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# Asymmetry of tensor product of asymmetric and invariant vectors arising from
Schur-Weyl duality based on hypergeometric orthogonal polynomial

Masahito Hayashi, Akihito Hora, Shintarou Yanagida  Shenzhen Institute for
Quantum Science and Engineering, Southern University of Science and
Technology, Shenzhen 518055, China.
Guangdong Provincial Key Laboratory of Quantum Science and Engineering,
Southern University of Science and Technology, Shenzhen 518055, China.
Graduate School of Mathematics, Nagoya University,Furocho, Chikusaku, Nagoya,
464-8602, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Mathematics, Faculty of Science, Hokkaido University, Kita 10,
Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan.
<EMAIL_ADDRESS>Graduate School of Mathematics, Nagoya University,
Furocho, Chikusaku, Nagoya, 464-8602, Japan<EMAIL_ADDRESS>
(Date: 2021.04.25)

###### Abstract.

We introduce and study a certain discrete probability distribution
$P_{n,m,k,l}$ having non-negative integer parameters $n,m,k,l$, motivated by
the asymmetry problem in quantum information theory. Its analysis reveals the
number of orthogonal vectors among permuted vectors of the tensor product of
asymmetric and invariant vectors. The distribution is defined by irreducible
decomposition of the tensor product $\Xi_{n,m|k,l}$ of a certain asymmetric
state and the Dicke state in the
$\operatorname{SU}(2)$-$\mathfrak{S}_{n}$-bimodule $(\mathbb{C}^{2})^{\otimes
n}$ appearing in the classical Schur-Weyl duality. We derive several explicit
formulas of the probability mass function $p(x)=p(x|n,m,k,l)$ involving
hypergeometric orthogonal polynomials via representation theoretic methods.
Among them, Racah presentation is the most remarkable and useful formula,
which expresses the pmf $p(x)$ by a single Racah polynomial. We also derive a
formula expressing the cumulative distribution function in terms of a
terminating ${}_{4}F_{3}$-hypergeometric series. Using these explicit
formulas, we study asymptotic behavior of $P_{n,m,k,l}$ in two types of limit
$n\to\infty$. In the first limit, we fix $k$, $l$ and $m/n$, and show that the
asymptotic discrete distribution is the convolution of two binomial
distributions. In the second limit, we fix $m/n$, $k/n$ and $l/n$, and prove
the central limit theorem of $P_{n,m,k,l}$, of which the limit normal
distribution is derived by the three-term recurrence relation of $p(x)$ coming
from that of Racah polynomial. We also describe the asymptotic behavior of the
expectation and the variance beyond the central limit theorem. Based on these
asymptotic analysis, we discuss the degree of asymmetry of the tensor product
state $\Xi_{n,m|k,l}$. In the appendix, we discuss $\mathbf{q}$-analogue of
the computations on $P_{n,m,k,l}$, involving spherical functions on a Gelfand
pair of Chevalley groups over a finite field $\mathbb{F}_{\mathbf{q}}$ and
basic hypergeometric orthogonal polynomials.

###### Key words and phrases:

Group asymmetry, Schur-Weyl duality, zonal spherical functions, (basic)
hypergeometric orthogonal polynomials, Racah polynomials, three-term
recurrence relation, central limit theorem

M.H. is supported by Guangdong Provincial Key Laboratory under Grant
2019B121203002. A.H. is supported by JSPS KAKENHI Grant Number 19K03532. S.Y.
is supported by JSPS KAKENHI Grant Number 19K03399, and also by the JSPS
Bilateral Program “Elliptic algebras, vertex operators and link invariant”.

###### Contents

  1. 1 Introduction
  2. 2 Summary of the obtained result for the distribution $P_{n,m,k,l}$
    1. 2.1 Setting and mathematical background
      1. 2.1.1 Global notation
    2. 2.2 Explicit formulas of $P_{n,m,k,l}$
      1. 2.2.1 Presentation as summation of Hahn polynomials
      2. 2.2.2 Presentation via Racah polynomial
    3. 2.3 Asymptotics analysis
      1. 2.3.1 Asymptotic analysis of Type I limit
      2. 2.3.2 Asymptotic analysis of type II limit
      3. 2.3.3 Proofs of the law of large numbers and the central limit theorems
      4. 2.3.4 Asymptotic analysis beyond central limit theorem
  3. 3 Asymmetry of state $\Xi_{n,m|k,l}$
  4. 4 Formulas and properties of the distribution $P_{n,m,k,l}$
    1. 4.1 Hahn presentation
      1. 4.1.1 Projector formula
      2. 4.1.2 Zonal spherical function on Gelfand pair
      3. 4.1.3 Zonal spherical function on $(\mathfrak{S}_{n},\mathfrak{S}_{m}\times\mathfrak{S}_{n-m})$
      4. 4.1.4 Proof of Theorem 4.1.1
    2. 4.2 Racah presentation
    3. 4.3 ${}_{4}F_{3}$-presentation of cdf and recurrence relation of pmf
    4. 4.4 Relation of expectation and variance
    5. 4.5 Specialization of parameters
      1. 4.5.1 Case $MN=0$
      2. 4.5.2 Case $KL=0$
      3. 4.5.3 Remark on specialization in Askey scheme
  5. 5 Asymptotic analysis of the distribution $P_{n,m,k,l}$
    1. 5.1 Type I Limit $n\to\infty$ with $k,l$ and $m/n$ fixed
    2. 5.2 Type II Limit $n\to\infty$ with $m/n$, $k/n$ and $l/n$ fixed
      1. 5.2.1 Type II limit with $\alpha=\gamma=0$
      2. 5.2.2 Type II limit with $\beta\delta=0$
      3. 5.2.3 Generic Type II limit
    3. 5.3 Proof of law of large numbers
      1. 5.3.1 Recursion formula
      2. 5.3.2 Random variable $Z$ with bounded support
      3. 5.3.3 Proof of the law of large numbers
    4. 5.4 Proof of central limit theorem
      1. 5.4.1 Random variable $Y$ with non-compact support and estimation of tails
      2. 5.4.2 The limit distribution
    5. 5.5 Asymptotic analysis beyond central limit theorem
    6. 5.6 Further analysis in special case $\alpha\gamma=0$
  6. 6 Discussion
  7. A Single-shot bounds for cq-chanels
    1. A.1 Single-shot bound for general cq-channel
    2. A.2 Single-shot bound for pure state channel
    3. A.3 Single-shot bound for pure state covariant channel
  8. B Asymptotic expectation for $(n,m,k,l)=(2m,m,2l,l)$
    1. B.1 Calculating expectation for $n=2m$
    2. B.2 Asymptotic expectation for $(n,m,k,l)=(2m,m,2l,l)$
  9. C $\mathbf{q}$-analogue of the probability mass function $p(x)$
    1. C.0.1 Global $\mathbf{q}$-notation
    2. C.1 Gelfand pair $(\operatorname{GL}(n,\mathbb{F}_{\mathbf{q}}),P(m,n-m,\mathbb{F}_{\mathbf{q}}))$ and $\mathbf{q}$-Hahn polynomials
    3. C.2 $\mathbf{q}$-analogue of the pmf $p(x\,|\,n,m,k,l)$
    4. C.3 $\mathbf{q}$-Racah presentation

##  1\. Introduction

Symmetry is a fundamental property in quantum system, and its most powerful
tool is the theory of unitary representations of groups, which is one of the
most classical and most developed branches in representation theory. The
unitary representation theory has various connections with other areas of
mathematics and physics, and many of such connections are described by
explicit formulas of special functions. A typical example is the spherical
harmonic function of $\operatorname{SO}(3)$, which can be expressed by a
${}_{2}F_{1}$-hypergeometric series with orthogonal property. The classical
${}_{2}F_{1}$-hypergeometric orthogonal polynomials, in particular, Jacobi,
Laguerre, Bessel and Hermite polynomials, are ubiquitous, and appear in many
places of mathematical physics. Today, there are also known generalized
hypergeometric series with orthogonal property, but these polynomials are rare
compared to the classical ${}_{2}F_{1}$-polynomials.

In this paper, we address a new relation among unitary representation,
probability theory, and hypergeometric orthogonal polynomials. For this aim,
we focus on the asymmetry, which is the opposite concept of the invariance.
Although it is generally difficult to measure the degree of asymmetry, we can
measure it for a normalized vector in a Hilbert space $\mathcal{H}$, or a
state $\left|\psi\right>$ on $\mathcal{H}$. Consider a finite group $G$ and
its unitary representation $f$ on $\mathcal{H}$. Since the invariance of
$\left|\psi\right>\in\mathcal{H}$ under the $G$-action $f$ is expressed as
$f(g)\left|\psi\right>\\!\left<\psi\right|f(g)^{\dagger}=\left|\psi\right>\\!\left<\psi\right|$
for any $g\in G$, the asymmetry of $\left|\psi\right>$ is considered as the
degree of change caused by the $G$-action. The key concept for the asymmetry
is the averaged state with respect to the representation $f$, which is defined
as

$\displaystyle\operatorname{av}_{f}[\psi]:=\sum_{g\in
G}\frac{1}{\left|G\right|}f(g)\left|\psi\right>\\!\left<\psi\right|f(g)^{\dagger}.$
(1.0.1)

It is natural to employ the difference between
$\left|\psi\right>\\!\left<\psi\right|$ and $\operatorname{av}_{f}[\psi]$. As
a typical measure of the difference, we employ the relative entropy
$D(\rho\|\rho^{\prime}):=\operatorname{tr}\rho(\log\rho-\log\rho^{\prime})$
[48, (3.20)] although the paper [50] employs another measure. In this case,
the difference is simplified to the von Neumann entropy
$S(\rho):=-\operatorname{tr}\rho\log\rho$ of the averaged state
$\operatorname{av}_{f}[\psi]$. Explicitly, we have

$\displaystyle
S(\operatorname{av}_{f}[\psi]):=-\operatorname{tr}_{\mathcal{H}}\bigl{(}\operatorname{av}_{f}[\psi]\log\operatorname{av}_{f}[\psi]\bigr{)}.$
(1.0.2)

In fact, the quantity can be regarded as a mutual information between the
classical information $g$ and the quantum system $\mathcal{H}$ when $g$ is
considered as a random variable subject to the uniform distribution.

This paper investigates how an invariant state increases the amount of
asymmetry. As a typical case, we consider the $n$-fold tensor product
$\mathcal{H}=(\mathbb{C}^{2})^{\otimes n}$ of the qubit system
$\mathbb{C}^{2}=\mathbb{C}\left|0\right>\oplus\mathbb{C}\left|1\right>$, and
consider it as the representation space of the permutation group
$G=\mathfrak{S}_{n}$ acting by permutation of tensor factors as

$\displaystyle\pi(g)(\left|i_{1}\dotsm
i_{n}\right>):=\left|i_{g^{-1}(1)}\dotsm i_{g^{-1}(n)}\right>,\quad
g\in\mathfrak{S}_{n}.$ (1.0.3)

Consider the vector

$\displaystyle\left|1^{l}\,0^{k-l}\right>:=|\overbrace{1\dotsm
1}^{l}\,\overbrace{0\dotsm 0}^{k-l}\rangle=\left|1\right>^{\otimes
l}\otimes\left|0\right>^{\otimes(k-l)}\in(\mathbb{C}^{2})^{\otimes k}$ (1.0.4)

for non-negative integers $k$ and $l$. The averaged state
$\operatorname{av}_{\pi}\bigl{[}|1^{l}\,0^{k-l}\rangle\bigr{]}$ is given as
the mixture of $\binom{k}{l}$ orthogonal states, and its asymmetry
$S\bigl{(}\operatorname{av}_{\pi}[|1^{l},0^{k-l}\rangle]\bigr{)}$ is
$\log\binom{k}{l}$. When $k$ is large, it is approximated as
$kh(\frac{l}{k})$, where $h(x):=-x\log x-(1-x)\log(1-x)$ is the binary
entropy.

In contrast, for non-negative integers $N$ and $M$, the normalized vector

$\displaystyle\left|\Xi_{N+M,M}\right>:=\tbinom{N+M}{M}^{-1/2}\bigl{(}\left|1^{M}\,0^{N}\right>+\text{permuted
terms}\bigr{)}\in(\mathbb{C}^{2})^{\otimes(N+M)}$ (1.0.5)

is invariant under the action of the permutation group $\mathfrak{S}_{N+M}$,
and belongs to the symmetric subspace in $(\mathbb{C}^{2})^{\otimes(N+M)}$. In
quantum information theory, the vector $\left|\Xi_{N+M,M}\right>$ is often
called the Dicke state [15], and has been playing important roles: calculation
of entanglement measures [63, 26, 64, 62, 66], quantum communication and
quantum networking [38, 51] for example. Some typical Dicke states have been
realized in trapped atomic ions [33]. Recently, the multi-qubit Dicke state
with half-excitations has been employed to implement a scalable quantum search
based on Grover’s algorithm by using adiabatic techniques [34].

Now, we are interested in the asymmetry of the state given by the tensor
product of the above two states. Explicitly, for non-negative integers
$n,m,k,l$ and $M:=m-l$, $N:=n-m-k+l$, we consider

$\displaystyle\left|\Xi_{n,m|k,l}\right>:=\left|1^{l}\,0^{k-l}\right>\otimes\left|\Xi_{N+M,M}\right>\in(\mathbb{C}^{2})^{\otimes
n}.$ (1.0.6)

Note that the averaged von Neumann entropy of the corresponding state is not
less than that for the state $\left|1^{0},0^{k-l}\right>$. Indeed, since
$\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]=\sum_{g\in\mathfrak{S}_{n}}\pi(g)(\operatorname{av}_{\pi}[|1^{l},0^{k-l}\rangle]\otimes\left|\Xi_{N+M,M}\right>\\!\left<\Xi_{N+M,M}\right|)\pi(g)^{\dagger}$,
the concavity of the entropy function guarantees the relation

$\displaystyle S(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])\geq
S(\operatorname{av}_{\pi}[|1^{l},0^{k-l}\rangle]\otimes\left|\Xi_{N+M,M}\right>\\!\left<\Xi_{N+M,M}\right|)=S(\operatorname{av}_{\pi}[|1^{l},0^{k-l}\rangle]).$
(1.0.7)

To compute the asymmetry, we focus on the classical
$\operatorname{SU}(2)$-$\mathfrak{S}_{n}$ Schur-Weyl duality on the tensor
product system $\mathcal{H}=(\mathbb{C}^{2})^{\otimes n}$ depicted as

$\displaystyle\operatorname{SU}(2)\curvearrowright(\mathbb{C}^{2})^{\otimes
n}\curvearrowleft\mathfrak{S}_{n}.$ (1.0.8)

The Schur-Weyl duality claims that we have the following decomposition of
$(\mathbb{C}^{2})^{\otimes n}$ into irreducible representations (irreps for
short):

$\displaystyle(\mathbb{C}^{2})^{\otimes n}=\bigoplus_{x=0}^{\lfloor
n/2\rfloor}\mathcal{U}_{(n-x,x)}\boxtimes\mathcal{V}_{(n-x,x)}.$ (1.0.9)

Here $\boxtimes$ denotes the tensor product of linear spaces, equipped with
$\operatorname{SU}(2)$-action on the left and $\mathfrak{S}_{n}$-action on the
right factor, $\mathcal{V}_{(n-x,x)}$ denotes the $\mathfrak{S}_{n}$-irrep
corresponding to the partition $(n-x,x)$, and $\mathcal{U}_{(n-x,x)}$ denotes
the highest weight $\operatorname{SU}(2)$-irrep of dimension $n-2x+1$. Using
the hook formula (see [44, I.5, Example 2; I.7, (7.6)] for example), we can
compute the dimension of $\mathcal{V}_{(n-x,x)}$ as

$\displaystyle\dim\mathcal{V}_{(n-x,x)}=\frac{k!}{\prod_{\square\in(n-x,x)}h(\square)}=\frac{n-2x+1}{n-x+1}\binom{n}{x}=\binom{n}{x}-\binom{n}{x-1},$
(1.0.10)

where $\square\in(n-x,x)$ denotes a box in the Young diagram corresponding to
the partition $(n-x,x)$.

The highest weight $\operatorname{SU}(2)$-irrep of dimension $2j+1$ has the
standard basis $\\{\left|j,m\right>\mid m=-j,-j+1,\dotsc,j\\}$, where $j$ and
$m$ express the total and z-direction angular momenta, respectively. By this
basis and the completely mixed state
$\rho_{\text{mix},\mathcal{V}_{(n-x,x)}}=\frac{1}{\dim\mathcal{V}_{(n-x,x)}}\operatorname{\mathsf{id}}_{\mathcal{V}_{(n-x,x)}}$,
the averaged state $\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]$ is expressed as

$\displaystyle\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]=\sum_{x=0}^{\lfloor
n/2\rfloor}p(x\,|\,n,m,k,l)\left|\tfrac{n}{2}-x,\tfrac{n-2m}{2}\right>\\!\left<\tfrac{n}{2}-x,\tfrac{n-2m}{2}\right|\boxtimes\rho_{\text{mix},\mathcal{V}_{(n-x,x)}},$
(1.0.11)

where $p(x\,|\,n,m,k,l)$ is given by

$\displaystyle
p(x\,|\,n,m,k,l):=\left<\Xi_{n,m|k,l}\right|\mathsf{P}_{(n-x,x)}\left|\Xi_{n,m|k,l}\right>.$
(1.0.12)

with $\mathsf{P}_{(n-x,x)}$ denoting the projector onto the isotypical
component in the decomposition (1.0.9):

$\displaystyle\mathsf{P}_{(n-x,x)}\colon(\mathbb{C}^{2})^{\otimes
n}\relbar\joinrel\twoheadrightarrow\mathcal{U}_{(n-x,x)}\boxtimes\mathcal{V}_{(n-x,x)}.$
(1.0.13)

The non-negative value $p(x\,|\,n,m,k,l)$ is considered as a probability to
get outcome $x$, and we denote the corresponding probability distribution by
$P_{n,m,k,l}$. We also denote by $X$ a discrete random variable distributed by
the probability mass function (pmf for short) $p(x\,|\,n,m,k,l)$.

Then, the von Neumann entropy is given by

$\displaystyle
S\bigl{(}\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]\bigr{)}=\sum_{x}p(x\,|\,n,m,k,l)\bigl{(}\log\dim\mathcal{V}_{(n-x,x)}-\log
p(x\,|\,n,m,k,l)\bigr{)}.$ (1.0.14)

Furthermore, we can also consider the number of approximately orthogonal
elements among the states
$\\{\pi(g)\left|\Xi_{n,m|k,l}\right>\\}_{g\in\mathfrak{S}_{n}}$, which is
characterized by a slightly different quantity. As explained in §3, the number
of distinguishable elements with error probability $\varepsilon>0$ is
approximated by the following value:

$\displaystyle\begin{split}&\overline{H}_{s}^{\varepsilon}(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])\\\
&:=\max\bigl{\\{}\lambda\mid
P_{n,m,k,l}[\\{x\mid\log\dim\mathcal{V}_{(n-x,x)}-\log
p(x\,|\,n,m,k,l)\leq\lambda\\}]\leq\epsilon\bigr{\\}}.\end{split}$ (1.0.15)

In fact, it is known that this type of permutation enhances the number of
distinguishable states in the entanglement-assisted communication [61].

The aim of this paper is the calculation of the distribution $P_{n,m,k,l}$.
Surprisingly, our calculation reveals interesting relations between the
distribution $P_{n,m,k,l}$ and hypergeometric orthogonal polynomials. That is,
we discover two types of expressions of the pmf $p(x\,|\,n,m,k,l)$ based on
Hahn and Racah polynomials, which are ${}_{3}F_{2}$\- and
${}_{4}F_{3}$-hypergeometric orthogonal polynomials, respectively. The first
expression is derived by the representation theory of $\mathfrak{S}_{n}$.
Precisely speaking, we use zonal spherical functions $\omega_{(n-x,x)}$ of the
Gelfand pair $(\mathfrak{S}_{n},\mathfrak{S}_{m}\times\mathfrak{S}_{n-m})$ to
expand $p(x\,|\,n,m,k,l)$. By [11, 12], the values of $\omega_{(n-x,x)}$ are
given by Hahn polynomials, and thus we can express $p(x\,|\,n,m,k,l)$ as a
summation of Hahn polynomials, which we call Hahn presentation. See Theorem
2.2.1 for the precise statement. In Figure 1.0.1, we show plots of the pmf
$p(x\,|\,n,m,k,l)$ calculated by Hahn presentation.

Figure 1.0.1. Plots of $p(x\,|\,n,m,k,l)$ with $(n,m,k,l)=(100,30,40,20)$ in
left and $(n,m,k,l)=(100,40,60,30)$ in right.

