B Proof of Eq. (4.14)

We start by recalling the characteristic function as defined in Eq. (4.13) by

$\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)=\int\limits_{0}^{R^{2}}e^{iu\lambda}\mathfrak{Q}_{j}^{B,R,m}\left(\lambda\right)d\lambda.$
(B.1)

By Eq. (4.8) it can also be written as

$\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)=\left(2(BR^{2}-m)-1\right)\tau_{j}^{B,R,m}I_{R}$
(B.2)

where $\tau_{j}^{B,R,m}$ is given in Eq. (4.4) and

$I_{R}:=\int_{0}^{1}e^{iuR^{2}x}\left(1-x\right)^{2(BR^{2}-m)-2}\left(x\right)^{|m-j|}\left(P_{m\wedge
j}^{(|m-j|,2(BR^{2}-m)-1)}\left(1-2x\right)\right)^{2}dx.$ (B.3)

Next, we introduce the notations $b=2(BR^{2}-m)-1$, $a=|m-j|$ and $l=m\wedge
j$ with the variable change $v=2x-1$ in order to rewrite $I_{R}$ as

$I_{R}=\frac{1}{2^{a+b}}\int_{-1}^{1}(v+1)^{a}(1-v)^{b-1}e^{\frac{iuR^{2}}{2}(v+1)}\left(P_{l}^{(a,b)}(-v)\right)^{2}dv.$
(B.4)

For the square of the Jacobi polynomial in the last integral we may use the
following Clebsh-Gordan type linearization formula ([44, p.611]):

$P_{i}^{(\lambda,\delta)}(x)P_{j}^{(\mu,\gamma)}(x)=\sum_{k=0}^{i+j}C_{k}P_{k}^{(\alpha,\beta)}(x)$
(B.5)

where the coefficients

$\begin{split}C_{k}=&\frac{(i+j)!(\alpha+1)_{i+j}(\lambda+\delta+1)_{2i}(\mu+\gamma+1)_{2j}}{i!j!(\lambda+\delta+1)_{i}(\mu+\gamma+1)_{j}}\frac{(-1)^{i+j-k}(\alpha+\beta+1)_{k}(2k+\alpha+\beta+1)}{(i+j-k)!(\alpha+1)_{k}(\alpha+\beta+1)_{i+j+k+1}}\\\
&\times\digamma_{2:1:1}^{2:2:2}\left(\begin{array}[]{c}-(i+j)+k,-\alpha-\beta-1-(i+j)-k;-i,-\lambda-i;-j,-\mu-j\\\
-(i+j),-\alpha-(i+j);-2i-\lambda-\delta;-2j-\mu-\gamma\end{array}\big{|}1,1\right)\end{split}$
(B.6)

are chosen with parameters $\lambda=\mu=a$, $\delta=\gamma=b$ and $i=j=l$.
Therefore, $I_{R}$ reads

$I_{R}=\frac{1}{2^{a+b}}\sum_{k=0}^{2l}C_{k}(-1)^{k}\int\limits_{-1}^{+1}(v+1)^{a}(1-v)^{b-1}e^{\frac{1}{2}iuR^{2}(v+1)}P_{k}^{(\beta,\alpha)}(v)dv$
(B.7)

with

$\begin{split}C_{k}=&\frac{(2l)!(\alpha+1)_{2l}((a+b+1)_{2l})^{2}}{(l!)^{2}((a+b+1)_{l})^{2}}\frac{(-1)^{2l-k}(\alpha+\beta+1)_{k}(2k+\alpha+\beta+1)}{(2l-k)!(\alpha+1)_{k}(\alpha+\beta+1)_{2l+k+1}}\\\
&\times\digamma_{2:1:1}^{2:2:2}\left(\begin{array}[]{c}-2l+k,-\alpha-\beta-1-2l-k;-l,-a-l;-l,-a-l\\\
-2l,-\alpha-2l;-2l-a-b;-2l-a-b\end{array}\big{|}1,1\right).\end{split}$ (B.8)

Now, the integral in Eq. (B.7) can be computed by applying the formula (see
[45, p.572]):

$\int_{-\omega}^{\omega}(x+\omega)^{\gamma-1}(\omega-x)^{\rho}e^{-p(x+\omega)^{r}}P_{n}^{(\rho,\sigma)}(\frac{x}{\omega})dx=\frac{(-1)^{n}}{n!}(2\omega)^{\gamma+\rho}\Gamma(\rho+n+1)$
$\times\sum_{s\geq 0}\frac{1}{s!}(\sigma-\gamma-
rs+1)_{n}\frac{\Gamma(\gamma+rs)}{\Gamma(\gamma+\rho+rs+n+1)}(-2^{r}\omega^{r}p)^{s},$
(B.9)