Hahn presentation gives an explicit description of $P_{n,m,k,l}$, but since
each Hahn polynomial is a summation, the presentation itself is a double sum,
and not suitable for further analysis. To avoid such a summation, we focus on
Racah polynomial, which is a ${}_{4}F_{3}$-hypergeometric orthogonal
polynomial sitting in the top line of the Askey scheme (see Figure 2.1.1 in
later §2.1). Due to the highly complicated structure, it is totally nontrivial
that Racah polynomial has a connection with the probability distribution
$P_{n,m,k,l}$. After a careful observation, we obtain an expression of the pmf
$p(x\,|\,n,m,k,l)$ by a single Racah polynomial, which we call Racah
presentation. See Theorem 2.2.2 for the precise statement. It is derived from
Hahn presentation by somewhat straightforward but nontrivial calculation. In
fact, in the course of the derivation, we obtain a new summation formula of
hypergeometric series (see Corollary 2.2.3). Furthermore, we derive a notable
recursion (Lemma 2.2.6) among the adjacent three probabilities
$p(x+i\,|\,n,m,k,l)$ for $i=-1,0,1$ from the three-term recurrence relation of
Racah polynomial. We also express the cumulative distribution function (cdf
for short) as a terminating ${}_{4}F_{3}$-hypergeometric series (Theorem
2.2.4), which brings another new summation formula of hypergeometric series.
In Figure 1.0.2, we show some plots of the cdf $P_{n,m,k,l}[X\leq x]$.

Figure 1.0.2. Plots of the cdf $P_{n,m,k,l}[X\leq x]$ with
$(m/n,k/n,l/n)=(0.4,0.6,0.3)$ fixed and $n=100$ (left), $1000$ (middle),
$10000$ (right).

However, the obtained hypergeometric expressions are still complicated for
intuitive understanding of our distribution. To get a broad trend of
$S\bigl{(}\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]\bigr{)}$ and
$\overline{H}_{s}^{\varepsilon}(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])$, we
consider the asymptotic behavior when $n$ is sufficiently large. That is, we
address their approximations, which need some asymptotic approximation of the
distribution $P_{n,m,k,l}$ in the limit $n\to\infty$. Several studies [58, 57,
9, 10] discussed the asymptotic behavior of the distribution related to
representation theory, although their distributions do not depend on a state
on the quantum system while our distribution depends on a state on the quantum
system. In the context of quantum information, various studies [49, 1, 2, 21,
27, 37, 36, 23] discussed the central limit theorem related to unitary
representation in the sense of quantum Gaussian state dependently of a state
on the quantum system. However, since their analysis assume the independent
and identical tensor product structure, these existing results cannot be
applied to our setting.

This paper focuses on two types of asymptotics, i.e., Type I and Type II
limits. Type I limit assumes that $k$ and $l$ are fixed and $m$ is linear for
$n$, i.e., $m=\xi n$ with fixed ratio $\xi$, under $n\to\infty$. Type II limit
does that $k,l$ and $m$ are linear for $n$ under $n\to\infty$.

In Type I limit, using Hahn presentation (Theorem 2.2.1), we show that the
distribution $P_{n,m,k,l}$ converges to a convolution $P_{R|\xi;k,l}$ of two
binomial distributions whose expectation is $u:=(k-l)\xi+l(1-\xi)$. See
Theorem 2.3.1 for the precise statement. Returning to the estimation of the
asymmetry, we will show in §3, (3.0.2) that the von Neumann entropy
$S\bigl{(}\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]\bigr{)}$ is approximated to

$\displaystyle a(n,\xi,k,l):=u\log
n+u-\log\sqrt{2\pi}+\sum_{x=0}^{k}q(x\,|\,\xi;k,l)\bigl{(}(x+\tfrac{1}{2})\log
x-\log q(x\,|\,\xi;k,l)\bigr{)}$ (1.0.16)

up to $o(n^{0})$, where $q(x\,|\,\xi;k,l)$ denotes the pmf of the limit
distribution $P_{R|\xi;k,l}$. In particular, the von Neumann entropy
$S\bigl{(}\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]\bigr{)}$ behaves as $u\log
n+o(\log n)$ in Type I limit. In Figure 1.0.3 we show an example of this
approximation. Also, we can approximately calculate the quantity
$\overline{H}_{s}^{\varepsilon}(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])$ using
the limit distribution $P_{R|\xi;k,l}$. The same scaling $\log n$ appears in
the bosonic coherent channel when the total power is fixed and the number $n$
of modes increases [24].

Figure 1.0.3. Plots of
$S\bigl{(}\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]\bigr{)}/\log n$ (blue dots)
and the approximation $a(n,\xi,k,l)/\log n$ (red dots) for
$(\xi=m/n,k,l)=(0.5,2,1)$. The horizontal axis is $\log_{10}n$, and the black
line shows the limit value $u:=(k-l)\xi+l(1-\xi)=1$.

In Type II limit, i.e., the limit $n\to\infty$ with $k,l$ and $m$ linear for
$n$, we need to treat many ratios and parameters, which are summarized in
Table 1.0.1.

fixed ratio set 1 | $\alpha=\frac{l}{n}$ |  $\beta=\frac{m-l}{n}$ $\gamma=\frac{k-l}{n}$ $\delta=\frac{n-m-k+l}{n}$
---|---|---
fixed ratio set 2 | $\alpha=\frac{l}{n}$ |  $\xi=\frac{m}{n}$ -l $\kappa=\frac{k}{n}$
limit pdf parameters | $\mu:=\frac{1-\sqrt{D}}{2}$ |  $\nu:=\frac{1+\sqrt{D}}{2}$ $\sigma^{2}:=\frac{(\alpha+\gamma)\beta\delta}{D}$
| $D:=1-4\eta$ | $\eta:=\alpha\gamma+\alpha\delta+\beta\gamma$
Table 1.0.1. Ratios and parameters for Type II limit

Using the recurrence relation of the pmf (Lemma 2.2.6) mentioned above, we
derive the law of large numbers (Theorem 2.3.2) and the central limit Theorem
2.3.4. That is, when $X$ is subject to the distribution $P_{n,m,k,l}$, the
variable $\frac{X}{n}$ converges to $\mu$ and the variable
$\frac{X-n\mu}{\sqrt{n}\sigma}$ asymptotically obeys the standard Gaussian
distribution, where $\mu$ and $\sigma$ are given in Table 1.0.1. The previous
Figure 1.0.2 displays the law of large numbers. As for the central limit
theorem, we show in Figure 1.0.4 some examples comparing the distribution
$P_{n,m,k,l}$ and the limit normal distribution. There we fix
$(m/n,k/n,l/n)=(0.4,0.6,0.3)$, in which case we have $\mu=0.3$ and
$\sigma=0.3354...$.

This result means that
$S\bigl{(}\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]\bigr{)}$ is approximated to
$h(\mu)n+o(n)$ because $\log\dim\mathcal{V}_{(x,n-x)}$ is approximated to
$h(\frac{x}{n})n+O(\log n)$ when $x$ behaves linearly to $n$. Also,
$\overline{H}_{s}^{\varepsilon}(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])$ can
be approximately calculated up to the second order $\sqrt{n}$.

Figure 1.0.4. Plots of $P_{n,m,k,l}[X=x]$ by blue dots and the limit normal
distribution by red lines with $(m/n,k/n,l/n)=(0.4,0.6,0.3)$ fixed and $n=100$
(left), $1000$ (middle), $10000$ (right).

Our proofs of the law of large numbers and the central limit theorem are not
straightforward. One can naively expect that a sequence satisfying a three-
term recurrence relation is approximated by a solution of the asymptotic
differential equation. However, there are two problems to justify this
approximation in the case of probability distribution. The first problem is
the evaluation of the tail probability. That is, for the above justification,
it is needed to be shown that the tail probability approaches zero. Further,
as the second problem, even though the limiting distribution exists, we cannot
say that the cumulative distribution function is differentiable, in general.
To resolve the second problem, we convert the three-term recursion relation to
a difference equation for the cumulative distribution function, which leads to
the desired Gaussian distribution.

The remaining part of this paper is organized as follows. §2 summarizes the
formulas and results on the distribution $P_{n,m,k,l}$ including its
asymptotic behavior, which will be proved in the later sections §4 and §5.
Using these results, §3 discusses the detail analysis on the degree of the
asymmetry of the state $\Xi_{n,m|k,l}$, which shows further impact to the area
of quantum information. §4 is devoted to the technical analysis for the
distribution $P_{n,m,k,l}$ with finite parameters $n,m,k,l$. §5 proceeds to
the asymptotic analysis for the distribution $P_{n,m,k,l}$ with Type I and
Type II limits.

##  2\. Summary of the obtained result for the distribution $P_{n,m,k,l}$

###  2.1. Setting and mathematical background

Let us mathematically rephrase the physical explanation in §1 along the well-
known connection between representation theory and probability theory.
Consider a unitary representation of a group $G$ with the representation space
$\mathcal{H}$, which is assumed to be of finite dimension for simplicity. The
irreducible decomposition is encoded by a family of projectors
$\\{\mathsf{P}_{x}\\}_{x\in J}$, each of which projects $\mathcal{H}$ to the
isotypical component of the $G$-irrep parametrized by $x\in J$. The family
$\\{\mathsf{P}_{x}\\}_{x\in J}$ is a positive operator-valued measure (POVM
for short), and for each normalized vector $\left|\psi\right>\in\mathcal{H}$,
we have a discrete probability distribution on $J$ whose pmf is given by

$\displaystyle\left<\psi\right|\mathsf{P}_{x}\left|\psi\right>.$ (2.1.1)

Here we used bra-ket notation. This principle gives us a bunch of discrete
probability distributions.

The purpose of this note is, following the above principle, to introduce and
study a new interesting discrete probability distribution with the motivation
explained in §1. The representation space is set to be the $n$-fold tensor
space $\mathcal{H}=(\mathbb{C}^{2})^{\otimes n}$ of
$\mathbb{C}^{2}=\mathbb{C}\left|0\right>\oplus\mathbb{C}\left|1\right>$
equipped with the standard Hermitian form. The choice of normalized vector
$\left|\psi\right>\in\mathcal{H}$ is the vector $\left|\Xi_{n,m|k,l}\right>$
in (1.0.6). Let us write it again as:

######  Definition 2.1.1.

Let $n,m,k$ and $l$ be integers satisfying

$\displaystyle 0\leq m,k,l\leq n,\quad k-l,m-l,n-m-k+l\geq 0.$ (2.1.2)

For such $n,m,k$ and $l$, we define
$\left|\Xi_{n,m|k,l}\right>\in(\mathbb{C}^{2})^{\otimes n}$ by

$\displaystyle\left|\Xi_{n,m|k,l}\right>:=\left|1^{l}\,0^{k-l}\right>\otimes\left|\Xi_{N+M,M}\right>\in(\mathbb{C}^{2})^{\otimes
n}.$ (2.1.3)

The concerned POVM is given by the projectors
$\\{\mathsf{P}_{(n-x,x)}\\}_{x=0}^{\lfloor n/2\rfloor}$ in (1.0.13). The
corresponding pmf (1.0.12) is the main object to study in this note. Let us
write it again as:

######  Definition 2.1.2.

Let $n,m,k,l$ be integers satisfying (2.1.2), and $x$ be an integer with
$0\leq x\leq\lfloor n/2\rfloor$. For these parameters, we define

$\displaystyle
p(x)=p(x\,|\,n,m,k,l):=\left<\Xi_{n,m|k,l}\right|\mathsf{P}_{(n-x,x)}\left|\Xi_{n,m|k,l}\right>.$
(2.1.4)

The setting implies a few but basic properties of $p(x)$. First, we can switch
$\left|0\right>\leftrightarrow\left|1\right>$ in the system, and $p(x)$
inherits the symmetry

$\displaystyle p(x\,|\,n,m,k,l)=p(x\,|\,n,n-m,k,k-l).$ (2.1.5)

Second, since $p(x)$ comes from the decomposition (1.0.9) of unitary
representation, $p(x)$ is a probability mass function for a discrete
probability distribution $\\{0,1,\dotsc,\lfloor
n/2\rfloor\\}\to\mathbb{R}_{\geq 0}$. Now, the following definition makes
sense.

######  Definition 2.1.3.

We denote by $P_{n,m,k,l}$ the discrete probability distribution obtained by
the above argument. We call $p(x)$ in Definition 2.1.2 _the_ probability mass
function, and denote by $X$ a discrete random variable distributed by $p(x)$.

Our distribution enjoys unexpectedly rich properties in mathematical
viewpoints such as representation theory, hypergeometric functions, orthogonal
polynomials, asymptotic analysis and $q$-analysis. Let us display one of its
features: The pmf $p(x)=p(x\,|\,n,m,k,l)$ has the expression

$\displaystyle
p(x)=\binom{n-k}{m-l}\frac{\binom{n}{x}}{\binom{n}{m}}\frac{n-2x+1}{n-x+1}{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mux-n-1{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}$ (2.1.6)

with $M:=m-l$ and $N:=n-m-k+l$ (Theorem 2.2.2). Here ${}_{4}F_{3}$ denotes the
generalized hypergeometric series (see (2.1.8) for the notation).

The ${}_{4}F_{3}$-hypergeometric series appearing in (2.1.6) is a
distinguished one, belonging to the family of Racah polynomials. Let us give a
brief introduction to Racah polynomial, some of which are quotes from [39,
Foreword, Preface]:

  * •

Racah polynomial is a ${}_{4}F_{3}$-hypergeometric orthogonal polynomial,
introduced by Askey and Wilson [5] as a generalization of Racah coefficient or
Wigner’s 6-j symbol. By this work, we can now understand various formulas of
6-j symbols in a uniform way from properties of Racah polynomials such as
orthogonality, recurrence relations and generating functions.

  * •

Classification theory of orthogonal polynomials has a long history, and we
refer [39, Preface] for the precise account. Here let us only mention the work
of Leonard [42], which shows that a finite system of orthogonal polynomials of
which the dual is again a finite system of orthogonal polynomials, must be
($q$-)Racah or one of its limit cases.

  * •

As for classification theory, we should mention the work of Askey and Wilson
[6], which is a first attempt to unify the continuous and discrete classical
orthogonal polynomials. The variety of orthogonal polynomials treated there is
depicted in the Askey scheme of hypergeometric orthogonal polynomials, for
which we explain in the next item. They showed that all the discrete classical
orthogonal polynomials can be obtained as limiting cases of Racah polynomials

  * •

In Figure 2.1.1, we show the modified version [39, p.183] of the Askey scheme.
See [43, p.XXXVII] for the original figure. Racah polynomial sits in the top
line, having the most four parameters. Thus, it can be considered as one of
the “mother orthogonal polynomials”.

We refer [39] for a comprehensive explanation on the theory of hypergeometric
orthogonal polynomials.

Figure 2.1.1. Askey scheme of hypergeometric orthogonal polynomials

The hypergeometric orthogonal polynomials in “lower level” of the Askey
scheme, in particular, Jacobi, Laguerre, Bessel and Hermite polynomials, are
ubiquitous, and appear in many places of mathematics and physics. Compared to
them, the Wilson and Racah polynomials sitting in the top line are rare. Going
back to our distribution, the appearance of Racah polynomial in (2.1.6) is
very surprising, in particular in the context of probability theory.

Let us mention another remarkable property of our distribution: The cumulative
distribution function (cdf for short) $\sum_{u=0}^{x}p(u)$ is also expressed
as a ${}_{4}F_{3}$-hypergeometric series (Theorem 2.2.4):

$\displaystyle\sum_{u=0}^{x}p(u)=\binom{n-k}{m-l}\frac{\binom{n}{x}}{\binom{n}{m}}{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mux-n{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}.$ (2.1.7)

These strong ${}_{4}F_{3}$-nature is peculiar to our distribution, and such a
distribution has not been discovered to the best of out knowledge.

We will explain several presentations of the pmf $p(x)$ and related results in
§2.2. These are not only interesting in their own, but also very useful for
the analysis of our distribution. As will be explained in §2.3, we study the
asymptotic behavior of the distribution in the limit $n\to\infty$. One of the
main results on the asymptotic analysis is a certain form of central limit
theorem (Theorem 2.3.4), which is proved using the recurrence relation of the
pmf $p(x)$ (Lemma 2.2.6) obtained from Racah presentation (2.1.6).

The following §2.2 and §2.3 are summary of our main results.

  * •

§2.2 explains various properties of the distribution $P_{n,m,k,l}$ for fixed
$n$, in particular, explicit formulas in terms of hypergeometric orthogonal
polynomials.

  * •

§2.3 explains the asymptotic analysis of the distribution $P_{n,m,k,l}$ in the
limit $n\to\infty$.

####  2.1.1. Global notation

Let us explain the notation used throughout the text.

  1. (1)

We denote by $\mathbb{N}:=\mathbb{Z}_{\geq 0}=\\{0,1,2,\dotsc\\}$ the set of
non-negative integers.

  2. (2)

We denote by $\left|S\right|$ the number of elements of a finite set $S$.

  3. (3)

Notation for numerical functions.

    1. (a)

For $a,b\in\mathbb{Z}$, we denote $a\vee b:=\max(a,b)$ and $a\wedge
b:=\min(a,b)$.

    2. (b)

For $r\in\mathbb{N}$ and a letter $a$, we denote the raising factorial by

$\displaystyle(a)_{r}:=a(a+1)\dotsm(a+r-1).$

Note that $(1)_{r}=r!$. For commuting letters $a,b,\dotsc,c$, we set

$\displaystyle(a,b,\dotsc,c)_{r}:=(a)_{r}(b)_{r}\dotsm(c)_{r}.$

    3. (c)

For $r\in\mathbb{N}$ and a letter $a$, we denote the falling factorial by

$\displaystyle a^{\downarrow r}:=a(a-1)\dotsm(a-r+1).$

For commuting letters $a,b,\dotsc,c$, we set

$\displaystyle(a,b,\dotsc,c)^{\downarrow r}:=a^{\downarrow r}b^{\downarrow
r}\dotsm c^{\downarrow r}.$

    4. (d)

For $r\in\mathbb{Z}$ and a letter $a$, we denote the binomial coefficient by

$\displaystyle\binom{a}{r}:=\begin{cases}a^{\downarrow r}/r^{\downarrow
r}=a(a-1)\dotsm(a-r+1)/r!&(r\geq 0)\\\ 0&(r<0)\end{cases}.$

In particular, for $s\in\mathbb{N}$ and $r\in\mathbb{Z}$ with $r<0$ or $r>s$,
we have $\binom{s}{r}=0$.

    5. (e)

The generalized hypergeometric series with parameters $a_{1},\dotsc,a_{r}$ and
$b_{1},\dotsc,b_{s}$ is denoted by

$\displaystyle{}_{r}F_{s}{\left[\genfrac{.}{.}{0.0pt}{}{a_{1}{\mathchar
44\relax}\mskip 8.0mu\dotsc{\mathchar 44\relax}\mskip
8.0mua_{r}}{b_{1}{\mathchar 44\relax}\mskip 8.0mu\dotsc{\mathchar
44\relax}\mskip 8.0mub_{s}};z\right]}:=\sum_{i\geq
0}\frac{(a_{1},\dotsc,a_{r})_{i}}{(1,b_{1},\dotsc,b_{s})_{i}}z^{i}.$ (2.1.8)

  4. (4)

For a group $G$, we denote its unit by $e\in G$.

  5. (5)

Notation for linear spaces.

    1. (a)

Linear spaces are defined over $\mathbb{C}$ unless otherwise stated.

    2. (b)

We use bra-ket notation, as already did in the introduction. In particular,
the paring $\left<\cdot\,\middle|\,\cdot\right>$ is anti-linear in the first
variable (bra), and linear in the second variable (ket).

For brevity, we denote the range (2.1.2) of the parameters $n,m,k,l$ by

$\displaystyle\mathfrak{N}:=\\{(n,m,k,l)\in\mathbb{Z}_{\geq 0}^{4}\mid m,k\leq
n,\ m+k-n\leq l\leq m\wedge k\\}.$ (2.1.9)

Hereafter we always assume that $(n,m,k,l)$ belongs to $\mathfrak{N}$ unless
otherwise stated.

###  2.2. Explicit formulas of $P_{n,m,k,l}$

Our distribution $P_{n,m,k,l}$ is defined abstractly, i.e., through the
irreducible decomposition of the unitary representation
$(\mathbb{C}^{2})^{\otimes n}$, and it is non-trivial to find properties of
$p(x)$ as a function of $x$. What we should do first is to obtain some
explicit description of $p(x)$, which turn out to be very non-trivial by
virtue of the choice of our vector $\left|\Xi_{n,m|k,l}\right>$ . The first
group of our main results is concerned with this description problem:

  * •

Presentation of $p(x)$ as a summation of Hahn polynomials (Theorem 2.2.1).

  * •

Presentation of $p(x)$ by a single Racah polynomial (Theorem 2.2.2).

  * •

Formula of the cdf $P_{n,m,k,l}[X\leq x]$ via terminating
${}_{4}F_{3}$-hypergeometric series (Theorem 2.2.4).

The details will be explained in §2.2.1 and §2.2.2.

Theorems 2.2.2 and 2.2.4 claim that both the pmf and cdf are given by
terminating ${}_{4}F_{3}$-hypergeometric series. In this sense we may say that
our distribution $P_{n,m,k,l}$ has a strong ${}_{4}F_{3}$-hypergeometric
nature. To the best of our knowledge, such a distribution has never appeared
in literature.

In the course of the derivation of these theorems, we also obtain:

  * •

(Seemingly) new summation formulas of terminating ${}_{3}F_{2}$\- and
${}_{4}F_{3}$-hypergeometric series (Corollaries 2.2.3 and 2.2.5).