$\omega,r,\Re\gamma>0,\Re\rho>-1$ for parameters:
$\omega=1,\gamma=a+1=|m-j|+1,\rho=\beta=b-1=2(BR^{2}-m)-2,p=-iuR^{2}/2,r=1,\sigma=\alpha=\gamma-1=|m-j|,n=k$.
Summarizing the above calculations, Eq. (B.2) reads

$\begin{split}\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)=&\,\left(2(BR^{2}-m)-1\right)\tau_{j}^{B,R,m}\sum_{k=0}^{2(m\wedge
j)}C_{j,k}^{B,R,m}\frac{(-1)^{k}}{k!}\Gamma(2(BR^{2}-m)-1+k)\\\
&\times\sum_{s\geq
0}\frac{1}{s!}\frac{\Gamma(|m-j|+s+k+1)}{\Gamma(2(BR^{2}-m)+|m-j|+s+2k)}(iuR^{2})^{s+k}\end{split}$
(B.10)

where

$\begin{split}&C_{j,k}^{B,R,m}=\frac{(2(m\wedge j))!(|m-j|+1))_{2(m\wedge
j)}((2(BR^{2}-m)+|m-j|)_{2(m\wedge j)})^{2}}{((m\wedge
j)!)^{2}((2(BR^{2}-m)+|m-j|)_{m\wedge j})^{2}}\\\
&\times\frac{(-1)^{k}(2(BR^{2}-m)+|m-j|-1)_{k}(2k+2(BR^{2}-m)+|m-j|-1)}{(2(m\wedge
j)-k)!(|m-j|+1)_{k}(2(BR^{2}-m)+|m-j|-1)_{2(m\wedge j)+k+1}}\\\
&\times\digamma_{2:1:1}^{2:2:2}\left(\begin{array}[]{c}-2(m\wedge
j)+k,-2(BR^{2}-m)-m-j+1-k;-m,-j;-m,-j\\\ -2(m\wedge
j),-m-j;-2(BR^{2}-m)-m-j+1;-2(BR^{2}-m)-m-j+1\end{array}\bigg{|}1,1\right)\end{split}$
(B.11)

Finally, the series over $s$ in the R.H.S of Eq. (B.10) can also be written in
terms of a ${}_{1}\digamma_{1}$-series. This ends the proof. $\square$

##  Appendix C Proof of Eq. (4.19)

First, we note that by using the asymptotic expansion Eq. (A.1)

$\tau_{m,j}^{B,R}(2(BR^{2}-m)-1)\simeq\frac{(m\wedge j)!}{(m\vee
j)!}(2BR^{2})^{|m-j|+1}\quad\text{as}\quad R\rightarrow\infty$ (C.1)

and

$\frac{\Gamma(2(BR^{2}-m)-1+k)}{\Gamma(2(BR^{2}-m)+|m-j|+s+2k)}{\simeq}\frac{1}{(2BR^{2})^{1+|m-j|+s+k}}\quad\text{as}\quad
R\rightarrow\infty.$ (C.2)

Next, we make us of the approximation formula which derive from Eq. (A.1)

$(z)_{a}=z^{a}\left(1+\frac{a(a-1)}{2z}+\mathcal{O}\left(z^{-2}\right)\right),\;\;|z|\rightarrow\infty,\
|arg(z)|\leq\pi-\varepsilon,\ 0<\varepsilon<\pi$ (C.3)

to obtain

$\displaystyle C_{j,k}^{B,R,m}$ $\displaystyle\simeq$
$\displaystyle\frac{(2(m\wedge j))!}{((m\wedge
j)!)^{2}}\frac{(|m-j|+1)_{2(m\wedge j)}(-1)^{k}}{(2(m\wedge
j)-k)!(|m-j|+1)_{k}}$ (C.7) $\displaystyle\times
F_{2:1:1}^{2:2:2}\left(\begin{array}[]{c}-2(m\wedge
j)+k,-2BR^{2};-m,-j;-m,-j\\\ -2(m\wedge
j),-m-j;-2BR^{2};-2BR^{2}\end{array}\bigg{|}1,1\right)\quad\text{as}\quad
R\rightarrow\infty.$

From the definition of $\digamma_{l:m:n}^{p:q:k}$ in Eq. (2.13), we may
rewrite Eq. (C.7) as

$\displaystyle C_{j,k}^{B,R,m}$ $\displaystyle\simeq$
$\displaystyle\frac{(2(m\wedge j))!}{((m\wedge
j)!)^{2}}\frac{(|m-j|+1)_{2(m\wedge j)}(-1)^{k}}{(2(m\wedge
j)-k)!(|m-j|+1)_{k}}$
$\displaystyle\times\digamma_{2:0:0}^{1:2:2}\left({\genfrac{}{}{0.0pt}{}{-2(m\wedge
j)+k;-m,-j;-m,-j}{-2(m\wedge
j),-m-j;-;-}}\bigg{|}1,1\right)\quad\text{as}\quad R\rightarrow\infty.$

Using equations Eq. (C.1), Eq. (C.2) and Eq. (C) we get, after simplification,
that