We could not find or derive them from known summation formulas in the
literature such as [53] and [20]. Let us also mention one of the interesting
features: In Corollary 2.2.5, the case $x=m$ is equivalent to the identity
$\sum_{x=0}^{m}p(x)=1$. The latter identity is is trivial in
representation/probability theoretic viewpoint, but gives a non-trivial
summation formula of hypergeometric series.

####  2.2.1. Presentation as summation of Hahn polynomials

In the main text, we give several expressions of the pmf

$\displaystyle
p(x)=p(x\,|\,n,m,k,l):=\left<\Xi_{n,m|k,l}\right|\mathsf{P}_{(n-x,x)}\left|\Xi_{n,m|k,l}\right>.$
(2.2.1)

The first expression is derived in §4.1 with the help of the representation
theory of $\mathfrak{S}_{n}$. The result is:

######  Theorem 2.2.1 (Hahn presentation, Theorem 4.1.1).

Assume $m\leq n-m$. Then, we have

$\displaystyle
p(x)=\frac{\binom{n}{x}}{\binom{n}{m}}\,\frac{n-2x+1}{n-x+1}\,\sum_{i=0}^{M\wedge
N}\binom{M}{i}\binom{N}{i}\,\omega_{(n-x,x)}(i),$ (2.2.2)

where $M:=m-l$, $N:=n-m-k+l$ and

$\displaystyle\omega_{(n-x,x)}(i)=\binom{m}{i}^{-1}\binom{n-m}{i}^{-1}\sum_{r=0}^{i\wedge
x}(-1)^{r}\binom{x}{r}\binom{m-x}{i-r}\binom{n-m-x}{i-r}.$ (2.2.3)

See Figure 1.0.1 for some plots of the pmf $p(x)$.

The function $\omega_{(n-x,x)}$ in Theorem 2.2.1 is in fact the zonal
spherical function on the Gelfand pair
$(G,K)=(\mathfrak{S}_{n},\mathfrak{S}_{m}\times\mathfrak{S}_{n-m})$ belonging
to the irreducible character $\chi^{(n-x,x)}$, and $\omega_{(n-x,x)}(i)$ is
the value at the double coset $KgK$ whose distance is $i$. See §4.1.3 for the
detail.

Let us briefly explain the naming “Hahn presentation” of Theorem 2.2.1. The
details will be explained in §4.1.3, Remark 4.1.3. Regarding the binomial
coefficient $\binom{x}{r}$ as a polynomial of $x$, we can consider the
alternating sum

$\displaystyle E_{i}(x):=\sum_{r=0}^{i\wedge
x}(-1)^{r}\binom{x}{r}\binom{m-x}{i-r}\binom{n-m-x}{i-r}$ (2.2.4)

in (2.2.3) as a polynomial of $x$. The polynomial $E_{i}(x)$ is called
Eberlein polynomial in [11], and can be rewritten by a
${}_{3}F_{2}$-hypergeometric series [12], [7, III.2]. Using hypergeometric
notation in §2.1.1, we have

$\displaystyle\omega_{(n-x,x)}(i)=\frac{E_{i}(x)}{\binom{m}{i}\binom{n-m}{i}}={}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-i{\mathchar
44\relax}\mskip 8.0mu-x{\mathchar 44\relax}\mskip 8.0mux-n-1}{-m{\mathchar
44\relax}\mskip 8.0mum-n};1\right]}:=\sum_{r\geq
0}\frac{(-i,-x,x-n-1)_{r}}{(1,-m,m-n)_{r}}.$ (2.2.5)

The above ${}_{3}F_{2}$-hypergeometric polynomial $E_{i}(x)$ is equal to Hahn
and dual Hahn polynomials [12], [7, III.2], which will be also reviewed in
Remark 4.1.3. Using the symbols in [39], we have

$\displaystyle\tbinom{m}{i}^{-1}\tbinom{n-m}{i}^{-1}E_{i}(x)=Q_{x}(i;m-n-1,-m-1,m)=R_{i}(\lambda(x);m-n-1,-m-1,m),$
(2.2.6)

where $Q_{x}(i)$ denotes Hahn polynomial of degree $x$ and variable $i$, and
$R_{i}(\lambda(x))$ denotes dual Hahn polynomial of degree $i$ and variable
$\lambda(x):=x(x-n-1)$. The families of Hahn and dual Hahn polynomials enjoy
orthogonality, sitting in the Askey scheme of hypergeometric orthogonal
polynomials (Figure 2.1.1).

####  2.2.2. Presentation via Racah polynomial

Hahn presentation of the pmf $p(x)$ in Theorem 2.2.1 is a double sum, and is
not useful for analysis. For example, it is difficult to check the trivial
identity $\sum_{x=0}^{\lfloor n/2\rfloor}p(x)=1$ using the double sum formula.
By some hypergeometric calculation, we can rewrite this double sum by a single
sum. The result is:

######  Theorem 2.2.2 (Racah presentation, Theorem 4.2.1).

Assume $m\leq n-m$ and $x\leq m$. Then, we have

$\displaystyle
p(x\,|\,n,m,k,l)=\binom{n-k}{m-l}\frac{\binom{n}{x}}{\binom{n}{m}}\frac{n-2x+1}{n-x+1}{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mux-n-1{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}$ (2.2.7)

with $M:=m-l$ and $N:=n-m-k+l$.

Note that the symmetry (2.1.5) of $p(x)$ is manifest in this presentation,
since the map $(n,m,k,l)\mapsto(n,n-m,k,k-l)$ is equivalent to
$(m,n-m,M,N)\mapsto(n-m,m,N,M)$.

The terminating ${}_{4}F_{3}$-hypergeometric series appearing in Theorem 2.2.2
is a Racah polynomial, which is the origin of the naming “Racah presentation”.
Using the symbol in [39] (see Remark 4.2.2), we have

$\displaystyle{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mux-n-1{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}=R_{x}(\lambda(M);-m-1,-n+m-1,-n+k-1,0),$
(2.2.8)

where $R_{x}(\lambda(M))$ denotes Racah polynomial of degree $x$ and variable
$\lambda(M):=-MN$. Similarly as Hahn and dual Hahn polynomials, Racah
polynomials also form a family of hypergeometric orthogonal polynomials, and
sit in the top line of Askey scheme (Figure 2.1.1). See [39] for more
information, some of which will be recalled in §4.2.

Our proof of Theorem 2.2.2 is just to rewrite the right hand side of (2.2.2)
in Theorem 2.2.1 using the hypergeometric presentation (2.2.5) of
$\omega_{(n-x,x)}(i)$. However, it is a non-trivial calculation, and yields
the following identity of hypergeometric series:

######  Corollary 2.2.3 (Equivalent to Theorem 2.2.2).

Assume $m\leq n-m$ and $x\leq m$. Then, we have

$\displaystyle\begin{split}\sum_{i=0}^{M\wedge
N}\binom{M}{i}\binom{N}{i}\,&{}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-i{\mathchar
44\relax}\mskip 8.0mu-x{\mathchar 44\relax}\mskip 8.0mux-n-1}{-m{\mathchar
44\relax}\mskip 8.0mum-n};1\right]}\\\
=\binom{n-k}{m-l}\,&{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mux-n-1{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}\end{split}$ (2.2.9)

with $M:=m-l$ and $N:=n-m-k+l$.

The identity (2.2.9) seems to be a new kind of hypergeometric summation
formula, on which we give some comments in the next paragraph.

Racah presentation of $p(x)$ in Theorem 2.2.2 is more suitable for analysis
than Hahn presentation in Theorem 2.2.1 from several points of view. One of
them is the simpleness as already mentioned. Another is that we may use
various formulas on Racah polynomials as orthogonal polynomials such as
orthogonal relations, generating functions, and recurrence relations. We have
one more expectation: Note that Racah polynomials are balanced hypergeometric
series, i.e., the parameters satisfy
$(-x)+(x-n-1)+(-M)+(-N)+1=(-m)+(m-n)+(-M-N)$. There often happens that a
summation of such balanced series is equal to another hypergeometric series,
and the resulting formula is called a (hypergeometric) summation formula. See
[3, Chap. 2,3] and [53, §2.3] for the detail, and also [20, Chap. 2,3] for
explanation in the spirit of basic ($q$-analogue of) hypergeometric series. A
prototypical example of summation formula is the Chu-Vandermonde formula of
the Gauss hypergeometric series (see [20, (1.2.9)] for example):

$\displaystyle{}_{2}F_{1}{\left[\genfrac{.}{.}{0.0pt}{}{-a{\mathchar
44\relax}\mskip
8.0mub}{c};1\right]}=\frac{(c-b)_{a}}{(c)_{a}}\quad(a=0,1,\dotsc)$ (2.2.10)

for any $b\in\mathbb{C}$ and $c\in\mathbb{C}\setminus\\{0,-1,-2,\dotsc\\}$.
Rewrite $A:=a-1+c$ and $B:=-b$, and assume $A,B$ are non-negative integers.
Then, (2.2.10) is equivalent to the following classical equality:

$\displaystyle\sum_{r\geq 0}\binom{A}{a-r}\binom{B}{r}=\binom{A+B}{a}.$
(2.2.11)

The Chu-Vandermonde formulas (2.2.10) and (2.2.11) will be used repeatedly in
the main text.

We may expect that such summation formulas are useful for the analysis of
$p(x)$. As an illustration, we will show in Theorem 4.3.1 the identity
$\sum_{x=0}^{m}p(x)=1$ by direct calculation using Racah presentation of
$p(x)$. Here we note that $p(x)=0$ for $x>m$ by (2.2.18). It is indeed non-
trivial from the hypergeometric point of view. More generally, we will show in
the same theorem that the cdf $\sum_{u=0}^{x}p(u)$ can be expressed by a
terminating ${}_{4}F_{3}$-hypergeometric series. The statements is as follows.

######  Theorem 2.2.4 (Theorem 4.3.1).

Assume $m\leq n-m$, and denote $M:=m-l$ and $N:=n-k-m+l$. Then, for
$x=0,1,\dotsc,m$, the cdf is given by

$\displaystyle P_{n,m,k,l}[X\leq
x]=\binom{n-k}{m-l}\frac{\binom{n}{x}}{\binom{n}{m}}{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mux-n{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}.$ (2.2.12)

See Figure 1.0.2 for some plots of the cdf computed by the above formula. By
Theorem 2.2.2, the identity (2.2.12) is equivalent to the following statement.

######  Corollary 2.2.5 (Equivalent to Theorem 2.2.4).

Assume $m\leq n-m$ and $x\leq m$. Then, we have

$\displaystyle\begin{split}\sum_{u=0}^{x}\binom{n}{u}\frac{n-2u+1}{n-u+1}&{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-u{\mathchar
44\relax}\mskip 8.0muu-n-1{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}\\\
=\binom{n}{x}&{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mux-n{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}.\end{split}$ (2.2.13)

The identity (2.2.13) seems to be another new kind of hypergeometric summation
formula (recall (2.2.9)). Specializing the parameters, we can obtain some
interesting formulas. For example, setting $M=0$ or $N=0$ in (2.2.13), we
obtain

$\displaystyle\sum_{u=0}^{x}\binom{n}{u}\frac{n-2u+1}{n-u+1}=\binom{n}{x},$
(2.2.14)

which is checked directly by
$\binom{n}{u}\frac{n-2u+1}{n-u+1}=\binom{n}{u}-\binom{n}{u-1}$. See also
§4.5.2, Remark 4.5.6, where the case $l=0$ or $k-l=0$ is treated.

In §§4.3–4.5, we derive various results on the properties of $p(x)$ which will
be used in the study of asymptotic behavior under the limit $n\to\infty$ in
§5. For the proof of those properties, we make use of known formulas of
hypergeometric orthogonal polynomials. Here let us display only one of them,
which turns out to be a key lemma in the asymptotic analysis of $p(x)$.

######  Lemma 2.2.6 (Three-term recurrence relation of $p(x)$, Lemma 4.3.3).

In the case $m\leq n-m$ and $0\leq x\leq m$, we have

$\displaystyle\begin{split}a_{x}\frac{n-2x-1}{n-x}\binom{n}{x+1}^{-1}p(x+1)&-(a_{x}+c_{x}-MN)\frac{n-2x+1}{n-x+1}\binom{n}{x}^{-1}p(x)\\\
&+c_{x}\frac{n-2x+3}{n-x+2}\binom{n}{x-1}^{-1}p(x-1)=0\end{split}$ (2.2.15)

with $M:=m-l$ and $N:=n-m-k+l$. The coefficients $a_{x}$ and $c_{x}$ are given
by

$\displaystyle a_{x}$
$\displaystyle:=\frac{(m-x)(n-m-x)(n-k-x)(n-x+1)}{(n-2x)(n-2x+1)},$ (2.2.16)
$\displaystyle c_{x}$
$\displaystyle:=\frac{x(x-k-1)(m-x+1)(n-m-x+1)}{(n-2x+1)(n-2x+2)}.$ (2.2.17)

This recursion can be immediately obtained from Racah presentation (Theorem
2.2.2) and the well-known three-term recurrence relation of Racah polynomials
[3, 39]. As an application of the recursion, we derive in Corollary 4.3.6 the
following support condition of our distribution $P_{n,m,k,l}$:

$\displaystyle p(x\,|\,n,m,k,l)=0\ \text{ for }\ x>m\wedge(n-m)\wedge k.$
(2.2.18)

###  2.3. Asymptotics analysis

Using the formulas explained in the previous §2.2, we study the asymptotics of
our distribution $P_{n,m,k,l}$ under the limit $n\to\infty$ in §5. We study
the following two cases.

  1. I:

The limit $n\to\infty$ with $k,l$ and $m/n$ fixed.

  2. II:

The limit $n\to\infty$ with $m/n$, $k/n$ and $l/n$ fixed.

We call them Type I and Type II limit, respectively. The details are given in
§2.3.1 and §§2.3.2–2.3.4.

####  2.3.1. Asymptotic analysis of Type I limit

Type I limit will be studied in §5.1, in which case we have a very clear
understanding:

######  Theorem 2.3.1 (Theorem 5.1.1).

In the limit $n\to\infty$ with $k,l$ and $\xi=m/n$ fixed, the distribution
$P_{n,m,k,l}$ is approximated to

$\displaystyle P_{R|\xi;k,l}:=B_{\xi,k-l}*B_{1-\xi,l}$ (2.3.1)

up to $O(1/n)$. Here $B_{\xi,j}$ denotes the binomial distribution
$\\{0,1,\dotsc,j\\}\to\mathbb{R}$ with parameters $0\leq\xi\leq 1$ and
$j\in\mathbb{N}$ whose pmf is given by
$x\mapsto\binom{j}{x}\xi^{x}(1-\xi)^{j-x}$, and $*$ denotes the convolution of
probability distributions. In particular, the asymptotic expectation is given
by

$\displaystyle\lim_{n\to\infty}\mathbb{E}[X]=(k-l)\xi+l(1-\xi).$ (2.3.2)

The pmf of the limit distribution $P_{R|\xi;k,l}$ is denoted by
$q(x\,|\,\xi;k,l)$, which has already appeared in §1, (1.0.16). See (5.1.2)
for its explicit expression. Theorem 2.3.1 will be derived from Hahn
presentation of $p(x)$ (Theorem 2.2.1) with some non-trivial hypergeometric
computation (Proposition 5.1.2).

####  2.3.2. Asymptotic analysis of type II limit

In §5.2–§5.4, we discuss Type II limit. Using the parametrization $\xi=m/n$ in
the previous Theorem 2.3.1, we can label the fixed ratios as

$\displaystyle\xi=\frac{m}{n},\quad\kappa=\frac{k}{n},\quad\alpha=\frac{l}{n}.$
(2.3.3)

Let us also introduce another parametrization of the fixed ratios:

$\displaystyle\alpha=\frac{l}{n},\quad\beta=\frac{m-l}{n},\quad\gamma=\frac{k-l}{n},\quad\delta=\frac{n-m-k+l}{n}.$
(2.3.4)

The latter parameters (2.3.4) are more convenient than (2.3.3) in the
following discussion. For example, the condition $(n,m,k,l)\in\mathfrak{N}$ in
(2.1.9) is equivalent to

$\displaystyle\alpha,\beta,\gamma,\delta\geq 0.$ (2.3.5)

Also, the following quantities are fundamental for the discussion:

$\displaystyle\mu:=\frac{1-\sqrt{D}}{2},\quad\sigma:=\sqrt{\frac{(\alpha+\gamma)\beta\delta}{D}},\quad
D:=1-4(\alpha\gamma+\alpha\delta+\beta\gamma).$ (2.3.6)

Hereafter we denote by $\operatorname{Pr}_{n}$ the probability for the
distribution $P_{n,n\xi,n\kappa,n\alpha}$, and by $X$ a random variable
distributed by it.

######  Theorem 2.3.2 (Law of large numbers, Theorem 5.2.3).

Consider the limit $n\to\infty$ with fixed ratios $\alpha=\frac{l}{n}$,
$\beta=\frac{m-l}{n}$, $\gamma=\frac{k-l}{n}$ and $\delta=\frac{n-m-k+l}{n}$.
Then, for any $\varepsilon\in\mathbb{R}_{>0}$, we have

$\displaystyle\lim_{n\to\infty}\operatorname{Pr}_{n}\Bigl{[}\left|\tfrac{X}{n}-\mu\right|>\varepsilon\Bigr{]}=0.$
(2.3.7)

In particular, the expectation $\mathbb{E}[X]$ behaves as

$\displaystyle\mathbb{E}[X]=n\mu+o(n).$ (2.3.8)

For further analysis, we need to divide the discussion into subclasses. After
a technical observation in Lemma 5.2.1, we find that the following division is
natural:

######  Definition 2.3.3 (Definition 5.2.2).

We divide Type II limit into the three classes as follows.

  1. (a)

$\sigma>0$, i.e., $\alpha+\gamma,\beta,\delta>0$.

  2. (b)

$\sigma=0$, i.e., $\alpha=\gamma=0$ or ($\beta\delta=0$ and
$\xi=\alpha+\beta\neq 1/2$).

  3. (c)

$\sigma$ is not defined, i.e., $\beta\delta=0$ and $\xi=\alpha+\beta=1/2$.

The case (a) will be called the generic Type II limit.

The cases (b) and (c) are special, and will be treated in §5.2.1 and §5.2.2.
The case $\alpha=\gamma=0$ is trivial, as explained in §5.2.1. The case
$\beta\delta=0$ is treated in §5.2.2, where we will see that if moreover
$\xi\neq 1/2$ then the distribution asymptotically behaves as a geometric
distribution (Theorem 5.2.4), and if moreover $\xi=1/2$ then it asymptotically
behaves as a Rayleigh distribution (Theorem 5.2.5).

The genetic Type II limit (a) turns out to enjoy quite rich properties, which
will be discussed in §5.2.3 and the following subsections. One of the main
results is the following form of central limit theorem:

######  Theorem 2.3.4 (Central limit Theorem 5.2.9).

In the generic Type II limit, i.e., in the limit $n\to\infty$ with fixed
ratios (2.3.4) satisfying

$\displaystyle\alpha+\gamma,\beta,\delta>0,$ (2.3.9)

we have for any real $t<u$ that

$\displaystyle\lim_{n\to\infty}\operatorname{Pr}_{n}\Bigl{[}t\leq\frac{X-n\mu}{\sqrt{n}\sigma}\leq
u\Bigr{]}=\frac{1}{\sqrt{2\pi}}\int_{t}^{u}e^{-s^{2}/2}\,ds,$ (2.3.10)

In particular, the pmf $p(x)$ behaves as

$\displaystyle
p(x)=\Psi(x)\cdot\bigl{(}1+o(n^{0})\bigr{)}\quad(n\to\infty),\quad\Psi(x):=\frac{1}{\sqrt{2\pi
n}\sigma}\exp\biggl{(}-\frac{1}{2}\Bigl{(}\frac{x-n\mu}{\sqrt{n}\sigma}\Bigr{)}^{2}\biggr{)}.$
(2.3.11)

See Figure 1.0.4 in §1, where we show some examples comparing the pmf $p(x)$
and the normal distribution $\Psi(x)$. At this moment we do not have good
understanding or interpretation of the parameters $\mu$ and $\sigma$. See
Remark 5.2.10 for related comments.

An ad-hoc derivation of the normal distribution $\Psi(x)$, or the parameters
$\mu$ and $\sigma$, is given in §5.2. We start with the recurrence relation of
Lemma 2.2.6, regarding it as a difference equation characterizing the pmf
$p(x)$. Assuming that there exists a limit distribution of $p(x)$ for
$n\to\infty$ which is differentiable, we derive a differential equation of the
limit distribution from the difference equation of $p(x)$ by taking
$n\to\infty$. The obtained differential equation turns out to be the one
characterizing $\Psi(x)$.

####  2.3.3. Proofs of the law of large numbers and the central limit theorems

In §5.3 and §5.4, we prove the law of large numbers (Theorem 2.3.2) and the
central limit Theorem 2.3.4, respectively. In both proofs, the key tool is the
recurrence relation of the pmf $p(x)$ in Lemma 2.2.6, and the main idea is to
convert it to the recurrence relation of the cdf $\sum_{u=0}^{x}p(x)$.

Let us sketch the outline of these proofs. For Theorem 2.3.2, we consider the
random variable $Z:=\frac{X-n\mu}{n}$. Let $G_{n}(z)$ be the cdf for $Z$.
Since the support of $Z$ is compact in the limit $n\to\infty$, we have the
limit distribution $G:=\lim_{n\to\infty}G_{n}$. Using the recursion of $p(x)$,
we can prove that the limit distribution $G$ has positive probability only at
$Z=0$ (Theorem 5.3.4), which is nothing but the law of large numbers.

For the central limit Theorem 2.3.4, we consider the random variable
$Y:=\frac{X-n\mu}{\sqrt{n}}$, whose support is non-compact in the limit
$n\to\infty$. We denote by $F_{n}$ the cdf for $Y$. Using Helly lemma (Fact
5.4.1), we have a limit function $F$ of $F_{n}$, which might not be a
probability distribution a priori. In order to show that $F$ is actually a
probability distribution, we give a preliminary estimation of the tails
(Proposition 5.4.2). Then, we can make a detailed analysis on the the
differential equation of $F$ obtained from the recurrence relation of the pmf
$p(x)$, culminating in Lemma 5.4.4. The obtained differential equation turns
out to be the one characterizing the standard normal distribution, and we have
the central limit Theorem 2.3.4.

####  2.3.4. Asymptotic analysis beyond central limit theorem

As a concluding material of the main text, we discuss in §5.5 and §5.6 the
asymptotic analysis beyond the central limit Theorem 2.3.4 in Type II limit.
Using the relation of the expectation and variance (Theorem 4.4.1), and making
a refined analysis on the tail probability, we can derive the following
result.

######  Theorem 2.3.5 (Theorem 5.5.1).

Consider Type II limit with the condition (2.3.9), under which the central
limit Theorem 2.3.4 holds. Then, the expectation and the variance
asymptotically behave as

$\displaystyle\mathbb{E}[X]=n\mu+\phi+o(n^{0}),\quad\phi:=\frac{\sigma^{2}-\mu}{1-2\mu},$
(2.3.12)
$\displaystyle\lim_{n\to\infty}\mathbb{V}\Bigl{[}\frac{X}{\sqrt{n}}\Bigr{]}=\lim_{n\to\infty}\mathbb{E}\Bigl{[}\Big{(}\frac{X-n\mu}{\sqrt{n}}\Big{)}^{2}\Bigr{]}=\sigma^{2}:=\frac{(\alpha+\gamma)\beta\delta}{D},$
(2.3.13)

where we used $\mu=(1-\sqrt{D})/2$ as before.

In the special case $\alpha\gamma=0$, we can further discuss the tight
exponential evaluation of the tail probability. See Proposition 5.6.1 for the
precise statement.

##  3\. Asymmetry of state $\Xi_{n,m|k,l}$

Using the results for the distribution $P_{n,m,k,l}$ explained in §2, we
discuss the asymmetry of the state $\Xi_{n,m|k,l}$ which was introduced in §1.

For a while, we consider a more general situation than the setting for our
distribution $P_{n,m,k,l}$. Let us given a quantum system described by a
Hilbert space $\mathcal{H}$ on which a finite group $G$ acts by a unitary
representation $f$. For the aim of studying asymmetry of the system, we
consider how many states can be distinguished among the states $\\{f(g)\rho
f(g)^{\dagger}\\}_{g\in G}$ when a state $\rho$ is given on $\mathcal{H}$. To
address this problem, we consider the classical-quantum channel $G\ni g\mapsto
f(g)\rho f(g)^{\dagger}$. The classical case with an abelian group $G$ is
called a regular channel [13] or a conditional additive channel [29, Section
4.1], and has been studied in the area of information theory. In the
classical-quantum channel, $g\in G$ is considered as the input variable and
the state $f(g)\rho f(g)^{\dagger}$ on $\mathcal{H}$ is considered as the
output quantum state. When the input is generated subject to a distribution
$P_{G}$ on $G$, the mutual information between the input and the output is
given by $S(\sum_{g\in G}P_{G}(g)f(g)\rho f(g)^{\dagger})-S(\rho)$, where
$S(\rho):=-\operatorname{tr}\rho\log\rho$ denotes the von Neumann entropy. Due
to the concavity of the von Neumann entropy, the maximum of this value is
realized when $P_{G}$ is the uniform distribution on $G$. Hence, the maximum
mutual information is $S(\operatorname{av}_{f}[\rho])-S(\rho)$. This quantity
can be considered as the transmission rate when the above classical-quantum
channel is repetitively used for the message transmission. In particular, when
$\rho$ is a pure state, this quantity is simplified to
$S(\operatorname{av}_{f}[\psi])$.

As an interlude, let us derive the asymptotic formula (1.0.16) of
$S(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])$ for the state $\Xi_{n,m|k,l}$ with
respect to the distribution $P_{n,m,k,l}$ in Type I limit $n\to\infty$ with
fixed $\xi=m/n$, $k$ and $l$. We denote the pmf by $p(x):=p(x\,|\,n,m,k,l)$,
and the limit pmf by $q(x):=q(x\,|\,\xi;k,l)$ for simplicity. Thus, we have
$p(x)=q(x)+O(1/n)$ by Theorem 2.3.1. Now, recall the expression (1.0.14) of
$S(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])$, which includes the dimension
formula $\dim\mathcal{V}_{(n-x,x)}=\binom{n}{x}\frac{n-2x+1}{n-x+1}$ in
(1.0.10). Let us cite from [54, Chap. 5, (5.12)] the asymptotic formula of
binomial coefficient $\binom{n}{x}$ for fixed $x$:

$\displaystyle\log\tbinom{n}{x}=x\log n+x-(x+\tfrac{1}{2})\log
x-\log\sqrt{2\pi}+o(n^{0}).$ (3.0.1)

Applying it to the dimension formula, we show (1.0.16) as

$\displaystyle\begin{split}S(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])&=\sum_{x=0}^{k}p(x)\bigl{(}\log\dim\mathcal{V}_{(n-x,x)}-\log
p(x)\bigr{)}\\\ &=\sum_{x=0}^{k}q(x)\bigl{(}x\log n+x-(x+\tfrac{1}{2})\log
x-\log\sqrt{2\pi}-\log q(x)\bigr{)}+o(n^{0})\\\ &=u\log
n+u-\log\sqrt{2\pi}+\sum_{x=0}^{k}q(x)\bigl{(}(x+\tfrac{1}{2})\log x-\log
q(x)\bigr{)}+o(n^{0}),\end{split}$ (3.0.2)

where we used the asymptotic expectation $u:=(k-l)\xi+l(1-\xi)$ in (2.3.2).
The last line in (3.0.2) is the conclusion (1.0.16).

Continuing the interlude, we consider Type II limit of
$S(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])$. Since $-\sum_{x=0}^{k}p(x)\log
p(x)\leq\log k$, the corresponding part in
$S(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])$ has the order $\log n$ at most.
Hence, Theorem 2.3.2 (the law of large numbers) implies that

$\displaystyle S(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])=nh(\mu)+o(n).$
(3.0.3)

Special cases have detailed analysis as follows. In the cases (b) and (c) of
Definition 2.3.3, the state $\Xi_{n,m|k,l}$ equals $\Xi_{n,m}$. Hence,
$S(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])$ equals $\log\binom{n}{m}$, whose
asymptotic expansion follows from (3.0.1). As another special case, we focus
on the case with $\alpha\gamma=0$, which is a special case of the generic Type
II limit (a) in Definition 2.3.3. In this case, as will be shown in
Proposition 5.6.3, we have

$\displaystyle S(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])=nh(\mu)+C_{0}+o(1),$
(3.0.4)

where the the precise form of the constant $C_{0}$ is given in Proposition
5.6.3. This is the end of the interlude.

Next, we address how many states $\\{f(g)\rho f(g)^{\dagger}\\}_{g\in G}$ can
be distinguished. For this aim, we return to the general setting, which will
end at (3.0.10). Our interest is the case when the above classical-quantum
channel is used only once. That is, we consider the number of distinguished
state without repetitive use of the same channel. We evaluate the number
$M(\rho,\epsilon)$ of distinguished states among $\\{f(g)\rho
f(g)^{\dagger}\\}_{g\in G}$ with error probability $\epsilon$.

For this aim, we prepare the following notation. Given a Hermitian matrix $A$,
the symbols $\\{A\geq 0\\}$ and $\\{A>0\\}$ express the projections to the
eigenspace of $A$ corresponding to non-negative and strictly positive
eigenvalues, respectively. For two Hermitian matrices $A$ and $B$, the symbol
$\\{A\geq B\\}$ denotes the projection $\\{A-B\geq 0\\}$. Other projections
such as $\\{A>B\\}$ and $\\{A\leq B\\}$ are similarly defined. Then, we
consider the information spectrum relative entropy $D_{s}^{\delta}$, which is
given by [28]:

$\displaystyle D_{s}^{\delta}(\rho\|\rho^{\prime}):=\max\bigl{\\{}\lambda\
\mid\operatorname{tr}\rho\\{\rho\leq
e^{\lambda}\rho^{\prime}\\}\leq\delta\bigr{\\}}.$ (3.0.5)

Since $\operatorname{av}_{f}[\rho]$ is invariant for $f$, applying [28, Lemma
4] to the case $\sigma=\operatorname{av}_{f}[\rho]$, we have

$\displaystyle\log M(\rho,\epsilon)\leq
D_{s}^{\epsilon+\delta}(\rho\|\operatorname{av}_{f}[\rho])+\log\frac{1}{\delta}.$
(3.0.6)

The paper [40, Theorem 2] derived the following evaluation:

$\displaystyle\log M(\rho,\epsilon)\geq
D_{s}^{\epsilon-\delta}(\rho\|\operatorname{av}_{f}[\rho]))-\log\frac{1}{\delta},$
(3.0.7)

which is slightly better than [28, Lemma 3] in this special case. We define

$\displaystyle
H_{s}^{\epsilon}(\rho):=\max\bigl{\\{}\lambda\mid\operatorname{tr}\rho\\{-\log\rho\leq\lambda\\}\leq\epsilon\bigr{\\}}=\max\bigl{\\{}\lambda\mid\operatorname{tr}\rho\\{\rho\geq
e^{-\lambda}\\}\leq\epsilon\bigr{\\}},$ (3.0.8)

When $\rho$ is a pure state, as shown in Appendix A, Theorem A.3.1, we have

$\displaystyle M(\rho,\epsilon)$ $\displaystyle\geq
H_{s}^{\epsilon-\delta_{1}-\delta_{2}}(\operatorname{av}_{f}[\rho])-\log\frac{1}{\delta_{1}\delta_{2}},$
(3.0.9) $\displaystyle M(\rho,\epsilon)$ $\displaystyle\leq
H_{s}^{\epsilon+\delta_{1}+2\delta_{2}}(\operatorname{av}_{f}[\rho])+\log\frac{1}{\delta_{1}\delta_{2}^{2}}.$
(3.0.10)

Now, we apply the above discussion to our setting. Since the state
$\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]$ is characterized as (1.0.11),
$H_{s}^{\epsilon}(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])$ equals
$\overline{H}_{s}^{\varepsilon}(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])$ which
is defined in (1.0.15). Hence,

$\displaystyle
H_{s}^{\varepsilon}(\operatorname{av}_{\pi}[\Xi_{n,m|k,l}])=\max\bigl{\\{}\lambda\mid
P_{n,m,k,l}[\\{x\mid\log\dim\mathcal{V}_{(n-x,x)}-\log
p(x\,|\,n,m,k,l)\leq\lambda\\}]\leq\epsilon\bigr{\\}}.$ (3.0.11)

To discuss Type I limit, we denote the cdf of $B_{\xi,k-l}*B_{1-\xi,l}$ by
$F[B_{\xi,k-l}*B_{1-\xi,l}]$. Recall the dimension formula (1.0.10) of
$\dim\mathcal{V}_{(n-x,x)}$. Then, $\log\dim\mathcal{V}_{(n-x,x)}$ is
approximated to $x\log n+o(\log n)$ with a fixed value $x$ by (3.0.1), and we
have

$\displaystyle\log
M(\rho,\epsilon)=F[B_{\xi,k-l}*B_{1-\xi,l}]^{-1}(\epsilon)\log n+o(\log n).$
(3.0.12)

To discuss Type II limit, we denote by $\Phi$ the cdf of the standard Gaussian
distribution. Also, by [4, Lemma 4.7.1], we have under the limit $n\to\infty$
with fixed ratio $t=x/n$ that

$\displaystyle\log\binom{n}{x}=h(t)n+O(\log n),\quad h(t):=-t\log
t-(1-t)\log(1-t).$ (3.0.13)

Then, we can approximate $\log\dim\mathcal{V}_{(n-x,x)}$ to
$h(\frac{x}{n})n+O(\log n)$ when $x$ behaves linearly with respect to $n$,
which yields

$\displaystyle\log
M(\rho,\epsilon)=h(\mu)n+\frac{\Phi^{-1}(\epsilon)}{\sigma}\sqrt{n}+O(\log
n).$ (3.0.14)

That is, (3.0.12) and (3.0.14) show the asymptotic behavior of the degree of
the asymmetry.

##  4\. Formulas and properties of the distribution $P_{n,m,k,l}$

We start the computation on our distribution $P_{n,m,k,l}$. The main object is
the pmf

$\displaystyle
p(x)=p(x\,|\,n,m,k,l):=\left<\Xi_{n,m|k,l}\right|\mathsf{P}_{(n-x,x)}\left|\Xi_{n,m|k,l}\right>.$
(4.0.1)

Hereafter the setting and symbols in §2.1 will be assumed. In particular, the
set of parameters $(n,m,k,l)$ belongs to $\mathfrak{N}$ in (2.1.9). Let us
write it again:

$\displaystyle\mathfrak{N}:=\\{(n,m,k,l)\in\mathbb{N}^{4}\mid m,k\leq n,\
m+k-n\leq l\leq m\wedge k\\}.$ (4.0.2)

Additionally, we will often use the symbols

$\displaystyle M:=m-l,\quad N:=n-m-k+l,$ (4.0.3)

which already appeared in §2.1. We have $M,N\geq 0$ by the condition
$(n,m,k,l)\in\mathfrak{N}$. We will also encounter in many places the
following binomial coefficient:

$\displaystyle\binom{n-k}{m-l}=\binom{M+N}{M}.$ (4.0.4)

Before starting the discussion, let us recall that the pmf $p(x)$ enjoys the
system symmetry

$\displaystyle p(x\,|\,n,m,k,l)=p(x\,|\,n,n-m,k,k-l),$ (4.0.5)

as mentioned in §2.1, (2.1.5).

###  4.1. Hahn presentation

In this subsection, we use the representation theory of $\mathfrak{S}_{n}$ to
give an explicit formula of the pmf $p(x)$. The result is:

######  Theorem 4.1.1 (Hahn presentation).

Assume $m\leq n-m$. Then, $p(x)=p(x\,|\,n,m,k,l)$ is equal to

$\displaystyle
p(x)=\frac{\binom{n}{x}}{\binom{n}{m}}\,\frac{n-2x+1}{n-x+1}\,\sum_{i=0}^{M\wedge
N}\binom{M}{i}\binom{N}{i}\,\omega_{(n-x,x)}(i),$ (4.1.1)

where $M:=m-l$, $N:=n-m-k+l$, and

$\displaystyle\omega_{(n-x,x)}(i)=\binom{m}{i}^{-1}\binom{n-m}{i}^{-1}\sum_{r=0}^{i\wedge
x}(-1)^{r}\binom{x}{r}\binom{m-x}{i-r}\binom{n-m-x}{i-r}.$ (4.1.2)

######  Remark 4.1.2.

A few comments are in order.

  1. (1)

By (4.1.1), we have $p(x)=0$ for $x>m$ in the case $m\leq n-m$.

  2. (2)

A caution is that the value $\omega_{(n-x,x)}(i)$ might be negative, for
example in the case $(n,m,x,i)=(3,1,1,1)$. So we cannot immediately deduce
from Theorem 4.1.1 that $p(x)\geq 0$.

For the proof of Theorem 4.1.1, we use the following known results in
representation theory:

  * •

Formula of the orthogonal projector onto isotypical component in terms of
irreducible characters (see §4.1.1).

  * •

Formula connecting irreducible characters and the values of zonal spherical
functions on a Gelfand pair $(G,K)$ of finite groups (see §4.1.2).

  * •

Formula of the values of zonal spherical functions on the Gelfand pair
$(\mathfrak{S}_{n},\mathfrak{S}_{m}\times\mathfrak{S}_{n-m})$ in terms of Hahn
polynomials (see §4.1.3).

After these preliminary parts, we will derive Theorem 4.1.1 in §4.1.4.

####  4.1.1. Projector formula

Let $G$ be a finite group, and let $(\varphi,\mathcal{W})$ be a unitary
representation of $G$ with $\mathcal{W}$ the representation space and
$\varphi\colon G\to\operatorname{Aut}(\mathcal{W})$ the structure group
homomorphism. Denoting by $\widehat{G}$ the set of equivalence classes of
$G$-irreps, we can express the irreducible decomposition of $\mathcal{W}$ as

$\displaystyle\mathcal{W}=\bigoplus_{\lambda\in\widehat{G}}\mathcal{W}_{\lambda}.$
(4.1.3)

Here $\mathcal{W}_{\lambda}$ is the isotypical component corresponding to
$\lambda\in\widehat{G}$. Then, in terms of the irreducible character
$\chi^{\lambda}$ of $G$, the orthogonal projector
$\mathsf{P}_{\lambda}\colon\mathcal{W}\twoheadrightarrow\mathcal{W}_{\lambda}$
is expressed as

$\displaystyle\mathsf{P}_{\lambda}=\sum_{g\in
G}\frac{\dim\mathcal{W}_{\lambda}}{\left|G\right|}\,\overline{\chi^{\lambda}(g)}\,\varphi(g).$
(4.1.4)

Here the overline on $\chi^{\lambda}(g)$ denotes complex conjugate. This
formula results from the fact that for any $\lambda\in\widehat{G}$, the
element $(\dim\mathcal{W}_{\lambda}/\left|G\right|)\overline{\chi^{\lambda}}$
is a primitive central idempotent in the group algebra $\mathbb{C}[G]$.

In the case $G=\mathfrak{S}_{n}$, we can identify
$\widehat{G}=\mathbb{Y}_{n}$, the set of Young diagrams with $n$ boxes. Then,
applying (4.1.4) to $G=\mathfrak{S}_{n}$ and
$(\varphi,\mathcal{W})=(\pi,(\mathbb{C}^{2})^{\otimes n})$ given in (1.0.3),
we have

$\displaystyle\mathsf{P}_{(n-x,x)}=\sum_{g\in\mathfrak{S}_{n}}\frac{\dim\mathcal{V}_{(n-x,x)}}{n!}\,\chi^{(n-x,x)}(g)\,\pi(g).$
(4.1.5)

Here we used the symbol $\mathcal{V}_{(n-x,x)}$ for the
$\mathfrak{S}_{n}$-irrep corresponding to the partition $(n-x,x)$ as before.
Also note that we suppressed the complex conjugate symbol since
$\chi^{(n-x,x)}$ is $\mathbb{R}$-valued.

####  4.1.2. Zonal spherical function on Gelfand pair

Here we give a brief recollection of zonal spherical functions on Gelfand
pairs of finite groups. See [44, VII.1] and [7, II.11] for the detail of the
theory.

Let $G$ be a finite group and $K$ its subgroup. Then, $(G,K)$ is called a
Gelfand pair if the space

$\displaystyle\mathcal{L}(K\backslash G/K):=\\{\phi\colon
G\to\mathbb{C}\mid\phi(k_{1}gk_{2})=\phi(g)\ (g\in G,\ k_{1},k_{2}\in K)\\}$
(4.1.6)

of $K$-bi-invariant functions on $G$ is commutative with respect to the
convolution product
$(\phi_{1}*\phi_{2})(g):=\sum_{fh=g}\phi_{1}(f)\phi_{2}(h)$. Equivalently, the
induced representation $\operatorname{Ind}_{K}^{G}1_{K}$ of the trivial
$K$-representation $1_{K}$ has a multiplicity-free irreducible decomposition:

$\displaystyle\operatorname{Ind}_{K}^{G}1_{K}=\sum_{\lambda\in
I}\mathcal{V}_{\lambda},\quad I\subset\widehat{G}.$ (4.1.7)

For each $\lambda\in I$, there exists a unique (up to scalar multiple)
simultaneous eigenfunction $\omega_{\lambda}\in\mathcal{L}(K\backslash G/K)$
for convolution operators
$\\{\phi\ast\,\cdot\;|\,\phi\in\mathcal{L}(K\backslash G/K)\\}$. The
eigenfunction $\omega_{\lambda}$ normalized as $\omega_{\lambda}(e)=1$ is
called the zonal spherical function on the Gelfand pair $(G,K)$ belonging to
the irreducible character $\chi^{\lambda}=\chi(\mathcal{V}_{\lambda})$.