$\displaystyle\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)$
$\displaystyle\simeq$ $\displaystyle\frac{(m+j)!}{m!j!}\sum_{k=0}^{2(m\wedge
j)}\binom{2(m\wedge
j)}{k}\digamma_{2:0:0}^{1:2:2}\left({\genfrac{}{}{0.0pt}{}{-2(m\wedge
j)+k;-m,-j;-m,-j}{-2(m\wedge j),-m-j;-;-}}\bigg{|}1,1\right)$ (C.9)
$\displaystyle\times\left(\frac{iu}{2B}\right)^{k}\sum_{s\geq
0}(|m-j|+k+1)_{s}\frac{\left(\frac{iu}{2B}\right)^{s}}{s!}\quad\text{as}\quad
R\rightarrow\infty$

The infinite sum in Eq. (C.9) reduces to

$\sum_{s\geq
0}\frac{1}{s!}(|m-j|+k+1)_{s}\left(\frac{iu}{2B}\right)^{s}=\left(1-\frac{iu}{2B}\right)^{-|m-j|-k-1},\quad
u<2B.$ (C.10)

Then for $u<2B$, Eq. (C.9) becomes

$\begin{split}\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)&\simeq\frac{(m+j)!}{m!j!}\left(1-\frac{iu}{2B}\right)^{-|m-j|-1}\\\
&\times\sum_{k=0}^{2(m\wedge j)}\binom{2(m\wedge
j)}{k}\digamma_{2:0:0}^{1:2:2}\left({\genfrac{}{}{0.0pt}{}{-2(m\wedge
j)+k;-m,-j;-m,-j}{-2(m\wedge
j),-m-j;-;-}}\bigg{|}1,1\right)\left(\frac{iu}{2B-iu}\right)^{k},\end{split}$
(C.11)

as $R\to\infty$. Applaying Proposition 6.1 for parameters $n=2(m\wedge j)$,
$a=-m$, $b=-j$, $c=-m-j$ and $x=\frac{iu}{2B-iu}$, Eq. (C.11) reduces to

$\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right){\simeq}\frac{(m+j)!}{m!j!}\left(1-\frac{iu}{2B}\right)^{-(m+j+1)}{}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{-m,-j}{-m-j}}\bigg{|}1+\left(\frac{u}{2B}\right)^{2}\right),\quad
u<2B$ (C.12)

as $R\rightarrow\infty$ for $m\wedge j>0$. Then, we recover the announced
limiting formula in Eq. (4.19) for $m\wedge j>0$ using the connexion formula
in Eq. (4). Now for $m\wedge j=0$, we get directly

$\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)={}_{1}\digamma_{1}\left({m\vee
j+1\atop 2(BR^{2}-m)+m\vee
j}\bigg{|}iuR^{2}\right){\simeq}\left(1-\frac{iu}{2B}\right)^{-(m\vee
j+1)},\quad u<2B$ (C.13)

as $R\to\infty$, that coincides to Eq. (4.19) putting $m\wedge j=0$. This
completes the proof. $\square$

##  Appendix D Proof of Eqs. (4.16)-(4.17)

Let us write the characteristic function
$\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)$ as

$\displaystyle\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)$
$\displaystyle=$ $\displaystyle\sum_{k=0}^{2(m\wedge j)}\sum_{s\geq
0}\delta_{j,k,s}^{B,R,m}(iuR^{2})^{s+k}$ (D.1) $\displaystyle=$
$\displaystyle\sum_{s\geq 0}\delta_{j,0,s}^{B,R,m}(iuR^{2})^{s}+\sum_{s\geq
0}\delta_{j,1,s}^{B,R,m}(iuR^{2})^{s+1}+\sum_{k=2}^{2(m\wedge j)}\sum_{s\geq
0}\delta_{j,k,s}^{B,R,m}(iuR^{2})^{s+k}$

where

$\delta_{j,k,s}^{B,R,m}:=\tau_{m,j}^{B,R}(2(BR^{2}-m)-1)C_{j,k}^{B,R,m}\frac{(-1)^{k}}{k!s!}\frac{\Gamma(2(BR^{2}-m)-1+k)\Gamma(|m-j|+s+k+1)}{\Gamma(2(BR^{2}-m)+|m-j|+s+2k)}$
(D.2)

$\tau_{m,j}^{B,R}$ and $C_{j,k}^{B,R,m}$ being given respectively by Eq. (4.4)
and Eq. (4.15). Then, the first derivative of
$\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)$ reads

$\begin{split}\frac{d}{du}\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)=&\left(\delta_{j,0,1}^{B,R,m}+\delta_{j,1,0}^{B,R,m}\right)(iR^{2})+\sum_{s\geq
2}\delta_{j,0,s}^{B,R,m}s(iR^{2})(iuR^{2})^{s-1}\\\ &+\sum_{s\geq
1}\delta_{j,1,s}^{B,R,m}(s+1)(iR^{2})(iuR^{2})^{s}+\sum_{k=2}^{2(m\wedge
j)}\sum_{s\geq
0}\delta_{j,k,s}^{B,R,m}(s+k)(iR^{2})(iuR^{2})^{s+k-1}.\end{split}$ (D.3)