Let us explain the relation of the functions $\omega_{\lambda}$ and
$\chi^{\lambda}$. Define the function $e_{K}$ on $G$ by $e_{K}(k):=1$ for
$k\in K$ and $e_{K}(g):=0$ for $g\notin K$. Identifying the linear space
$\mathcal{L}(G)$ of $\mathbb{C}$-valued functions on $G$ with the group
algebra $\mathbb{C}[G]$ by mapping the function $\phi\in\mathcal{L}(G)$ to
$\sum_{g\in G}\phi(g)g\in\mathbb{C}[G]$, we have
$e_{K}=\left|K\right|^{-1}\sum_{k\in K}k$. Then, for $\lambda\in I$ appearing
in the decomposition (4.1.7), we have

$\displaystyle\omega_{\lambda}=\overline{\chi^{\lambda}}*e_{K}.$ (4.1.8)

We can consider this equality as an alternating definition of
$\omega_{\lambda}$. See [7, II.11, Theorem 11.3] and [44, VII.1, (1.3)] for
the detail. As a consequence, for any $g\in G$ we have

$\displaystyle\omega_{\lambda}(g)=\frac{1}{\left|K\right|}\sum_{k\in
K}\overline{\chi^{\lambda}(gk)}=\frac{1}{\left|K\right|}\sum_{k\in
K}\chi^{\lambda}(kg^{-1}).$ (4.1.9)

####  4.1.3. Zonal spherical function on
$(\mathfrak{S}_{n},\mathfrak{S}_{m}\times\mathfrak{S}_{n-m})$

Here we recall an explicit formula for the values of zonal spherical functions
for the Gelfand pair
$(\mathfrak{S}_{n},\mathfrak{S}_{m}\times\mathfrak{S}_{n-m})$. See [7, III.2]
and [44, VII.1, Exercise 13] for the detail.

For a finite set $A$, we denote by $\mathfrak{S}_{A}$ the group of bijections
on $A$. Let $n,m\in\mathbb{N}$ with $m\leq n-m$, and regard
$K_{m}:=\mathfrak{S}_{m}\times\mathfrak{S}_{n-m}$ as a subgroup of
$\mathfrak{S}_{n}$ by $\mathfrak{S}_{m}=\mathfrak{S}_{\\{1,\dotsc,m\\}}$,
$\mathfrak{S}_{n-m}=\mathfrak{S}_{\\{m+1,\dotsc,n\\}}$ and
$\mathfrak{S}_{n}=\mathfrak{S}_{\\{1,\dotsc,n\\}}$. Then,
$(\mathfrak{S}_{n},K_{m})=(\mathfrak{S}_{n},\mathfrak{S}_{m}\times\mathfrak{S}_{n-m})$
is a Gelfand pair, since the Littlewood-Richardson rule (or the Pieri formula)
yields

$\displaystyle\operatorname{Ind}_{K_{m}}^{\mathfrak{S}_{n}}1_{K_{m}}\cong\bigoplus_{x=0}^{m}\mathcal{V}_{(n-x,x)}.$
(4.1.10)

In this case, the formula (4.1.9) is rewritten as

$\displaystyle\omega_{(n-x,x)}(g)=\frac{1}{m!(n-m)!}\sum_{k\in
K_{m}}\chi^{(n-x,x)}(kg^{-1}),\quad g\in\mathfrak{S}_{n},\
x\in\\{0,1,\dotsc,m\\}.$ (4.1.11)

Let us observe the structure of the double coset
$K_{m}\backslash\mathfrak{S}_{n}/K_{m}$. We put

$\displaystyle\Gamma:=\\{J\subset\\{1,2,\dotsc,n\\}\mid\left|J\right|=m\\}\,\ni\,J_{0}:=\\{1,2,\dotsc,m\\}.$
(4.1.12)

The group $\mathfrak{S}_{n}$ acts transitively on $\Gamma$ by permutation, and
the stabilizer subgroup of $J_{0}$ is
$K_{m}=\mathfrak{S}_{m}\times\mathfrak{S}_{n-m}$. Thus,
$\mathfrak{S}_{n}/K_{m}\cong\Gamma$. A distance $\partial$ on $\Gamma$ is
defined by

$\displaystyle\partial(J_{1},J_{2}):=m-\left|J_{1}\cap J_{2}\right|,\quad
J_{1},J_{2}\in\Gamma,$ (4.1.13)

and $(\Gamma,\partial)$ is a distance-regular graph called the Johnson graph
$J(n,m)$ in [11, §4.2] and [7, III.2]. Since $J_{1},J_{2}\in\Gamma$ belong to
the same $K_{m}$-orbit if and only if
$\partial(J_{0},J_{1})=\partial(J_{0},J_{2})$, we have the bijective
correspondence

$\displaystyle
K_{m}\backslash\mathfrak{S}_{n}/K_{m}\cong\\{\text{$K_{m}$-orbits of
$\Gamma$}\\}\cong\\{0,1,\dotsc,m\\}.$ (4.1.14)

In particular, we have $K_{m}\leftrightarrow\\{J_{0}\\}\leftrightarrow 0$
under (4.1.14).

Let $\omega_{(n-x,x)}$ be the zonal spherical function on
$(\mathfrak{S}_{n},K_{m})$ belonging to $\chi^{(n-x,x)}$ for
$x\in\\{0,1,\dotsc,m\\}$. For $g\in\mathfrak{S}_{n}$, let
$i=i(g)\in\\{0,1,\dotsc,m\\}$ be the distance which the correspondence
(4.1.14) assigns to the double coset containing $g$, and set

$\displaystyle\omega_{(n-x,x)}(i):=\omega_{(n-x,x)}(g).$ (4.1.15)

By Delsarte [11, 12], the value $\omega_{(n-x,x)}(i)$ is equal to

$\displaystyle\omega_{(n-x,x)}(i)=\tbinom{m}{i}^{-1}\tbinom{n-m}{i}^{-1}E_{i}(x),\quad
x,i\in\\{0,1,\dotsc,m\\},$ (4.1.16)

where $E_{i}(x)$ is the Eberlein polynomial [11, §4.2]:

$\displaystyle E_{i}(x):=\sum_{r=0}^{i\wedge
x}(-1)^{r}\binom{x}{r}\binom{m-x}{i-r}\binom{n-m-x}{i-r}.$ (4.1.17)

See also [16], [18, §4] and [7, III.2, Theorems 2.9, 2.10] for the detail.

######  Remark 4.1.3.

As mentioned in §2.2.1, $E_{i}(x)$ is essentially equal to Hahn and dual Hahn
polynomials. Let us explain the detail, using the notation of hypergeometric
orthogonal polynomials given in [39].

  1. (1)

For $D\in\mathbb{N}$, $d=0,1,\dotsc,D$, a variable $z$ and parameters
$\alpha,\beta$, Hahn polynomial [39, §9.5] is the polynomial of $z$ of degree
$d$ given by

$\displaystyle
Q_{d}(z;\alpha,\beta,D):={}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-d{\mathchar
44\relax}\mskip 8.0mu-z{\mathchar 44\relax}\mskip
8.0muz+\alpha+\beta+1}{\alpha+1{\mathchar 44\relax}\mskip 8.0mu-D};1\right]}.$
(4.1.18)

On the other hand, for $D\in\mathbb{N}$, $d=0,1,\dotsc,D$, a variable $z$ and
parameters $a,b$, dual Hahn polynomial [39, §9.6] is the polynomial of $z$ of
degree $2d$ given by

$\displaystyle
R_{d}(\lambda(z);\alpha,\beta,D):={}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-d{\mathchar
44\relax}\mskip 8.0mu-z{\mathchar 44\relax}\mskip
8.0muz+\alpha+\beta+1}{\alpha+1{\mathchar 44\relax}\mskip
8.0mu-D};1\right]},\quad\lambda(z):=z(z+\alpha+\beta+1).$ (4.1.19)

Thus, Hahn and dual Hahn polynomials are the same hypergeometric series. The
difference is the choice of variable and degree, and for each choice we have a
family of orthogonal polynomials.

  2. (2)

By [7, III.2., Theorem 2.10 (1)]111There is a typo in the proof. The final
line in Page 221 should be read as “Apply (2.23) with $n=j$, $a=k-j+1$,
$b=-i$, $c=k-i-j+1$, $d=-v+k$”., the value
$\omega_{(n-x,x)}(i)=\binom{m}{i}^{-1}\binom{n-m}{i}^{-1}E_{i}(x)$ of zonal
spherical function is equal to Hahn and dual Hahn polynomials with the
following parameters.

$\displaystyle\begin{split}\omega_{(n-x,x)}(i)&={}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-i{\mathchar
44\relax}\mskip 8.0mu-x{\mathchar 44\relax}\mskip 8.0mux-n-1}{m-n{\mathchar
44\relax}\mskip 8.0mu-m};1\right]}\\\
&=Q_{x}(i;m-n-1,-m-1,m)=R_{i}(\lambda(x);m-n-1,-m-1,m).\end{split}$ (4.1.20)

####  4.1.4. Proof of Theorem 4.1.1

We start from the formula (4.1.5) of the projector $\mathsf{P}_{(n-x,x)}$. In
Definition 2.1.1 of $\left|\Xi_{n,m|k,l}\right>$, let us denote the positions
of $\left|1\right>$ lying after $(k+1)$-th by
$k+p_{1},k+p_{2},\dotsc,k+p_{m-l}$. Then, using the notation
$\left|1^{a}0^{b}\right>$ in (1.0.4), we have

$\displaystyle\left|\Xi_{n,m|k,l}\right>=\binom{n-k}{m-l}^{-1/2}\sum_{1\leq
p_{1}<\dotsb<p_{m-l}\leq
n-k}|1^{l}0^{k-l}\dotsm\underbrace{1}_{\text{$(k+p_{1})$-th}}\dotsm\underbrace{1}_{\text{$(k+p_{m-l})$-th}}\dotsm\bigr{\rangle}.$
(4.1.21)

We proceed as

$\displaystyle\left<\Xi_{n,m|k,l}\right|\mathsf{P}_{(n-x,x)}\left|\Xi_{n,m|k,l}\right>=\sum_{g\in\mathfrak{S}_{n}}\,\frac{\dim\mathcal{V}_{(n-x,x)}}{n!}\,\chi^{(n-x,x)}(g)\cdot\left<\Xi_{n,m|k,l}\right|\pi(g)\left|\Xi_{n,m|k,l}\right>$
$\displaystyle=\sum_{1\leq p_{1}<\dotsb<p_{m-l}\leq n-k}\ \sum_{1\leq
q_{1}<\dotsb<q_{m-l}\leq n-k}\
\sum_{g\in\mathfrak{S}_{n}}\,\frac{\dim\mathcal{V}_{(n-x,x)}}{n!}\,\chi^{(n-x,x)}(g)\cdot\frac{1}{\binom{n-k}{m-l}}$
$\displaystyle\qquad\qquad\cdot\langle
1^{l}0^{k-l}\dotsm\underbrace{1}_{\text{$(k+p_{1})$-th}}\dotsm\underbrace{1}_{\text{$(k+p_{m-l})$-th}}\dotsm|\pi(g)|1^{l}0^{k-l}\dotsm\underbrace{1}_{\text{$(k+q_{1})$-th}}\dotsm\underbrace{1}_{\text{$(k+q_{m-l})$-th}}\dotsm\rangle$
$\displaystyle=\frac{\dim\mathcal{V}_{(n-x,x)}}{n!}\,\frac{1}{\binom{n-k}{m-l}}\sum_{1\leq
p_{1}<\dotsb<p_{m-l}\leq n-k}\ \sum_{1\leq q_{1}<\dotsb<q_{m-l}\leq n-k}\
\sum_{g\in\mathfrak{S}_{n},\,(\star)}\chi^{(n-x,x)}(g).$ (4.1.22)

Let us investigate condition $(\star)$ which makes the sum in $g$ effective.
Set

$\displaystyle
g_{0}:=\begin{pmatrix}1&\dotsm&l&k+q_{1}&\dotsm&k+q_{m-l}&\text{the other
letters increasing}\\\ 1&\dotsm&l&k+p_{1}&\dotsm&k+p_{m-l}&\text{the other
letters increasing}\end{pmatrix}$ (4.1.23)

in the two-line notation of a permutation. Then,

$\displaystyle g\text{ satisfies }(\star)$
$\displaystyle\iff\\{1,\dotsc,l,k+q_{1},\dotsc,k+q_{m-l}\\}=\\{g^{-1}(1),\dotsc,g^{-1}(l),g^{-1}(k+p_{1}),\dotsc,g^{-1}(k+p_{m-l})\\}$
$\displaystyle\iff
g\bigl{(}\\{1,\dotsc,l,k+q_{1},\dotsc,k+q_{m-l}\\}\bigr{)}=\\{1,\dotsc,l,k+p_{1},\dotsc,k+p_{m-l}\\}$
$\displaystyle\iff
gg_{0}^{-1}\in\mathfrak{S}_{\\{1,\dotsc,l,k+p_{1},\dotsc,k+p_{m-l}\\}}\times\mathfrak{S}_{\\{1,\dotsc,l,k+p_{1},\dotsc,k+p_{m-l}\\}^{c}},$

where the complement c is taken in $\\{1,\dotsc,n\\}$. We set

$\displaystyle\sigma_{0}:=\begin{pmatrix}1&\dotsm&l&l+1&\dotsm&m&m+1\quad\dotsm\quad
n\qquad\\\ 1&\dotsm&l&k+p_{1}&\dotsm&k+p_{m-l}&\text{the other letters
increasing}\end{pmatrix},$ (4.1.24)

which yields

$\displaystyle\sigma_{0}^{-1}\bigl{(}\mathfrak{S}_{\\{1,\dotsc,l,k+p_{1},\dotsc,k+p_{m-l}\\}}\times\mathfrak{S}_{\\{1,\dotsc,l,k+p_{1},\dotsc,k+p_{m-l}\\}^{c}}\bigr{)}\sigma_{0}=\mathfrak{S}_{\\{1,\dotsc,m\\}}\times\mathfrak{S}_{\\{m+1,\dotsc,n\\}}=\mathfrak{S}_{m}\times\mathfrak{S}_{n-m}.$
(4.1.25)

We continue (4.1.22) as

$\displaystyle\text{\eqref{eq:4-1}}=\frac{\dim\mathcal{V}_{(n-x,x)}}{n!}\,\frac{1}{\binom{n-k}{m-l}}\sum_{1\leq
p_{1}<\dotsb<p_{m-l}\leq n-k}\ \sum_{1\leq q_{1}<\dotsb<q_{m-l}\leq n-k}\
\sum_{h\in\mathfrak{S}_{m}\times\mathfrak{S}_{n-m}}\chi^{(n-x,x)}(\sigma_{0}h\sigma_{0}^{-1}g_{0}).$
(4.1.26)

In this sum, we set

$\displaystyle
s:=\left|\\{k+p_{1},\dotsc,k+p_{m-l}\\}\cap\\{k+q_{1},\dotsc,k+q_{m-l}\\}\right|,\quad
i:=m-l-s.$ (4.1.27)

Since $s+2(m-l-s)\leq n-k$ holds, we have $0\leq i\leq(m-l)\wedge(n-k-m+l)$.

The element $\sigma_{0}^{-1}g_{0}^{-1}\sigma_{0}\in\mathfrak{S}_{n}$ has the
following properties:

  * •

Fixing $1,\dotsc,l,m+1,\dotsc,m+(k-l)$.

  * •

$\left|\sigma_{0}^{-1}g_{0}^{-1}\sigma_{0}(\\{l+1,\dotsc,m\\})\cap\\{l+1,\dotsc,m\\}\right|=s$
($=m-l-i$).

Hence, there exist
$h^{\prime},h^{\prime\prime}\in\mathfrak{S}_{n}\times\mathfrak{S}_{n-m}$ such
that

$\displaystyle\sigma_{0}^{-1}g_{0}^{-1}\sigma_{0}=h^{\prime}th^{\prime\prime},\quad
t:=(l+1,m+k-l+1)(l+2,m+k-l+2)\dotsm(l+i,m+k-l+i).$ (4.1.28)

Here $t$ is a product of $i$ transpositions. Then, the inner sum of (4.1.26)
equals to

$\displaystyle\begin{split}\sum_{h\in\mathfrak{S}_{m}\times\mathfrak{S}_{n-m}}\chi^{(n-x,x)}\bigl{(}h(\sigma_{0}^{-1}g_{0}^{-1}\sigma_{0})^{-1}\bigr{)}&=\sum_{h\in\mathfrak{S}_{m}\times\mathfrak{S}_{n-m}}\chi^{(n-x,x)}(h^{\prime-1}hh^{\prime\prime-1}t^{-1})\\\
&=\sum_{h\in\mathfrak{S}_{m}\times\mathfrak{S}_{n-m}}\chi^{(n-x,x)}(ht^{-1}).\end{split}$
(4.1.29)

The expression of the rightmost side in (4.1.29) depends only on $s$, hence on
$i$, and not on individual $p$’s nor $q$’s satisfying (4.1.27). For each $i$,
the way of choosing $p$’s and $q$’s satisfying (4.1.27) amounts to

$\displaystyle\binom{n-k}{m-l}\binom{m-l}{s}\binom{n-k-m+l}{m-l-s}=\binom{n-k}{m-l}\binom{m-l}{i}\binom{n-k-m+l}{i}=\binom{n-k}{m-l}\binom{M}{i}\binom{N}{i},$
(4.1.30)

where we used $M:=m-l$ and $N:=n-m-k+l$. Combining it with (4.1.29), we have

$\displaystyle\text{\eqref{eq:4-2}}=\frac{\dim\mathcal{V}_{(n-x,x)}}{n!}\sum_{i=0}^{M\wedge
N}\binom{M}{i}\binom{N}{i}\sum_{g\in\mathfrak{S}_{m}\times\mathfrak{S}_{n-m}}\chi^{(n-x,x)}(gt^{-1}).$
(4.1.31)

Then, the hook formula (1.0.10) and the zonal spherical function
$\omega_{(n-x,x)}$ of (4.1.11) yields

$\displaystyle\text{\eqref{eq:4-5}}=\frac{m!(n-m)!}{n!}\,\frac{n-2x+1}{n-x+1}\binom{n}{x}\sum_{i=0}^{M\wedge
N}\binom{M}{i}\binom{N}{i}\omega_{(n-x,x)}(t).$ (4.1.32)

Since $t=(l+1,m+k-l+1)\dotsm(l+i,m+k-l+i)\in\mathfrak{S}_{n}$ corresponds to
distance $i$ by (4.1.14), we have

$\displaystyle\omega_{(n-x,x)}(t)=\omega_{(n-x,x)}(i),$ (4.1.33)

whose value is given by (4.1.16) and (4.1.17). Thus, we have

$\displaystyle\text{\eqref{eq:Hora:fin}}=\frac{\binom{n}{x}}{\binom{n}{m}}\frac{n-2x+1}{n-x+1}\sum_{i=0}^{M\wedge
N}\binom{M}{i}\binom{N}{i}\omega_{(n-x,x)}(i),$ (4.1.34)
$\displaystyle\omega_{(n-x,x)}(i)=\binom{m}{i}^{-1}\binom{n-m}{i}^{-1}\,\sum_{r=0}^{i\wedge
x}(-1)^{r}\binom{x}{r}\binom{m-x}{i-r}\binom{n-m-x}{i-r}.$ (4.1.35)

This completes the proof of Theorem 2.2.1.

###  4.2. Racah presentation

Let us recall Theorem 4.1.1: For $m\leq n-m$, we have

$\displaystyle\begin{split}p(x)&=p(x\,|\,n,m,k,l):=\left<\Xi_{n,m|k,l}\right|\mathsf{P}_{(n-x,x)}\left|\Xi_{n,m|k,l}\right>\\\
&=\frac{\binom{n}{x}}{\binom{n}{m}}\,\frac{n-2x+1}{n-x+1}\,\sum_{i=0}^{M\wedge
N}\frac{\binom{M}{i}\binom{N}{i}}{\binom{m}{i}\binom{n-m}{i}}\,\sum_{r=0}^{i\wedge
x}(-1)^{r}\binom{x}{r}\binom{m-x}{i-r}\binom{n-m-x}{i-r}\end{split}$ (4.2.1)

with $M:=m-l$ and $N:=n-m-k+l$. As noted in §2.2.2, this formula is a double
sum, and is not suitable for further analysis. The result of this subsection
is a presentation of $p(x)$ by a single sum.

######  Theorem 4.2.1 (Racah presentation).

If $m\leq n-m$ and $x\leq m$, then we have

$\displaystyle
p(x)=\binom{n-k}{m-l}\frac{\binom{n}{x}}{\binom{n}{m}}\frac{n-2x+1}{n-x+1}{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mux-n-1{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}.$ (4.2.2)

######  Remark 4.2.2.

Let us explain the origin of the naming “Racah presentation”.

  1. (1)

The ${}_{4}F_{3}$-hypergeometric series in (4.2.2) is a specialization of
Racah polynomial [39, §9.2]. Let us recall the definition. For
$D\in\mathbb{N}$, $d=0,1,\dotsc,D$, a variable $y$ and parameters
$\alpha,\beta,\gamma,\delta$ which satisfy $\alpha+1=-D$ or
$\beta+\delta+1=-D$ or $\gamma+1=-D$, Racah polynomial is defined to be

$\displaystyle
R_{d}(\lambda(z);\alpha,\beta,\gamma,\delta):={}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-d{\mathchar
44\relax}\mskip 8.0mud+\alpha+\beta+1{\mathchar 44\relax}\mskip
8.0mu-z{\mathchar 44\relax}\mskip 8.0muz+\gamma+\delta+1}{\alpha+1{\mathchar
44\relax}\mskip 8.0mu\beta+\delta+1{\mathchar 44\relax}\mskip
8.0mu\gamma+1};1\right]}$ (4.2.3)

with $\lambda(z):=z(z+\gamma+\delta+1)$. Specializing
$(D,d,z,\alpha,\beta,\gamma,\delta)=(m,x,M,-m-1,-n+m-1,-M-N-1,0)$, we can
rewrite (4.2.2) as

$\displaystyle
p(x)=\binom{n-k}{m-l}\frac{\binom{n}{x}}{\binom{n}{m}}\frac{n-2x+1}{n-x+1}\,R_{x}(\lambda(M);-m-1,-n+m-1,-n+k-1,0)$
(4.2.4)

with $\lambda(M)=M\bigl{(}M+(-M-N-1)+0+1\bigr{)}=-MN$.

  2. (2)

Among various properties of Racah polynomial, let us only mention that it is
balanced or Saalschützian, i.e., it satisfies

$\displaystyle(-d)+(d+\alpha+\beta+1)+(-z)+(z+\gamma+\delta+1)+1=(\alpha+1)+(\beta+\delta+1)+(\gamma+1).$
(4.2.5)

As mentioned in §2.2.2, the balancing condition often yields various summation
formulas of hypergeometric series. We will derive such a summation formula in
§4.3. See [3, Definition 2.2.7], [20, §1.2, p.5] and [53, §2.3.1] for more
information on the balancing condition.

For the proof of Theorem 4.2.1, we use Remark 4.1.3 (2): The value

$\displaystyle\omega_{(n-x,x)}(i)=\binom{m}{i}^{-1}\binom{n-m}{i}^{-1}\sum_{r=0}^{i\wedge
x}(-1)^{r}\binom{x}{r}\binom{m-x}{i-r}\binom{n-m-x}{i-r}$ (4.2.6)

of zonal spherical function is equal to

$\displaystyle\omega_{(n-x,x)}(i)={}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-i{\mathchar
44\relax}\mskip 8.0mu-x{\mathchar 44\relax}\mskip 8.0mux-n-1}{-m{\mathchar
44\relax}\mskip 8.0mum-n};1\right]}.$ (4.2.7)

###### Proof of Theorem 4.2.1.

By (4.2.7) and $\binom{M}{i}=(-1)^{i}(-M)_{i}/(1)_{i}$, we have

$\displaystyle\sum_{i\geq
0}\binom{M}{i}\binom{N}{i}\omega_{(n-x,x)}(i)=\sum_{i\geq 0}\sum_{r\geq
0}\frac{(-i,x,x-n-1)_{r}}{(1,-m,m-n)_{r}}\frac{(-M,-N)_{i}}{(1,1)_{i}}.$
(4.2.8)

Changing the order of summations, we have

$\displaystyle\sum_{i\geq 0}(-i)_{r}\frac{(-M,-N)_{i}}{(1,1)_{i}}$
$\displaystyle=\sum_{i\geq
r}(-i)_{r}\frac{(-M,-N)_{i}}{(1,1)_{i}}=(-1)^{r}\sum_{u\geq
0}\frac{(M,N)^{\downarrow u+r}}{u!\,(u+r)!}$
$\displaystyle=(-1)^{r}\frac{(M,N)^{\downarrow r}}{r!}\sum_{u\geq
0}\frac{(M-r,N-r)^{\downarrow
u}}{u!\,(r+1)_{u}}=(-1)^{r}\frac{(M,N)^{\downarrow r}}{r!}\sum_{u\geq
0}\frac{(r-M,r-N)_{u}}{(1,r+1)_{u}}$
$\displaystyle=(-1)^{r}\frac{(M,N)^{\downarrow
r}}{r!}{}_{2}F_{1}{\left[\genfrac{.}{.}{0.0pt}{}{r-M{\mathchar 44\relax}\mskip
8.0mur-N}{r+1};1\right]}.$ (4.2.9)

Using Gauss’s summation formula
${}_{2}F_{1}{\left[\genfrac{.}{.}{0.0pt}{}{a{\mathchar 44\relax}\mskip
8.0mub}{c};1\right]}=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$
for $\mathop{\mathrm{Re}}\nolimits(c-a-b)>0$ (see [20, (1.2.11)] for example),
we have

$\displaystyle\eqref{eq:Hahn:2213}$
$\displaystyle=(-1)^{r}\frac{(M,N)^{\downarrow
r}}{r!}\cdot\frac{\Gamma(r+1)\Gamma(M+N-r+1)}{\Gamma(M+1)\Gamma(N+1)}=(-1)^{r}\frac{(M+N-r)!}{(M-r)!\,(N-r)!}.$
(4.2.10)

Thus, we have

$\displaystyle\begin{split}\sum_{i\geq
0}\binom{M}{i}\binom{N}{i}\omega_{(n-x,x)}(i)&=\sum_{r\geq
0}(-1)^{r}\frac{(-x,x-n-1)_{r}}{(1,-m,m-n-1)_{r}}\frac{(M+N-r)!}{(M-r)!\,(N-r)!}\\\
&=\frac{(M+N)!}{M!\,N!}\sum_{r\geq
0}(-1)^{r}\frac{(-x,x-n-1)_{r}}{(1,-m,m-n-1)_{r}}\frac{(M,N)^{\downarrow
r}}{(M+N)^{\downarrow r}}\\\ &=\binom{M+N}{M}\sum_{r\geq
0}\frac{(-x,x-n-1,-M,-N)_{r}}{(1,-m,m-n-1,-M-N)_{r}}\\\
&=\binom{n-k}{m-l}{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mux-n-1{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n-1{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}.\end{split}$ (4.2.11)

Since $p(x)=\binom{n}{m}^{-1}\binom{n}{x}\frac{n-2x+1}{n-x+1}\sum_{i\geq
0}\binom{M}{i}\binom{N}{i}\omega_{(n-x,x)}(i)$, we have the conclusion. ∎

###  4.3. ${}_{4}F_{3}$-presentation of cdf and recurrence relation of pmf

In this subsection, we give a few observations on the formulas in §4.1 and
§4.2.

First, let us show that the summation of $p(x)$ is equal to one directly by
the explicit formulas obtained before. As explained in §2.2.2, the equality
$\sum_{x\geq 0}p(x)=1$ is obvious from representation/probability theoretic
points of view, but it will yield a non-trivial summation formula. At this
moment we have a proof using Racah presentation (Theorem 4.2.1), but cannot
prove it by Hahn presentation (Theorem 4.1.1).

######  Theorem 4.3.1.

In the case $m\leq n-m$, the cdf

$\displaystyle s(x):=\sum_{u=0}^{x}p(u)=\sum_{u=0}^{x\wedge m}p(u)$ (4.3.1)

up to $x\in\mathbb{N}$ is given by

$\displaystyle
s(x)=\binom{n-k}{m-l}\frac{\binom{n}{x}}{\binom{n}{m}}{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mux-n{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}$ (4.3.2)

with $M:=m-l$ and $N:=n-m-k+l$. Moreover, in the case $x=m$, we have

$\displaystyle s(m)=1.$ (4.3.3)

###### Proof.

We can assume $x\leq m$ by Remark 4.1.2 (1). By Theorem 4.2.1 and the
inequality $m\geq M\wedge N$, we have

$\displaystyle s(x)=\sum_{u=0}^{x}p(u)$
$\displaystyle=\frac{\binom{n-k}{m-l}}{\binom{n}{m}}\sum_{u=0}^{x}\frac{n-2u+1}{n-u+1}\binom{n}{u}\sum_{r=0}^{u\wedge
M\wedge N}\frac{(-u,u-n-1,-M,-N)_{r}}{(-m,m-n,-M-N,1)_{r}}$
$\displaystyle=\frac{\binom{n-k}{m-l}}{\binom{n}{m}}\sum_{r=0}^{M\wedge
N}\frac{(-M,-N)_{r}}{(-m,m-n,-M-N,1)_{r}}\sum_{u=r}^{x}\frac{n-2u+1}{n-u+1}\binom{n}{u}(-u,u-n-1)_{r}.$
(4.3.4)

Let us compute the inner summation $\sum_{u=r}^{x}$. We have

$\displaystyle\frac{n-2u+1}{n-u+1}=\frac{u-n+r-1}{u-n-1}-\frac{u}{n-u+1}\frac{u-r}{u},$
(4.3.5)

which yields

$\displaystyle\frac{n-2u+1}{n-u+1}\binom{n}{u}(-u,u-n-1)_{r}=f(u)-f(u-1),\quad
f(u):=\binom{n}{u}(-u,u-n)_{r}.$ (4.3.6)

Thus, we have

$\displaystyle\sum_{u=r}^{x}\frac{n-2u+1}{n-u+1}\binom{n}{u}(-u,u-n-1)_{r}=f(x)-f(r-1)=\binom{n}{x}(-x,x-n)_{r}.$
(4.3.7)

Returning to the original summation, we have

$\displaystyle\eqref{eq:S:237}$
$\displaystyle=\frac{\binom{n-k}{m-l}}{\binom{n}{m}}\sum_{r=0}^{M\wedge
N}\frac{(-M,-N)_{r}}{(-m,m-n,-M-N,1)_{r}}\cdot\binom{n}{x}(-x,x-n)_{r}$
$\displaystyle=\frac{\binom{n-k}{m-l}}{\binom{n}{m}}\binom{n}{x}{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mux-n{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}.$ (4.3.8)

For $x=m$, we have

$\displaystyle s(m)$
$\displaystyle=\binom{n-k}{m-l}{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-m{\mathchar
44\relax}\mskip 8.0mum-n{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-
M-N};1\right]}=\binom{M+N}{M}{}_{2}F_{1}{\left[\genfrac{.}{.}{0.0pt}{}{-M{\mathchar
44\relax}\mskip 8.0mu-N}{-M-N};1\right]}=1,$ (4.3.9)

where the last equality follows from the Chu-Vandermonde formula (2.2.10). ∎

######  Remark 4.3.2.

We have some comments on the balancing condition in Theorem 4.3.1.

  1. (1)

The hypergeometric series
${}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-M{\mathchar 44\relax}\mskip
8.0mu-N{\mathchar 44\relax}\mskip 8.0mu-x{\mathchar 44\relax}\mskip
8.0mux-n}{-M-N{\mathchar 44\relax}\mskip 8.0mu-m{\mathchar 44\relax}\mskip
8.0mum-n};1\right]}$ is $0$-balanced in the sense of [20, §1.2, p.5], i.e.,
satisfies $(-M)+(-N)+(-b)+(b-n)+0=(-M-N)+(-m)+(m-n)$, whereas Racah
polynomials are $1$-balanced as mentioned in Remark 4.2.2. Hence, the
summation on $a$ lowers the balancing condition, while preserving the
hypergeometric level ${}_{4}F_{3}$.

  2. (2)

For general parameters $(n,m,k,l)$, we don’t have nice explicit formula for
expectation or higher moments. This is somehow related to (1): Regarding the
total sum as $0$-th moment, the calculation for $d$-th moment will need a
hypergeometric summation formula for $(1-d)$-balanced series. However, such a
summation formula is rare in the case $d>0$, and we cannot expect to have a
nice formula of $d$-th moment for general parameters $(n,m,k,l)$. Instead, if
we specialize parameters $(n,m,k,l)$ nicely, then we might have good formula
of higher moments. As an illustration, we will give a calculation of the
expectation for the special case $n=2m$ in §B.1.

The next lemma will be used repeatedly in §§5.2–5.4 to study the asymptotic
behavior of our distribution $P_{n,m,k,l}$ in the limit $n\to\infty$.

######  Lemma 4.3.3.

In the case $m\leq n-m$ and $0\leq x\leq m$, we have

$\displaystyle\begin{split}a_{x}\frac{n-2x-1}{n-x}\binom{n}{x+1}^{-1}p(x+1)&-(a_{x}+c_{x}-MN)\frac{n-2x+1}{n-x+1}\binom{n}{x}^{-1}p(x)\\\
&+c_{x}\frac{n-2x+3}{n-x+2}\binom{n}{x-1}^{-1}p(x-1)=0\end{split}$ (4.3.10)

with $M:=m-l$ and $N:=n-m-k+l$. Here we regard the third term as $0$ in the
case $x=0$, and regard the first term as $0$ in the case $x=m$. The
coefficients $a_{x}$ and $c_{x}$ are given by

$\displaystyle a_{x}$
$\displaystyle:=\frac{(m-x)(n-m-x)(n-k-x)(n-x+1)}{(n-2x)(n-2x+1)},$ (4.3.11)
$\displaystyle c_{x}$
$\displaystyle:=\frac{x(x-k-1)(m-x+1)(n-m-x+1)}{(n-2x+1)(n-2x+2)}.$ (4.3.12)

For the proof, we need the recurrence relation of Racah polynomials (see
Remark 4.2.2):

$\displaystyle
R_{d}(\lambda(z);\alpha,\beta,\gamma,\delta):={}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-d{\mathchar
44\relax}\mskip 8.0mud+\alpha+\beta+1{\mathchar 44\relax}\mskip
8.0mu-z{\mathchar 44\relax}\mskip 8.0muz+\gamma+\delta+1}{\alpha+1{\mathchar
44\relax}\mskip 8.0mu\beta+\delta+1{\mathchar 44\relax}\mskip
8.0mu\gamma+1};1\right]},\quad\lambda(z):=z(z+\gamma+\delta+1)$ (4.3.13)

for $D\in\mathbb{N}$, $d=0,1,\dotsc,D$, and $\alpha+1=-D$ or
$\beta+\delta+1=-D$ or $\gamma+1=-D$.

######  Fact 4.3.4 ([39, §9.2, pp.191–192]).

The Racah polynomial $R_{d}:=R_{d}(\lambda(z);\alpha,\beta,\gamma,\delta)$
satisfies the recurrence equation

$\displaystyle\lambda(z)R_{d}=\widetilde{A}_{d}R_{d+1}-(\widetilde{A}_{d}+\widetilde{C}_{d})R_{d}+\widetilde{C}_{d}R_{d-1}$
(4.3.14)

for $d=0,1,\dotsc,D$, where we set $R_{-1}=R_{D+1}:=0$ and

$\displaystyle\widetilde{A}_{d}$
$\displaystyle:=\frac{(d+\alpha+1)(d+\alpha+\beta+1)(d+\beta+\delta+1)(d+\gamma+1)}{(2d+\alpha+\beta+1)(2d+\alpha+\beta+2)},$
(4.3.15) $\displaystyle\widetilde{C}_{d}$
$\displaystyle:=\frac{d(d+\alpha+\beta-\gamma)(d+\alpha-\delta)(d+\beta)}{(2d+\alpha+\beta)(2d+\alpha+\beta+1)}.$
(4.3.16)

###### Proof of Lemma 4.3.3.

As mentioned in Remark 4.2.2, putting

$\displaystyle(D,d,z,\alpha,\beta,\gamma,\delta)=(m,x,M,-m-1,-n+m-1,-M-N-1,0),$
(4.3.17)

and $R_{x}:=R_{x}(\lambda(M);-m-1,-n+m-1,-M-N-1,0)$, we have

$\displaystyle
p(x)=\binom{n-k}{m-l}\frac{\binom{n}{x}}{\binom{n}{m}}\frac{n-2x+1}{n-x+1}\,R_{x}.$
(4.3.18)

Then, a direct calculation using Fact 4.3.4 yields

$\displaystyle\lambda(M)R_{a}=a_{x}R_{x+1}-(a_{x}+c_{x})R_{x}+c_{x}R_{x-1},$
(4.3.19)

where $a_{x}$ and $c_{x}$ are given in the statement. By $\lambda(M)=-MN$ and
$\binom{n-k}{m-l}\binom{n}{m}^{-1}R_{x}=\frac{n-x+1}{n-2x+1}\binom{n}{x}^{-1}p(x)$,
we have the consequence. ∎

As an application of the recurrence relation of Lemma 4.3.3, we study the
support of the distribution $P_{n,m,k,l}$.

######  Corollary 4.3.5.

In the case $m\leq n-m$, we have

$\displaystyle p(x)=0\quad(x>k).$ (4.3.20)

###### Proof.

We can assume $k+1\leq m$ since we have $p(x)=0$ for $a>m$ by the assumption
$n\geq 2m$. We will show $p(x)=0$ for $k+1\leq x\leq m$ by induction on $x$.

As for $p(k+1)$, the Racah polynomial in Theorem 4.2.1 is

$\displaystyle{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-(k+1){\mathchar
44\relax}\mskip 8.0mu(k+1)-n-1{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mum-n{\mathchar
44\relax}\mskip 8.0mu-
M-N};1\right]}={}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-(k+1){\mathchar
44\relax}\mskip 8.0mu-M{\mathchar 44\relax}\mskip 8.0mu-N}{-m{\mathchar
44\relax}\mskip 8.0mum-n};1\right]}.$ (4.3.21)

Let us recall one of Thomae’s ${}_{3}F_{2}$ transformation formulas (see [20,
(3.1.1)–(3.1.3)] and [3, Corollary 3.3.4, Proof]):

$\displaystyle{}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-d{\mathchar
44\relax}\mskip 8.0mu\alpha{\mathchar 44\relax}\mskip
8.0mu\beta}{\gamma{\mathchar 44\relax}\mskip
8.0mu\delta};1\right]}=\frac{(\delta-\beta)_{d}}{(\delta)_{d}}{}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-d{\mathchar
44\relax}\mskip 8.0mu\gamma-\alpha{\mathchar 44\relax}\mskip
8.0mu\beta}{\gamma{\mathchar 44\relax}\mskip
8.0mu1+\beta-\delta-d};1\right]},\quad d\in\mathbb{N}.$ (4.3.22)

Applying this formula to the case
$(d,\alpha,\beta,\gamma,\delta)=(k+1,-M,-N,-m,m-n)$, we have in the right hand
side the term $(\delta-\beta)_{d}=(k-l)_{k+1}$, which is $0$. Thus, we find
$p(k+1)=0$.

As for $p(k+2)$, we want to apply the recurrence formula in Lemma 4.3.3 to the
case $x=k+1$. On the coefficients of the recurrence formula, we have
$c_{k+1}=0$ and $a_{k+1}\neq 0$. The latter holds since if $a_{k+1}=0$ then
$n-k-(k+1)=0$, which would mean $k+1>n/2\geq m$. Thus, the recurrence formula
yields $p(k+2)=0$.

Now, the induction step holds by the recurrence formula, and have the
consequence. ∎

Combining the system symmetry (4.0.5), Remark 4.1.2 (1) and Corollary 4.3.5,
we have:

######  Corollary 4.3.6.

For any $(n,m,k,l)$, the support of the distribution $P_{n,m,k,l}$ is given by

$\displaystyle p(x)=0\ \text{ for }\ x>m\wedge(n-m)\wedge k.$ (4.3.23)

###  4.4. Relation of expectation and variance

In this subsection, we derive a relation between the expectation and the
variance of our probability distribution by using the representation theory of
$\operatorname{SU}(2)$, instead of $\mathfrak{S}_{n}$ used so far.
Specifically, we use the action of the Casimir element of the Lie algebra
$\mathfrak{sl}(2,\mathbb{C})$ on our space $(\mathbb{C}^{2})^{\otimes n}$. The
result is:

######  Theorem 4.4.1.

Let $X$ be a random variable distributed by $p(x)=p(x\,|\,n,m,k,l)$. Then, the
expectation $\mathbb{E}[X]$ and the variance $\mathbb{V}[X]$ are related as

$\displaystyle\frac{\mathbb{V}[X]}{n^{2}}=\frac{\mathbb{E}[X]}{n}\left(1-\frac{\mathbb{E}[x]}{n}\right)+\frac{\mathbb{E}[X]}{n^{2}}-\eta,$
(4.4.1)

where we put

$\displaystyle\eta:=\frac{n(n-m)-(m-l)(n-m-k+l)}{n^{2}}.$ (4.4.2)

Using the ratio $\alpha=l/n$, $\beta=(m-l)/n$, $\gamma=(k-l)/n$ and
$\delta=(n-m-k+l)/n$ in (2.3.4), we can rewrite the quantity $\eta$ in (4.4.2)
as

$\displaystyle\eta=\alpha\gamma+\alpha\delta+\beta\gamma,$ (4.4.3)

which appears in (2.3.6) of the central limit Theorem 2.3.4.