The mean value of $\mathfrak{X}_{m}^{(R)}$ is then given by

$\mathbb{E}\left(\mathfrak{X}_{m}^{\left(R\right)}\right):=-i\frac{d}{du}\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\bigg{|}_{u=0}=\left(\delta_{j,0,1}^{B,R,m}+\delta_{j,1,0}^{B,R,m}\right)R^{2}.$
(D.4)

where

$\delta_{j,0,1}^{B,R,m}=\tau_{m,j}^{B,R}\frac{(|m-j|+1)!\Gamma(2(BR^{2}-m))}{\Gamma(2(BR^{2}-m)+|m-j|+1)}C_{j,0}^{B,R,m}$
(D.5)

and

$\delta_{j,1,0}^{B,R,m}=-\tau_{m,j}^{B,R}\frac{(|m-j|+1)!\Gamma(2(BR^{2}-m))}{\Gamma(2(BR^{2}-m)+|m-j|+2)}(2(BR^{2}-m)-1)C_{j,1}^{B,R,m}.$
(D.6)

This completes the proof of Eq. (4.16). For $m=0$, it is clear that $k=0$ and
$C_{j,0}^{B,R,0}=1$. Then

$\mathbb{E}\left(\mathfrak{X}_{0}^{\left(R\right)}\right)=\kappa_{j}^{B,R,0}\frac{(j+1)!\Gamma(2BR^{2})}{\Gamma(2BR^{2}+j+1)}R^{2}\\\
=\frac{(j+1)R^{2}}{2BR^{2}+j}.$ (D.7)

Now, for the variance, we search for the second derivative of
$\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)$. We rewrite Eq. (D.3)
as

$\begin{split}\frac{d}{du}\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)=&\left(\delta_{j,0,1}^{B,R,m}+\delta_{j,1,0}^{B,R,m}\right)(iR^{2})+\left(\delta_{j,0,2}^{B,R,m}+\delta_{j,1,1}^{B,R,m}+\delta_{j,2,0}^{B,R,m}\right)2(iR^{2})(iuR^{2})\\\
&+\sum_{s\geq 3}\delta_{j,0,s}^{B,R,m}s(iR^{2})(iuR^{2})^{s-1}+\sum_{s\geq
2}\delta_{j,1,s}^{B,R,m}(s+1)(iR^{2})(iuR^{2})^{s}\\\ &+\sum_{s\geq
1}\delta_{j,2,s}^{B,R,m}(s+2)(iR^{2})(iuR^{2})^{s+1}+\sum_{k=3}^{2(m\wedge
j)}\sum_{s\geq
0}\delta_{j,k,s}^{B,R,m}(s+k)(iR^{2})(iuR^{2})^{s+k-1}.\end{split}$ (D.8)

Then,

$\begin{split}\frac{d^{2}}{du^{2}}\phi_{\mathfrak{X}_{m}^{\left(R\right)}}(u)=&\left(\delta_{j,0,2}^{B,R,m}+\delta_{j,1,1}^{B,R,m}+\delta_{j,2,0}^{B,R,m}\right)2(iR^{2})^{2}+\sum_{s\geq
3}\delta_{j,0,s}^{B,R,m}(s-1)s(iR^{2})^{2}(iuR^{2})^{s-2}\\\ &+\sum_{s\geq
2}\delta_{j,1,s}^{B,R,m}s(s+1)(iR^{2})^{2}(iuR^{2})^{s-1}+\sum_{s\geq
1}\delta_{j,2,s}^{B,R,m}(s+1)(s+2)(iR^{2})^{2}(iuR^{2})^{s}\\\
&+\sum_{k=3}^{2(m\wedge j)}\sum_{s\geq
0}\delta_{j,k,s}^{B,R,m}(s+k-1)(s+k)(iR^{2})^{2}(iuR^{2})^{s+k-2}.\end{split}$
(D.9)

The evaluation at $u=0$ with the mean value gives the variance of
$\mathfrak{X}_{m}^{\left(R\right)}$ as

$\mathbb{V}ar\left(\mathfrak{X}_{m}^{\left(R\right)}\right):=-\frac{d^{2}}{du^{2}}\phi_{\mathfrak{X}_{m}^{\left(R\right)}}\left(u\right)\bigg{|}_{u=0}-\mathbb{E}\left(\mathfrak{X}_{m}^{\left(R\right)}\right)^{2}=\left(\delta_{j,0,2}^{B,R,m}+\delta_{j,1,1}^{B,R,m}+\delta_{j,2,0}^{B,R,m}\right)2R^{4}-\mathbb{E}\left(\mathfrak{X}_{m}^{\left(R\right)}\right)^{2}.$
(D.10)

Replacing

$\delta_{j,0,2}^{B,R,m}=\tau_{m,j}^{B,R}\frac{(|m-j|+2)!\Gamma(2(BR^{2}-m))}{2\Gamma(2(BR^{2}-m)+|m-j|+2)}C_{j,0}^{B,R,m},$
(D.11)
$\delta_{j,1,1}^{B,R,m}=-\tau_{m,j}^{B,R}\frac{(|m-j|+2)!\Gamma(2(BR^{2}-m))}{\Gamma(2(BR^{2}-m)+|m-j|+3)}(2(BR^{2}-m)-1)C_{j,1}^{B,R,m}$
(D.12)

and

$\delta_{j,2,0}^{B,R,m}=\tau_{m,j}^{B,R}\frac{(|m-j|+2)!\Gamma(2(BR^{2}-m)+1)}{2\Gamma(2(BR^{2}-m)+|m-j|+4)}(2(BR^{2}-m)-1)C_{j,2}^{B,R,m}$
(D.13)

by theirs expressions into Eq. (D.10) gives the announced formula in Eq.
(4.17). Like above, we find for $m=0$

$\displaystyle\mathbb{V}ar\left(\mathfrak{X}_{0}^{(R)}\right)$
$\displaystyle=$
$\displaystyle\kappa_{j}^{B,R,0}\frac{(j+2)!\Gamma(2BR^{2})}{\Gamma(2BR^{2}+j+2)}\left(C_{j,0}^{B,R,0}\right)R^{4}-\left(\kappa_{j}^{B,R,0}\frac{(j+1)!\Gamma(2BR^{2})}{\Gamma(2BR^{2}+j+1)}R^{2}\right)^{2},$
(D.14) $\displaystyle=$
$\displaystyle\frac{(j+1)(2BR^{2}-1)R^{4}}{(2BR^{2}+j)^{2}(2BR^{2}+j+1)}.$

This ends the proof. $\square$

##  Appendix E Proof of Eq. (5.24)

With the approximation Eq. (A.3) we find

$\left(\frac{1-\frac{z\bar{z}}{R^{2}}}{1-\frac{z\bar{z}}{R^{2}}e^{-\beta}}\right)^{2BR^{2}}=\left(1-\frac{2Bz\bar{z}}{2BR^{2}}\right)^{2BR^{2}}\left(1+\frac{2Bz\bar{z}e^{-\beta}}{-2BR^{2}}\right)^{-2BR^{2}}\simeq\exp(-2Bz\bar{z}(1-e^{-\beta}))$
(E.1)

as $R\to\infty$. Direct calculation gives

$\left(\frac{\left(\frac{z\bar{z}}{R^{2}}-e^{-\beta}\right)\left(1-\frac{z\bar{z}}{R^{2}}e^{-\beta}\right)}{\left(1-\frac{z\bar{z}}{R^{2}}\right)^{2}}\right)^{m}\eqsim(-1)^{m}e^{-\beta
m}\quad\text{as}\quad R\to\infty,$ (E.2)

and

$P^{(2(BR^{2}-m)-1,0)}_{m}\left(1+\frac{2e^{-\beta}\left(1-\frac{z\bar{z}}{R^{2}}\right)^{2}}{\left(\frac{z\bar{z}}{R^{2}}-e^{-\beta}\right)\left(1-\frac{z\bar{z}}{R^{2}}e^{-\beta}\right)}\right)\simeq(-1)^{m}P^{(0,2BR^{2})}_{m}\left(1\right),\quad\text{as}\quad
R\to\infty$ (E.3)

that becomes

$P^{(2(BR^{2}-m)-1,0)}_{m}\left(1+\frac{2e^{-\beta}\left(1-\frac{z\bar{z}}{R^{2}}\right)^{2}}{\left(\frac{z\bar{z}}{R^{2}}-e^{-\beta}\right)\left(1-\frac{z\bar{z}}{R^{2}}e^{-\beta}\right)}\right)\simeq(-1)^{m}L_{m}^{(0)}\left(-8Bz\bar{z}\sinh^{2}\frac{\beta}{2}\right),\quad\text{as}\quad\to\infty$
(E.4)

by using the limiting formula Eq. (A.5). Then, it follow from Eq. (E.1), Eq.
(E.2) and Eq. (E.4) that

$Q_{\beta}^{B,R,m}(z)\simeq\left(1-e^{-\beta}\right)\exp\left(-2Bz\bar{z}(1-e^{-\beta})\right)e^{-\beta
m}L_{m}^{(0)}\left(-8Bz\bar{z}\sinh^{2}\frac{\beta}{2}\right),\quad\text{as}\quad
R\to\infty.$ (E.5)

Comparing Eq. (E.5) to [4, Eq. (3.14)], we recover the limiting formula
$Q_{\beta}^{B,R,m}(z)\to q_{m}(|\sqrt{2B}z|^{2},\beta)$ as $R\to\infty$.
$\square$