As a preliminary of the proof, let us recall the natural action of
$\mathfrak{sl}(2,\mathbb{C})$ on our tensor space

$\displaystyle(\mathbb{C}^{2})^{\otimes n}=\bigoplus_{x=0}^{\lfloor
n/2\rfloor}\mathcal{U}_{(n-x,x)}\boxtimes\mathcal{V}_{(n-x,x)}.$ (4.4.4)

First, the Lie algebra $\mathfrak{su}(2)$ of the Lie group
$\operatorname{SU}(2)$ acts on $(\mathbb{C}^{2})^{\otimes n}$ by the
differential representation $\varphi$ of the
$\operatorname{SU}(2)$-representation. Explicitly, we have

$\displaystyle\varphi(A)(\left|i_{1}\dotsm
i_{n}\right>)=\sum_{j=1}^{n}\left|i_{1}\dotsm i_{j-1}\right>\otimes
A\left|i_{j}\right>\otimes\left|i_{j+1}\dotsm i_{n}\right>$ (4.4.5)

for $A\in\mathfrak{su}(2)$ and $i_{j}\in\\{0,1\\}$, $j=1,\dotsc,n$. Since
$\mathfrak{sl}(2,\mathbb{C})$ is the complexification of $\mathfrak{su}(2)$,
the action $\varphi$ of $\mathfrak{su}(2)$ can be extended by
$\mathbb{C}$-linearly to that of $\mathfrak{sl}(2,\mathbb{C})$, which is the
desired action. We denote the obtained $\mathfrak{sl}(2,\mathbb{C})$-action by
the same symbol $\varphi$.

We denote the standard $\mathfrak{sl}(2,\mathbb{C})$-triple by

$\displaystyle E:=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix},\quad
F:=\begin{bmatrix}0&0\\\ 1&0\end{bmatrix},\quad H:=\begin{bmatrix}1&0\\\
0&-1\end{bmatrix},$ (4.4.6)

which satisfy $[E,F]=H$, $[H,E]=2E$ and $[H,F]=-2F$. The action $\varphi$
coincides with the one induced by the matrix product on
$\mathbb{C}^{2}=\mathbb{C}\left|1\right>\oplus\mathbb{C}\left|0\right>$ with
the identification $\left|1\right>=\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]$ and $\left|0\right>=\left[\begin{smallmatrix}0\\\
1\end{smallmatrix}\right]$. Explicitly, we have

$\displaystyle He_{1}=e_{1},\ Ee_{1}=0,\ Fe_{1}=e_{0},\quad He_{0}=-e_{0},\
Ee_{0}=e_{1},\ Fe_{0}=0.$ (4.4.7)

For $I=(i_{1},\dotsc,i_{n})\in\mathbb{F}_{2}^{n}=\\{0,1\\}^{n}$, we denote

$\displaystyle\left|I\right>:=\left|i_{1}\dotsm
i_{n}\right>,\quad\left|I\right|:=i_{1}+\dotsb+i_{n}=\left|\\{r=1,\dotsc,n\mid
i_{r}=1\\}\right|.$ (4.4.8)

Then, we have

$\displaystyle\varphi(H)\left|I\right>=(2\left|I\right|-n)\left|I\right>.$
(4.4.9)

The element

$\displaystyle\Omega:=H^{2}+2H+4FE$ (4.4.10)

in the universal enveloping algebra $U(\mathfrak{sl}(2,\mathbb{C}))$ is
central, called the (quadratic) Casimir element of
$\mathfrak{sl}(2,\mathbb{C})$. The $\operatorname{SU}(2)$-representation
$\mathcal{U}_{(n-x,x)}$ in (4.4.4) is the highest weight irrep of dimension
$n-2x+1$, and on which $\Omega$ acts by

$\displaystyle\left.\varphi(\Omega)\right|_{\mathcal{U}_{(n-x,x)}}=(n-2x)(n-2x+2)\operatorname{\mathsf{id}}_{\,\mathcal{U}_{(n-x,x)}}.$
(4.4.11)

Thus, $\Omega$ acts on the total space $(\mathbb{C}^{2})^{\otimes n}$ by

$\displaystyle\varphi(\Omega)=\sum_{x=0}^{\lfloor
n/2\rfloor}(n-2x)(n-2x+2)\mathsf{P}_{(n-x,x)},$ (4.4.12)

where $\mathsf{P}_{(n-x,x)}:(\mathbb{C}^{2})^{\otimes
n}\twoheadrightarrow\mathcal{U}_{(n-x,x)}\boxtimes\mathcal{V}_{(n-x,x)}$
denotes the projector (1.0.13).

Now, recall the vector $\left|\Xi_{n,m|k,l}\right>\in(\mathbb{C}^{2})^{\otimes
n}$.

######  Proposition 4.4.2.

We have

$\displaystyle\left<\Xi_{n,m|k,l}\right|\varphi(\Omega)\left|\Xi_{n,m|k,l}\right>=(n-2m)^{2}+2n+4(m-l)(n-m-k+l).$
(4.4.13)

###### Proof.

By (4.4.9), we have

$\displaystyle\left<\Xi_{n,m|k,l}\right|\varphi(H^{2}+2H)\left|\Xi_{n,m|k,l}\right>=(2m-n)^{2}+(2m-n).$
(4.4.14)

Thus, it is enough to show

$\displaystyle\left<\Xi_{n,m|k,l}\right|\varphi(FE)\left|\Xi_{n,m|k,l}\right>=k-l+(m-l+1)(n-k-m+l).$
(4.4.15)

Since $\varphi(E)$ and $\varphi(F)$ are adjoint with respect to the Hermitian
form, we have

$\displaystyle\left<\Xi_{n,m|k,l}\right|\varphi(FE)\left|\Xi_{n,m|k,l}\right>=\left<\Xi_{n,m|k,l}\right|\varphi(F)\varphi(E)\left|\Xi_{n,m|k,l}\right>=\left<\Xi_{n,m|k,l}\right|\varphi(E)^{\dagger}\varphi(E)\left|\Xi_{n,m|k,l}\right>$
$\displaystyle=\binom{n-k}{m-k}^{-1}\sum_{I,J\in\mathbb{F}_{2}^{n-k},\,\left|I\right|=\left|J\right|=m-l}\left<1^{k}0^{k-l}I\right|\varphi(E)^{\dagger}\varphi(E)\left|1^{k}0^{k-l}J\right>.$
(4.4.16)

Here we denoted by $\dagger$ the adjoint with respect to the Hermitian form,
and used the notation (4.4.8).

Now, we consider the three-factor decomposition

$\displaystyle(\mathbb{C}^{\otimes 2})^{\otimes n}=(\mathbb{C}^{\otimes
2})^{\otimes l}\otimes(\mathbb{C}^{\otimes
2})^{\otimes(k-l)}\otimes(\mathbb{C}^{\otimes 2})^{\otimes(n-k)},$ (4.4.17)

and denote by $\varphi^{\prime}(E)$ and $\varphi^{\prime\prime}(E)$ the action
of $\varphi(E)$ on the factor $(\mathbb{C}^{\otimes 2})^{\otimes(k-l)}$ and
$(\mathbb{C}^{\otimes 2})^{\otimes(n-k)}$ respectively. Then, by $Ee_{1}=0$ in
(4.4.7), we have

$\displaystyle\left<1^{k}0^{k-l}I\right|\varphi(E)^{\dagger}\varphi(E)\left|1^{k}0^{k-l}J\right>=\left<0^{k-l}I\right|\varphi^{\prime}(E)^{\dagger}\varphi^{\prime}(E)\left|0^{k-l}J\right>.$
(4.4.18)

As for the right hand side, we have

$\displaystyle\varphi^{\prime}(E)\left|0^{k-l}I\right>=\sum_{i=1}^{k-l}\left|0^{i-1}10^{k-l-i}I\right>+\left|0^{k-l}\right>\otimes\varphi^{\prime\prime}(E)\left|I\right>.$
(4.4.19)

Hence, we have

$\displaystyle\eqref{eq:EV:**}=(k-l)\delta(I,J)+\left<I\right|\varphi^{\prime\prime}(E)^{\dagger}\varphi^{\prime\prime}(E)\left|J\right>.$
(4.4.20)

Here and hereafter we denote
$\delta(I,J):=\delta_{i_{1},j_{1}}\dotsm\delta_{i_{n-k}j_{n-k}}$. Thus, we
have

$\displaystyle\eqref{eq:EV:*}=k-l+\binom{n-k}{m-l}^{-1}\sum_{I,J\in\mathbb{F}_{2}^{n-k},\,\left|I\right|=\left|J\right|=m-l}\left<I\right|\varphi^{\prime\prime}(E)^{\dagger}\varphi^{\prime\prime}(E)\left|J\right>.$
(4.4.21)

As for the second term in the right hand side of (4.4.21), we have

$\displaystyle\varphi^{\prime\prime}(E)\left|I\right>=\sum_{r=1}^{n-k}\delta_{i_{r},0}|I^{<r}\,1\,I^{>r}\rangle,\quad
I^{<r}:=(i_{1},\dotsc,i_{r-1}),\ I^{>r}:=(i_{r+1},\dotsc,i_{n-k}).$ (4.4.22)

Then, we have

$\displaystyle\begin{split}\left<I\right|\varphi^{\prime\prime}(E)^{\dagger}\varphi^{\prime\prime}(E)\left|J\right>&=\sum_{r=1}^{n-k}\sum_{s=1}^{n-k}\delta_{i_{r},0}\delta_{j_{s},0}\langle
I^{<r}\,1\,I^{>r}\,|\,J^{<s}\,1\,J^{>r}\rangle\\\
&=\sum_{r=1}^{n-k}\delta_{i_{r},0}\delta_{j_{r},0}\delta(I^{\neq r},J^{\neq
r})+\sum_{r\neq
s}\delta_{i_{r},0}\delta_{j_{s},0}\delta_{i_{s},1}\delta_{j_{r},1}\delta(I^{\neq
r,s}J^{\neq r,s}),\end{split}$ (4.4.23)

where we defined $I^{\neq r}:=(i_{1},\dotsc,i_{r-1},i_{r+1},\dotsc,i_{r})$ and
similarly $I^{\neq r,s}$ to be the sequence deleting the $r$\- and $s$-th
components of $I$. Going back to (4.4.21), we have

$\displaystyle\begin{split}\eqref{eq:EV:***}&=(k-l)+\binom{n-k}{m-l}^{-1}\left(\sum_{r=1}^{n-k}\binom{n-k-1}{m-l}+\sum_{r\neq
s}\binom{n-k-2}{m-l-1}\right)\\\
&=k-l+\binom{n-k}{m-l}^{-1}\left((n-k)\binom{n-k-1}{m-l}+(n-k)(n-k-1)\binom{n-k-2}{m-l-1}\right)\\\
&=k-l+N+MN=k-l+(M+1)N\end{split}$ (4.4.24)

with $M:=m-l$ and $N:=n-m-k+l$, which is the desired (4.4.15). ∎

Now, we show Theorem 4.4.1.

###### Proof of Theorem 4.4.1.

The spectral decomposition (4.4.12) of $\varphi(\Omega)$ yields

$\displaystyle\left<\Xi_{n,m|k,l}\right|\varphi(\Omega)\left|\Xi_{n,m|k,l}\right>=\sum_{x\geq
0}(n-2x)(n-2x+2)\left<\Xi_{n,m|k,l}\right|\mathsf{P}_{(n-x,x)}\left|\Xi_{n,m|k,l}\right>$
$\displaystyle=\sum_{x\geq
0}(n-2x)(n-2x+2)\,p(x)=n(n+2)+4\mathbb{E}[X^{2}]-4(n+1)\mathbb{E}[X].$
(4.4.25)

Comparing it with Proposition 4.4.2, we have

$\displaystyle\mathbb{E}[X^{2}]-(n+1)\mathbb{E}[X]=(m-l)(n-m-k+l)+m^{2}-mn,$
(4.4.26)

The expectation $\mathbb{E}[X]$ and the variance $\mathbb{V}[X]$ are related
by

$\displaystyle\mathbb{E}[X^{2}]-(n+1)\mathbb{E}[X]=(m-l)(n-m-k+l)+m^{2}-mn=-n^{2}\eta,$
(4.4.27)

where $\eta$ is given in (4.4.2). Since
$\mathbb{V}[X]=\mathbb{E}[X^{2}]-\mathbb{E}[X]^{2}$, we have

$\displaystyle\frac{\mathbb{V}[X]}{n^{2}}=\frac{\mathbb{E}[X]}{n}\left(1-\frac{\mathbb{E}[x]}{n}\right)+\frac{\mathbb{E}[X]}{n^{2}}-\eta$
(4.4.28)

which is the consequence. ∎

###  4.5. Specialization of parameters

In this subsection we discuss some special cases of the parameters
$(n,m,k,l)$. Recall the system symmetry (4.0.5):

$\displaystyle p(x\,|\,n,m,k,l)=p(x\,|\,n,n-m,k,k-l),$ (4.5.1)

which can be restated the invariance under the switch

$\displaystyle(K,L,M,N)\longleftrightarrow(L,K,N,M)$ (4.5.2)

with $l:=L$, $K:=k-l$, $M:=m-l$ and $N:=n-m-k+l$.

####  4.5.1. Case $MN=0$

In this part we treat the case when $M:=m-l$ or $N:=n-m-k+l$ is equal to $0$,
which is mentioned at §2.2.2, (2.2.14). In each case, the vector
$\left|\Xi_{n,m|k,l}\right>$ is given by

$\displaystyle\left|\Xi_{n,m|k,m}\right>=\left|\Xi_{n,m|k,m+k-n}\right>=\left|1^{m}0^{n-m}\right>.$
(4.5.3)

Thus, the pmf $p(x)$ is independent of the parameters $k$ and $l$.

######  Lemma 4.5.1.

Assume $m-l=0$ or $n-m-k+l=0$. Then, we have

$\displaystyle
p(x)=\frac{\binom{n}{x}}{\binom{n}{m}}\frac{n-2x+1}{n-x+1}=\frac{\binom{n}{x}-\binom{n}{x-1}}{\binom{n}{m}}$
(4.5.4)

for $x=0,1,\dotsc,m\wedge(n-m)$.

###### Proof.

Assume $m\leq n-m$. Then, by Theorem 4.2.1 we have

$\displaystyle
p(x)=\binom{M+N}{M}\frac{\binom{n}{x}}{\binom{n}{m}}\frac{n-2x+1}{n-x+1}{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mu-n+x-1{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mu-n+m{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}$ (4.5.5)

for $x\leq m$ with $M:=m-l$ and $N:=n-m-k+l$. The assumption is $M=0$ or
$N=0$, which yields $\binom{M+N}{M}=1$ and
${}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar 44\relax}\mskip
8.0mu-n+x-1{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar 44\relax}\mskip
8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mu-n+m{\mathchar 44\relax}\mskip
8.0mu-M-N};1\right]}=1$. Thus, we have the first equality. The second equality
follows from

$\displaystyle\binom{n}{x}\frac{n-2x+1}{n-x+1}=\binom{n}{x}-\binom{n}{x}\frac{x}{n-x+1}=\binom{n}{x}-\binom{n}{x-1}.$
(4.5.6)

Since both equalities are symmetric under $(m,n-m)\leftrightarrow(n-m,m)$,
they also hold in the case $m\geq n-m$. ∎

As mentioned at (2.2.14) in §2.2.2, we can show
$\sum_{x=0}^{m\wedge(n-m)}p(x)=1$ in this case directly from the rightmost
expression in (4.5.4). By a similar calculation, we can find that the
expectation is given by

$\displaystyle\mathbb{E}[X]:=\sum_{x=0}^{m\wedge(n-m)}x\,p(x)=\begin{cases}m-\binom{n}{m}^{-1}\sum_{x=0}^{m-1}\binom{n}{x}&(m\leq
n-m)\\\ (n-m)-\binom{n}{m}^{-1}\sum_{x=0}^{n-m-1}\binom{n}{x}&(m\geq
n-m)\end{cases}.$ (4.5.7)

We can further rewrite it by the formula of partial sum of binomial
coefficients in terms of incomplete beta function [59, p.52, (3-3)]. In the
case $m\leq n-m$, we have

$\displaystyle\mathbb{E}[X]=m-2^{n}m\int_{0}^{1/2}t^{n-m}(1-t)^{m-1}\,dt.$
(4.5.8)

######  Remark 4.5.2.

Let us consider the meaning of the condition $MN=0$ in the representation
theoretic viewpoint. Recall the hook formula of the $\mathfrak{S}_{n}$-irrep
$\mathcal{V}_{(n-a,a)}$ in (1.0.10):

$\displaystyle\dim\mathcal{V}_{(n-x,x)}=\frac{n-2x+1}{n-x+1}\binom{n}{x}=\binom{n}{x}-\binom{n}{x-1}.$
(4.5.9)

Hence, our probability distribution in the case $MN=0$ is just given by the
normalization of the above dimension. This dimension can be regarded as the
multiplicity of the $x$-th eigenvalue of the discrete Laplacian on the Johnson
graph $J(n,m)=(\Gamma,\partial)$ appearing in (4.1.12) and (4.1.13). There we
discussed zonal spherical functions for the Gelfand pair
$(G,K)=(\mathfrak{S}_{n},\mathfrak{S}_{m}\times\mathfrak{S}_{n-m})$, and the
Johnson graph is identified with the homogeneous space $G/K$. If we put the
weight (4.5.9) at the $x$-th eigenvalue of $L$ instead of at $x$, then we
obtain the spectral distribution of $L$. See [31] and [32, Chap. 6] for the
detail of this spectral distribution. We will return to this topic in Remark
5.2.7.

####  4.5.2. Case $KL=0$

In this part we study the case when $K:=k-l$ or $L:=l$ is equal to $0$. In
each case, the vector $\left|\Xi_{n,m|k,l}\right>$ is given by

$\displaystyle\left|\Xi_{n,m|l,l}\right>=\left|1^{l}\right>\otimes\left|\Xi_{n-l,m-l}\right>,\quad\left|\Xi_{n,m|k,0}\right>=\left|0^{k}\right>\otimes\left|\Xi_{n-k,m}\right>.$
(4.5.10)

Let us consider the case $k=l$ first.

######  Proposition 4.5.3.

Under the assumption $m\leq n-m$ and $k=l$, we have

$\displaystyle
p(x)=\frac{\binom{n}{x}}{\binom{n}{m}}\frac{\binom{l}{x}}{\binom{m}{x}}\frac{n-2x+1}{n-x+1}\binom{n-l-x}{m-l}$
(4.5.11)

for $x\leq l\wedge m$, and $p(x)=0$ otherwise.

Although it is not surprising if this distribution has been studied, we could
not find it in literature.

We give two proofs of Proposition 4.5.3.

###### Proof 1 of Proposition 4.5.3.

Recall the Racah presentation in Theorem 2.2.2:

$\displaystyle
p(x)=\binom{M+N}{M}\frac{\binom{n}{x}}{\binom{n}{m}}\frac{n-2x+1}{n-x+1}{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mu-n+x-1{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mu-n+m{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}$ (4.5.12)

with $M:=m-l$ and $N:=n-m-k+l$. The assumption $k=l$ yields $N=n-m$ and

$\displaystyle{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mu-n+x-1{\mathchar 44\relax}\mskip 8.0mu-M{\mathchar
44\relax}\mskip 8.0mu-N}{-m{\mathchar 44\relax}\mskip 8.0mu-n+m{\mathchar
44\relax}\mskip 8.0mu-
M-N};1\right]}={}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mu-n+x-1{\mathchar 44\relax}\mskip 8.0mu-M}{-m{\mathchar
44\relax}\mskip 8.0mu-M-N};1\right]}.$ (4.5.13)

Now, recall the Pfaff-Saalschutz formula for balanced terminating
${}_{3}F_{2}$ [3, Theorem 2.2.6], [20, (1.7.1)], [53, (2.3.1.4)]: Denoting a
product of Gamma functions by
$\Gamma(a,b,\dotsc,c):=\Gamma(a)\Gamma(b)\dotsm\Gamma(c)$, we have

$\displaystyle{}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{a{\mathchar
44\relax}\mskip 8.0mub{\mathchar 44\relax}\mskip 8.0muc}{d{\mathchar
44\relax}\mskip
8.0mue};1\right]}=\frac{\Gamma(d,1+a-e,1+b-e,1+c-e)}{\Gamma(1-e,d-a,d-b,d-c)},$
(4.5.14)

where one of the numerator parameters $a,b,c$ is a negative integer and
$a+b+c+1=d+e$. We can apply this formula to (4.5.13) with $d=-m$, and the
result is

$\displaystyle{}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-x{\mathchar
44\relax}\mskip 8.0mu-n+x-1{\mathchar 44\relax}\mskip 8.0mu-M}{-m{\mathchar
44\relax}\mskip 8.0mu-
M-N};1\right]}=\frac{\Gamma(-m,n-l-x+1,x-l,N+1)}{\Gamma(M+N+1,x-m,N-x+1,-l)}$
(4.5.15)
$\displaystyle=\binom{M+N}{M}^{-1}\frac{1}{M!}\frac{\Gamma(-m,n-l-x+1,x-l)}{\Gamma(x-m,n-m-x+1,-l)}=\binom{M+N}{M}^{-1}\binom{n-l-x}{m-l}\frac{l!}{m!}\frac{(m-x)!}{(l-x)!}.$
(4.5.16)

Multiplying the overall factor, we obtain Proposition 4.5.3. ∎

###### Proof 2 of Proposition 4.5.3.

We only give a sketch. Let us cite a lemma on dual Hahn polynomials (see
Remark 4.1.3).