##  Appendix F Proof of Eq. (5.28)

Identify $\alpha=2(BR^{2}-m)-1$ and $\xi=e^{-\beta}$ and set
$x=\frac{\lambda}{R^{2}}$. Then, the characteristic function reads from Eq.
(5.21) as

$\begin{split}\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)=&\alpha(1-\xi)\int\limits_{0}^{1}e^{iuxR^{2}}\left(\frac{1-x}{1-x\xi}\right)^{\alpha+2m+1}\left(\frac{(x-\xi)(1-x\xi)}{(1-x)^{2}}\right)^{m}\\\
&\times
P^{(\alpha,0)}_{m}\left(1+\frac{2\xi(1-x)^{2}}{(x-\xi)(1-x\xi)}\right)\frac{dx}{(1-x)^{2}}.\end{split}$
(F.1)

By using the Jacobi polynomial definition in Eq. (2.9) one obtains the
contraction

$\displaystyle\left(\frac{(x-\xi)(1-x\xi)}{(1-x)^{2}}\right)^{m}P^{(\alpha,0)}_{m}\left(1+\frac{2\xi(1-x)^{2}}{(x-\xi)(1-x\xi)}\right)$
$\displaystyle=$
$\displaystyle\xi^{m}\sum_{k=0}^{m}\binom{m+\alpha}{k}\binom{m}{k}\left(\frac{(1-\xi)^{2}}{\xi}\right)^{k}$
(F.2) $\displaystyle\times\left(\frac{x}{(1-x)^{2}}\right)^{k}.$

Then, Eq. (F.1) becomes after reorganization

$\begin{split}\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)=&\frac{\alpha(1-\xi)\xi^{m}}{(-\xi)^{\alpha+2m+1}}\\\
&\sum_{k=0}^{m}\binom{m+\alpha}{k}\binom{m}{k}\left(\frac{(1-\xi)^{2}}{\xi}\right)^{k}\int\limits_{0}^{1}e^{iuxR^{2}}x^{k}(1-x)^{\alpha+2m-2k-1}(x-\xi^{-1})^{-\alpha-2m-1}dx.\end{split}$
(F.3)

Thanks to the formula ([46, p.328]):

$\int\limits_{0}^{a}x^{\tau-1}(a-x)^{\beta-1}(x+z)^{-\rho}e^{-px}dx=\mathcal{B}(\tau,\beta)z^{-\rho}a^{\tau+\beta-1}\Phi_{1}\left({\genfrac{}{}{0.0pt}{}{\tau,\rho}{\tau+\beta}}\bigg{|}-\frac{a}{z},ap\right)$
(F.4)

where $a,\Re\tau,\Re\beta>0;|\arg(1+a/z)|<\pi$. It application on the integral
at the R.H.S of Eq. (F.3) for $a=1$, $z=-\xi^{-1}$, $\tau=k+1$,
$\beta=\alpha+2m-2k$, $\rho=\alpha+2m+1$ and $p=-iuR^{2}$ gives after
simplification

$\begin{split}\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)=&\,\alpha(1-\xi)\xi^{m}\sum_{k=0}^{m}\binom{m+\alpha}{k}\binom{m}{k}\left(\frac{(1-\xi)^{2}}{\xi}\right)^{k}\\\
&\mathcal{B}(k+1,\alpha+2m-2k)\Phi_{1}\left({\genfrac{}{}{0.0pt}{}{k+1,\alpha+2m+1}{\alpha+2m-k+1}}\Big{|}\xi,-iuR^{2}\right).\end{split}$
(F.5)

Replacing $\alpha$ and $\xi$ by their expressions ends the proof. $\square$

##  Appendix G Proof of Eqs. (5.29)-(5.30)

Due to the definition of the Humbert series in Eq. (2.14), it derivative reads

$\frac{\partial}{\partial
z}\Phi_{1}\left({\genfrac{}{}{0.0pt}{}{a,b}{c}}\Big{|}w,z\right)=\frac{a}{c}\Phi_{1}\left({\genfrac{}{}{0.0pt}{}{a+1,b}{c+1}}\Big{|}w,z\right).$
(G.1)

Then, the derivative of $\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)$ in Eq.
(F.5) is given by

$\displaystyle\frac{d}{du}\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)$
$\displaystyle=$
$\displaystyle\alpha(1-\xi)\xi^{m}(-iR^{2})\sum_{k=0}^{m}\binom{m+\alpha}{k}\binom{m}{k}\left(\frac{(1-\xi)^{2}}{\xi}\right)^{k}$
(G.2)
$\displaystyle\mathcal{B}(k+2,\alpha+2m-2k)\Phi_{1}\left({\genfrac{}{}{0.0pt}{}{k+2,\alpha+2m+1}{\alpha+2m-k+2}}\Big{|}\xi,-iuR^{2}\right).$

The evaluation at $u=0$ taking into account the formula Eq. (2.15) gives the
mean value of $\mathfrak{Y}_{m}^{\left(R\right)}$ as

$\displaystyle\mathbb{E}\left(\mathfrak{Y}_{m}^{\left(R\right)}\right)$
$\displaystyle=$
$\displaystyle-i\frac{d}{du}\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)\bigg{|}_{u=0}$
(G.3) $\displaystyle=$
$\displaystyle\alpha(\xi-1)\xi^{m}R^{2}\sum_{k=0}^{m}\binom{m+\alpha}{k}\binom{m}{k}\left(\frac{(1-\xi)^{2}}{\xi}\right)^{k}$
$\displaystyle\mathcal{B}(k+2,\alpha+2m-2k){}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{k+2,\alpha+2m+1}{\alpha+2m-k+2}}\Big{|}\xi\right).$

For the variance, we search for the second derivative of
$\phi_{\mathfrak{Y}_{m}^{(R)}}(u)$ that reads from Eq. (G.1) and Eq. (G.2) as

$\displaystyle\frac{d^{2}}{du^{2}}\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)$
$\displaystyle=$
$\displaystyle-\alpha(1-\xi)\xi^{m}R^{4}\sum_{k=0}^{m}\binom{m+\alpha}{k}\binom{m}{k}\left(\frac{(1-\xi)^{2}}{\xi}\right)^{k}$
(G.4)
$\displaystyle\mathcal{B}(k+3,\alpha+2m-2k)\Phi_{1}\left({\genfrac{}{}{0.0pt}{}{k+3,\alpha+2m+1}{\alpha+2m-k+3}}\Big{|}\xi,-iuR^{2}\right).$

The evaluation at $u=0$ taking into account the formula Eq. (2.15) gives

$\displaystyle\frac{d^{2}}{du^{2}}\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)\bigg{|}_{u=0}$
$\displaystyle=$
$\displaystyle-\alpha(1-\xi)\xi^{m}R^{4}\sum_{k=0}^{m}\binom{m+\alpha}{k}\binom{m}{k}\left(\frac{(1-\xi)^{2}}{\xi}\right)^{k}$
(G.5)
$\displaystyle\mathcal{B}(k+3,\alpha+2m-2k){}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{k+3,\alpha+2m+1}{\alpha+2m-k+3}}\Big{|}\xi\right).$

The variance of $\mathfrak{Y}_{m}^{\left(R\right)}$ that is defined by

$\mathbb{V}ar\left(\mathfrak{Y}_{m}^{\left(R\right)}\right):=-\frac{d^{2}}{du^{2}}\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)-\left(\mathbb{E}\left(\mathfrak{Y}_{m}^{\left(R\right)}\right)\right)^{2}$
(G.6)

reads then as

$\mathbb{V}ar\left(\mathfrak{Y}_{m}^{\left(R\right)}\right)=$
$\begin{split}&\,\alpha(1-\xi)\xi^{m}R^{4}\sum_{k=0}^{m}\binom{m+\alpha}{k}\binom{m}{k}\left(\frac{(1-\xi)^{2}}{\xi}\right)^{k}\mathcal{B}(k+3,\alpha+2m-2k){}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{k+3,\alpha+2m+1}{\alpha+2m-k+3}}\Big{|}\xi\right)-\\\
&\Bigg{[}\alpha(1-\xi)\xi^{m}R^{2}\sum_{k=0}^{m}\binom{m+\alpha}{k}\binom{m}{k}\left(\frac{(1-\xi)^{2}}{\xi}\right)^{k}\mathcal{B}(k+2,\alpha+2m-2k){}_{2}\digamma_{1}\left({\genfrac{}{}{0.0pt}{}{k+2,\alpha+2m+1}{\alpha+2m-k+2}}\Big{|}\xi\right)\Bigg{]}^{2}.\end{split}$
(G.7)

This ends the proof replacing $\alpha$ and $\xi$ by their expressions.
$\square$

##  Appendix H Proof of Eq. (5.31)

By using the formula Eq. (A.1), we obtain

$\binom{2BR^{2}-m-1}{k}\mathcal{B}(k+1,2BR^{2}-2k-1)=\frac{\Gamma(2BR^{2}-m)}{\Gamma(2BR^{2}-m-k)}\frac{\Gamma(2BR^{2}-2k-1)}{\Gamma(2BR^{2}-k)}\simeq\frac{1}{2BR^{2}}$
(H.1)

as $R\rightarrow+\infty$, that yields

$(2(BR^{2}-m)-1)\binom{2BR^{2}-m-1}{k}\mathcal{B}(k+1,2BR^{2}-2k-1)\simeq
1,\quad\text{as}\quad R\rightarrow\infty.$ (H.2)

Next, by the definition of $\Phi_{1}$ in Eq. (2.14) and the use of the
approximation formula Eq. (C.3) we obtain after simplification