######  Fact 4.5.4 ([39, §9.6, (9.6.12)]).

For $D\in\mathbb{N}$, we have the following generating function of dual Hahn
polynomials $R_{d}$, $d=0,1,\dotsc,D$.

$\displaystyle\begin{split}\sum_{d=0}^{D}\frac{(a+1,-D)_{d}}{(-b-D,1)_{d}}R_{d}(\lambda(z);a,b,D)t^{d}&=\sum_{d=0}^{D}\frac{(a+1,-D)_{d}}{(-b-D,1)_{d}}{}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-d{\mathchar
44\relax}\mskip 8.0mu-z{\mathchar 44\relax}\mskip 8.0muz+a+b+1}{a+1{\mathchar
44\relax}\mskip 8.0mu-D};1\right]}t^{d}\\\
&=(1-t)^{z}{}_{2}F_{1}{\left[\genfrac{.}{.}{0.0pt}{}{z-D{\mathchar
44\relax}\mskip 8.0muz+a+1}{-b-D};t\right]}.\end{split}$ (4.5.17)

By (4.2.7) and Fact 4.5.4, we can show the following:

######  Lemma 4.5.5.

We have the following generating function for $\omega_{(n-x,x)}(i)$.

$\displaystyle\sum_{i\geq
0}\binom{m}{i}\binom{n}{i}\omega_{(n-x,x)}(i)t^{i}=(1-t)^{x}\sum_{i\geq
0}\binom{m-a}{i}\binom{n-m-a}{i}t^{i}.$ (4.5.18)

For $u\in\mathbb{N}$ with $u\leq m$, we also have

$\displaystyle\sum_{i\geq
0}\binom{m-u}{i-u}\binom{n}{i}\omega_{(n-x,x)}(i)=(-1)^{x}\binom{n-u-x}{m-x}\binom{m-x}{u-x}\binom{m}{u}^{-1}.$
(4.5.19)

Then, using (4.5.19) and the Chu-Vandermonde identity (2.2.11), we can show

$\displaystyle\sum_{i\geq 0}\binom{m-l}{i}\binom{n-m}{i}\omega_{(n-x,x)}(i)$
$\displaystyle=(-1)^{x}\frac{l!}{m!}\cdot(-1)^{x}\frac{(n-2x)!}{(l-x)!\,(n-m-x)!}{}_{2}F_{1}{\left[\genfrac{.}{.}{0.0pt}{}{x-l{\mathchar
44\relax}\mskip 8.0mux+m-n}{2x-n};1\right]}.$ (4.5.20)

Finally, using Gauss’s summation formula
${}_{2}F_{1}{\left[\genfrac{.}{.}{0.0pt}{}{a{\mathchar 44\relax}\mskip
8.0mub}{c};1\right]}=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$
for $\mathop{\mathrm{Re}}\nolimits(c-a-b)>0$, we have

$\displaystyle\sum_{i\geq
0}\binom{m-l}{i}\binom{n-m}{i}\omega_{(n-x,x)}(i)=\frac{l!}{m!}\frac{(n-2x)!}{(l-x)!\,(n-m-x)!}\binom{n-l-x}{m-l}\binom{n-2x}{m-x}.$
(4.5.21)

Multiplying this equality by
$T(x)=\binom{n}{m}^{-1}\binom{n}{x}\frac{n-2x-1}{n-x-1}$, we obtain
Proposition 4.5.3. ∎

######  Remark 4.5.6.

By Theorem 4.3.1, we have $\sum_{x=0}^{l}p(x)=1$ in the case $k=l$. Combined
with the expression of $p(x)$ in Proposition 4.5.3, it yields a non-trivial
summation formula. One can check that the obtained formula is equivalent to

$\displaystyle{}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-n{\mathchar
44\relax}\mskip 8.0mu-l{\mathchar 44\relax}\mskip
8.0mu-(n-m)}{-(n-l){\mathchar 44\relax}\mskip
8.0mu-m};-1\right]}-\frac{l(n-m)}{m(n-l)}{}_{3}F_{2}{\left[\genfrac{.}{.}{0.0pt}{}{-n{\mathchar
44\relax}\mskip 8.0mu-l+1{\mathchar 44\relax}\mskip
8.0mu-(n-m)+1}{-(n-l)+1{\mathchar 44\relax}\mskip
8.0mu-m+1};-1\right]}=\frac{n^{\downarrow l}}{m^{\downarrow l}},$ (4.5.22)

which can be shown independently by the Wilf-Zeilberger method [3, §3.11]. We
omit the detail and leave it to the readers. Unfortunately we cannot find a
derivation from known summation formulas of hypergeometric functions.

The case $L:=l=0$ can be similarly treated, and we only record the result.

######  Proposition 4.5.7.

Assume $m\leq n-m$ and $l=0$. Then, we have

$\displaystyle
p(x)=\frac{\binom{n}{x}}{\binom{n}{m}}\frac{\binom{k}{x}}{\binom{n-m}{x}}\frac{n-2x+1}{n-x+1}\binom{n-k-x}{n-m-k}$
(4.5.23)

for $x\leq k\wedge m$, and $p(x)=0$ otherwise.

####  4.5.3. Remark on specialization in Askey scheme

One may wonder what will happen by applying the specializations in the Askey
scheme (Figure 2.1.1). One of the specializations from Racah polynomial
$R_{d}$ to Hahn polynomials $Q_{d}$ (see Remark 4.1.3) is given by [39,
(9.2.15)]:

$\displaystyle\lim_{\delta\to\infty}R_{d}(\lambda(z);\alpha,\beta,-D-1,\delta)=Q_{d}(z;\alpha,\beta,D).$
(4.5.24)

Unfortunately, we took $\delta=0$ in (4.2.4) for Racah presentation, and
cannot apply the above limit. Neither can we apply the other specializations
to Hahn and dual Hahn polynomials [39, (9.2.16)–(9.2.20)] by similar reason.

##  5\. Asymptotic analysis of the distribution $P_{n,m,k,l}$

In this section we study the asymptotic behavior of the distribution
$P_{n,m,k,l}$ in the limit $n\to\infty$. We consider the following two cases.

  1. I:

The limit $n\to\infty$ with $k,l$ and $m/n$ fixed (§5.1).

  2. II:

The limit $n\to\infty$ with $m/n$, $k/n$ and $l/n$ fixed (§§5.2–5.4).

We call them Type I and Type II limit, respectively. Hereafter the symbol
$F\sim G$ means $F-G=O(1/n)$ under $n\to\infty$.

###  5.1. Type I Limit $n\to\infty$ with $k,l$ and $m/n$ fixed

We have a clear understanding of Type I limit. Let us denote by $B_{\xi,j}(x)$
the binomial distribution $\\{0,1,\dotsc,j\\}\to\mathbb{R}$ with parameters
$0\leq\xi\leq 1$ and $j\in\mathbb{N}$ whose pmf is given by
$x\mapsto\binom{j}{x}\xi^{x}(1-\xi)^{j-x}$, and by $*$ the convolution of
probability distributions.

######  Theorem 5.1.1.

In the limit $n\to\infty$ with $k,l$ and $\xi=m/n$ fixed, we have

$\displaystyle P_{n,n\xi,k,l}\sim P_{R|\xi;k,l}:=B_{\xi,k-l}*B_{1-\xi,l}.$
(5.1.1)

The limit distribution $P_{R|\xi,k,l}$ is a discrete probability distribution
on $\\{0,1,\dotsc,k\\}$, and its expectation and variance are given by
$(k-l)\xi+l(1-\xi)$ and $k\xi(1-\xi)$, respectively.

We denote by $q(x|\xi;k,l)$ the pmf of the limit distribution $P_{R|\xi;k,l}$.
By definition of the convolution $*$, its explicit form is given by

$\displaystyle
q(x\,|\,\xi;k,l)=\xi^{l-x}(1-\xi)^{k-l-x}\sum_{u=0\vee(x-k+l)}^{x\wedge
l}\binom{k-l}{x-u}\binom{l}{u}\xi^{2(x-u)}(1-\xi)^{2u}.$ (5.1.2)

Note that the limit distribution $P_{R|\xi;k,l}$ inherits the system symmetry
(4.0.5) of the distribution $P_{n,m,k,l}$, and is invariant under the switch
$(\xi,k-l,l)\leftrightarrow(1-\xi,l,k-l)$, which can be checked by the
definition (5.1.1) and also by the expression (5.1.2) of limit pmf
$q(x\,|\,\xi;k,l)$.

Now, we show Theorem 5.1.1. The proof consists of two steps.

###### The first step of the proof of Theorem 5.1.1.

By the symmetry remarked after (5.1.2), it is enough to consider the case
$m\leq n-m$. Thus, we can use Hahn presentation of the pmf
$p(x)=p(x\,|\,n,m,k,l)$ in Theorem 4.1.1. Let us show it again: For $m\leq
n-m$, we have

$\displaystyle
p(x)=\frac{\binom{n}{x}}{\binom{n}{m}}\frac{n-2x+1}{n-x+1}\sum_{i\geq
0}\frac{\binom{M}{i}\binom{N}{i}}{\binom{m}{i}\binom{n-m}{i}}\sum_{r=0}^{i}(-1)^{r}\binom{x}{r}\binom{m-x}{i-r}\binom{n-m-x}{i-r}$
(5.1.3)

with $M:=m-l$ and $N:=n-m-k+l$. In this first step of the proof, we transform
the formula (5.1.3) without taking limit. The result is:

######  Proposition 5.1.2.

The formula (5.1.3) is equivalent to

$\displaystyle
p(x)=\frac{\binom{n-k}{m-l}}{\binom{n}{m}}\frac{n-2x+1}{n-x+1}\frac{n^{\downarrow
x}}{m^{\downarrow x}(n-m)^{\downarrow x}}\frac{1}{x!}w(x),$ (5.1.4)
$\displaystyle w(x):=\sum_{\begin{subarray}{c}0\leq u,v\leq x\\\ u+v\geq
x\end{subarray}}l^{\downarrow u}(k-l)^{\downarrow v}\frac{(M^{\downarrow
x-u})^{2}(N^{\downarrow x-v})^{2}}{(M+N)^{\downarrow
2x-u-v}}\frac{x!}{(u+v-x)!(x-u)!(x-v)!}.$ (5.1.5)

###### Proof.

Let us set $T:=\binom{n}{m}^{-1}\binom{n}{x}\frac{n-2x+1}{n-x+1}$ and

$\displaystyle W(x):=\binom{n-k}{m-l}^{-1}\sum_{i=0}^{M\wedge
N}\frac{\binom{M}{i}\binom{N}{i}}{\binom{m}{i}\binom{n-m}{i}}\,\sum_{r=0}^{i\wedge
x}(-1)^{r}\binom{x}{r}\binom{m-x}{i-r}\binom{n-m-x}{i-r}.$ (5.1.6)

Thus, the formula (5.1.3) is $p(x)=T\cdot\binom{n-k}{m-l}\cdot W(x)$, and the
consequence will be obtained if we show

$\displaystyle W(x)=\frac{1}{m^{\downarrow x}(n-m)^{\downarrow
x}}\sum_{u,v=0}^{x}l^{\downarrow u}(k-l)^{\downarrow v}\frac{(M^{\downarrow
x-u})^{2}(N^{\downarrow x-v})^{2}}{(M+N)^{\downarrow
2x-u-v}}\binom{x}{u+v-x}\binom{2x-u-v}{x-u}.$ (5.1.7)

We start with

$\displaystyle\frac{\binom{m-x}{i-r}}{\binom{m}{i}}=\frac{i^{\downarrow
r}(m-i)^{\downarrow x-r}}{m^{\downarrow
x}},\quad\frac{\binom{n-m-x}{i-r}}{\binom{n-m}{i}}=\frac{i^{\downarrow
r}(n-m-i)^{\downarrow x-r}}{(n-m)^{\downarrow x}}.$ (5.1.8)

Thus, we have

$\displaystyle W(x):=I(x)/\bigl{(}m^{\downarrow x}(n-m)^{\downarrow
x}\bigr{)},$ (5.1.9) $\displaystyle
I(x):=\binom{M+N}{N}^{-1}\sum_{i=0}^{M\wedge N}\sum_{r=0}^{i\wedge
x}\binom{M}{i}\binom{N}{i}(-1)^{r}\binom{x}{r}i^{\downarrow
r}(m-i)^{\downarrow x-r}i^{\downarrow r}(n-m-i)^{\downarrow x-r}.$ (5.1.10)

Next, using the identity

$\displaystyle(y+z)^{\downarrow j}=\sum_{U=0}^{j}y^{\downarrow U}z^{\downarrow
j-U}\binom{j}{U}$ (5.1.11)

for commuting letters $x,y$ and $j\in\mathbb{N}$, we have

$\displaystyle\begin{split}&i^{\downarrow r}\binom{M}{i}\cdot(m-i)^{\downarrow
x-r}=\frac{M^{\downarrow i}}{(i-r)!}\cdot\sum_{U=0}^{x-r}(M-i)^{\downarrow
U}l^{\downarrow x-r-U}\binom{x-r}{U}\\\ &=\sum_{U=0}^{x-r}\frac{M^{\downarrow
i+U}}{(i-r)!}l^{\downarrow x-r-U}\binom{x-r}{U}=\sum_{U=0}^{x-r}M^{\downarrow
r+U}\binom{M-r-U}{i-r}l^{\downarrow x-r-U}\binom{x-r}{U}.\end{split}$ (5.1.12)

Similarly we have

$\displaystyle i^{\downarrow r}\binom{N}{i}\cdot(n-m-i)^{\downarrow
x-r}=\sum_{V=0}^{x-r}N^{\downarrow r+V}\binom{N-r-V}{i-r}(k-l)^{\downarrow
x-r-V}\binom{x-r}{V}.$ (5.1.13)

Then, changing the order of summations, we can rewrite $I(x)$ as

$\displaystyle\begin{split}I(x)=&\sum_{U,V=0}^{x}\sum_{r=0}^{x}(-1)^{r}\binom{x}{r}l^{\downarrow
a-r-U}\binom{x-r}{U}(k-l)^{\downarrow x-r-V}\binom{x-r}{V}M^{\downarrow
r+U}N^{\downarrow r+V}\\\ &\cdot\binom{M+N}{N}^{-1}\sum_{i=0}^{M\wedge
N}\binom{M-r-U}{i-r}\binom{N-r-V}{i-r}.\end{split}$ (5.1.14)

Using the Chu-Vandermonde formula (2.2.11), we have

$\displaystyle\sum_{i=0}^{M\wedge
N}\binom{M-r-U}{i-r}\binom{N-r-V}{i-r}=\binom{M+N-2r-U-V}{N-r-V}=\binom{M+N}{N}\frac{M^{\downarrow
r+U}N^{\downarrow r+V}}{(M+N)^{\downarrow 2r+U+V}}.$ (5.1.15)

Replacing $u:=x-r-U$ and $v:=x-r-V$, we can further rewrite $I(x)$ as

$\displaystyle I(x)=\sum_{u,v=0}^{x}l^{\downarrow u}(k-l)^{\downarrow
v}\frac{(M^{\downarrow x-u})^{2}(N^{\downarrow a-v})^{2}}{(M+N)^{\downarrow
2x-u-v}}J(u,v),\quad
J(u,v):=\sum_{r=0}^{x}(-1)^{r}\binom{x}{r}\binom{x-r}{u}\binom{x-r}{v}.$
(5.1.16)

By (5.1.9) and (5.1.16), in order to prove (5.1.7), it remains to show

$\displaystyle J(u,v)=\begin{cases}0&(u+v<x)\\\
\frac{x!}{(u+v-x)!(x-u)!(x-v)!}&(u+v\geq x)\end{cases}.$ (5.1.17)

Note that $J(u,v)$ is equal to the coefficient of $A^{u}B^{v}$ in
$\bigl{(}(1+A)(1+B)-1)^{x}$ for commuting letters $A$ and $B$. Thus, it is
also equal to the coefficient of $(AB)^{u+v-x}A^{x-v}B^{x-u}$ in
$(AB+A+B)^{x}$, and we have the above equality (5.1.17). ∎

This is the end of the first step. ∎

###### The second step of the proof of Theorem 5.1.1.

Next, as the second step, we take Type I limit $n\to\infty$ of Proposition
5.1.2.

######  Proposition 5.1.3.

In the limit $n\to\infty$ with $k,l$ and $\xi=m/n$ fixed, the function $h(x)$
in Proposition 5.1.2 behaves as

$\displaystyle\frac{n^{\downarrow x}}{m^{\downarrow x}(n-m)^{\downarrow
x}}\frac{1}{x!}h(x)\sim\xi^{-x}(1-\xi)^{-x}\sum_{u=0}^{x}\binom{l}{u}\binom{k-l}{x-u}\xi^{2(x-u)}(1-\xi)^{2u}\quad(n\to\infty).$
(5.1.18)

###### Proof.

We find from the expression of $h(x)$ that the terms with $u+v=x$ are dominant
in the limit. Then, we can estimate the left hand side of the statement as

$\displaystyle\begin{split}\frac{n^{\downarrow x}}{m^{\downarrow
x}(n-m)^{\downarrow
x}}\frac{1}{x!}h(x)\sim&\frac{n^{x}}{m^{x}(n-m)^{x}}\frac{1}{x!}\sum_{\begin{subarray}{c}0\leq
u,v\leq x\\\ u+v=x\end{subarray}}l^{\downarrow u}(k-l)^{\downarrow
v}\frac{m^{2(x-u)}(n-m)^{2(x-v)}}{n^{2x-u-v}}\binom{x}{u}\\\
&=\xi^{-x}(1-\xi)^{-x}\sum_{u=0}^{x}\binom{l}{u}\binom{k-l}{u}\xi^{2(x-u)}(1-\xi)^{2u}.\end{split}$
(5.1.19)

∎

By Proposition 5.1.3 and the estimation $\frac{m^{\downarrow
l}(n-m)^{\downarrow k-l}}{n^{\downarrow
k}}\frac{n-2x+1}{n-x+1}\sim\xi^{l}(1-\xi)^{k-l}$, we have

$\displaystyle
p(x)\sim\xi^{l-x}(1-\xi)^{k-l-x}\sum_{u=0}^{x}\binom{k-l}{x-u}\binom{l}{u}\xi^{2(x-u)}(1-\xi)^{2u},$
(5.1.20)

which is equivalent to the expression (5.1.2) of $q(x\,|\,\xi;k,l)$. Thus, we
obtain the first half (5.1.1) of Theorem 5.1.1.

Next we show the latter half of Theorem 5.1.1, i.e, calculate the expectation
and the variance of the limit distribution $P_{R|\xi,k,l}$. For that, we
consider the cumulant generating function for $P_{R|\xi;k,l}$:

$\displaystyle K(s):=\log\sum_{x=0}^{k}e^{sx}q(x\,|\,\xi;k,l).$ (5.1.21)

Recall that, denoting by $K_{P}$ the cumulant generating function of a
probability distribution $P$, we have $K_{P_{1}*P_{2}}=K_{P_{1}}+K_{P_{2}}$.
Then, since $K_{B_{\xi,j}}(s)=j\log(1-\xi+\xi e^{s})$ for the binomial
distribution $B_{\xi,j}$, we have

$\displaystyle K(s)=(k-l)\log\bigl{(}1-\xi+\xi
e^{s}\bigr{)}+l\log\bigl{(}\xi+(1-\xi)e^{s}\bigr{)}.$ (5.1.22)

Hence, the expectation and the variance of $P_{R|\xi;k,l}$ are given by

$\displaystyle\frac{d}{ds}K(0)=(k-l)\xi+l(1-\xi),\quad\frac{d^{2}}{d^{2}s}K(0)=k\xi(1-\xi).$
(5.1.23)

Thus, we finished the proof of Theorem 5.1.1 ∎

###  5.2. Type II Limit $n\to\infty$ with $m/n$, $k/n$ and $l/n$ fixed

Next we consider Type II limit, i.e., the limit $n\to\infty$ with the ratios
$m/n$, $k/n$ and $l/n$ fixed. As summarized in Table 1.0.1 of §1, we will use
the parametrization (2.3.4) of the fixed ratios:

$\displaystyle\alpha=\frac{l}{n},\quad\beta=\frac{m-l}{n},\quad\gamma=\frac{k-l}{n},\quad\delta=\frac{n-m-k+l}{n}.$
(5.2.1)

Recall that the condition $(n,m,k,l)\in\mathfrak{N}$ in (2.1.9) is equivalent
to (2.3.5):

$\displaystyle\alpha,\beta,\gamma,\delta\geq
0,\quad\alpha+\beta+\gamma+\delta=1.$ (5.2.2)

We also use the parametrization (2.3.3):

$\displaystyle\xi=\frac{m}{n}=\alpha+\beta,\quad\kappa=\frac{k}{n}=\alpha+\gamma.$
(5.2.3)

Before starting the main argument, we introduce some fundamental quantities
and discuss their property in the next Lemma 5.2.1. As will be explained
later, the quantity $\mu$ gives the asymptotic expectation, and $\sigma$
describes the asymptotic variance. Since Lemma 5.2.1 is technical, we