$\Phi_{1}\left({\genfrac{}{}{0.0pt}{}{k+1,2BR^{2}}{2BR^{2}-k}}\Big{|}e^{-\beta},-iuR^{2}\right)\simeq\sum_{s,t=0}^{+\infty}(k+1)_{s+t}\frac{e^{-s\beta}}{s!}\frac{(\frac{iu}{2B})^{t}}{t!},\quad\text{as}\quad
R\rightarrow\infty.$ (H.3)

By the use of the fact $(a)_{n+m}=(a)_{n}(a+n)_{m}$, the series at the R.H.S
of Eq. (H.3) becomes

$\sum_{s,t=0}^{+\infty}(k+1)_{s+t}\frac{e^{-s\beta}}{s!}\frac{(\frac{iu}{2B})^{t}}{t!}=\sum_{s=0}^{+\infty}(k+1)_{s}\frac{e^{-s\beta}}{s!}\sum_{t=0}^{+\infty}(k+1+s)_{t}\frac{(\frac{iu}{2B})^{t}}{t!}.$
(H.4)

Identifying the series (over t) at the R.H.S of Eq. (H.4) as a
${}_{1}\digamma_{0}$, it follow for $u<2B$

$\displaystyle\sum_{s,t=0}^{+\infty}(k+1)_{s+t}\frac{e^{-s\beta}}{s!}\frac{(\frac{iu}{2B})^{t}}{t!}$
$\displaystyle=$
$\displaystyle\sum_{s=0}^{+\infty}(k+1)_{s}\frac{e^{-s\beta}}{s!}\left(1-\frac{iu}{2B}\right)^{-k-1-s}$
(H.5) $\displaystyle=$
$\displaystyle\left(1-\frac{iu}{2B}\right)^{-k-1}\sum_{s=0}^{+\infty}(k+1)_{s}\frac{[(1-\frac{iu}{2B})^{-1}e^{-\beta}]^{s}}{s!}$

that becomes identifying twice ${}_{1}\digamma_{0}$ as

$\displaystyle\sum_{s,t=0}^{+\infty}(k+1)_{s+t}\frac{e^{-s\beta}}{s!}\frac{(\frac{iu}{2B})^{t}}{t!}$
$\displaystyle=$
$\displaystyle\left(1-\frac{iu}{2B}\right)^{-k-1}\left(1-(1-\frac{iu}{2B})^{-1}e^{-\beta}\right)^{-k-1}$
(H.6) $\displaystyle=$
$\displaystyle\left(1-e^{-\beta}-\frac{iu}{2B}\right)^{-k-1}.$

Then, Eq. (H.3) becomes

$\Phi_{1}\left({\genfrac{}{}{0.0pt}{}{k+1,2BR^{2}}{2BR^{2}-k}}\Big{|}e^{-\beta},-iuR^{2}\right)\simeq\left(1-e^{-\beta}-\frac{iu}{2B}\right)^{-k-1},\quad\text{as}\quad
R\rightarrow\infty$ (H.7)

for $u<2B$. Thus, from Eq. (H.2) and Eq. (H.7),
$\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)$ has as limit when
$R\rightarrow\infty$

$\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)\simeq(1-e^{-\beta})e^{-m\beta}\sum_{k=0}^{m}\binom{m}{k}\left(\frac{(1-e^{-\beta})^{2}}{e^{-\beta}}\right)^{k}\left(1-e^{-\beta}-\frac{iu}{2B}\right)^{-k-1},\quad
u<2B$ (H.8)

that reads successively

$\displaystyle\phi_{\mathfrak{Y}_{m}^{\left(R\right)}}(u)$
$\displaystyle\simeq$
$\displaystyle\frac{(1-e^{-\beta})e^{-m\beta}}{1-e^{-\beta}-\frac{iu}{2B}}\sum_{k=0}^{m}\binom{m}{k}\left(\frac{\frac{(1-e^{-\beta})^{2}}{e^{-\beta}}}{1-e^{-\beta}-\frac{iu}{2B}}\right)^{k}$
(H.9) $\displaystyle\simeq$
$\displaystyle\frac{(1-e^{-\beta})e^{-m\beta}}{1-e^{-\beta}-\frac{iu}{2B}}\left(1+\frac{\frac{(1-e^{-\beta})^{2}}{e^{-\beta}}}{1-e^{-\beta}-\frac{iu}{2B}}\right)^{m}$
$\displaystyle\simeq$
$\displaystyle\frac{(1-e^{-\beta})}{1-e^{-\beta}-\frac{iu}{2B}}\left(\frac{1-e^{-\beta}-\frac{iu}{2B}e^{-\beta}}{1-e^{-\beta}-\frac{iu}{2B}}\right)^{m},\quad
u<2B.$

This ends the proof by comparing Eq. (H.9) with ([4, Eq. (3.24)]). $\square$

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