ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.08
https://doi.org/10.26493/1855-3974.2976.f76
(Also available at http://amc-journal.eu)
On the beta distribution, the nonlinear Fourier
transform and a combinatorial problem
Pavle Saksida *
Faculty of Mathematics and Physics, University of Ljubljana,
Jadranska ulica 21, Ljubljana, Slovenia
Received 19 October 2022, accepted 31 January 2023, published online 22 August 2023
Abstract
The paper describes some probabilistic and combinatorial aspects of the nonlinear
Fourier transform associated with the AKNS-ZS problems. We show that the volumes
of a family of polytopes that appear in a power expansion of the nonlinear Fourier trans-
form are distributed according to the beta probability distribution. We establish this result
by studying an Euler-type discretization of the nonlinear Fourier transform. This approach
leads to the combinatorial problem of finding the number of alternating ordered partitions
of an integer into a fixed number of distinct parts. We find the explicit formula for these
numbers and show that they are essentially distributed according to a novel discretization
of the beta distribution for a suitable choice of the shape parameters. We also find the
generating functions of the numbers of alternating sums. These functions are expressed in
terms of the our discrete nonlinear Fourier transform.
Keywords: Beta distribution, nonlinear Fourier transform, discretisation.
Math. Subj. Class. (2020): 37K15, 42A99, 60E05, 05A17
1 Introduction
As announced in the title, this paper investigates relations between three topics from differ-
ent parts of mathematics: probability distributions, combinatorics and the theory of nonlin-
ear partial differential equations, more concretely, the nonlinear Fourier transform. Despite
the apparent heterogeneity of the topics, the relations between them are rather natural.
*I am grateful to Matjaž Konvalinka for a very fruitful discussion. I am also grateful to both anonymous
reviewers for their comments and suggestions. The research for this paper was supported in part by the research
program Analysis and Geometry, P1- 0291, funded by the Slovenian Research Agency.
E-mail address: pavle.saksida@fmf.uni-lj.si (Pavle Saksida)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 24 (2024) #P1.08
The construction and the study of various versions of the nonlinear Fourier transform
stem from the theory of integrable nonlinear partial differential equations. The most fa-
mous examples of such equations include the Korteweg-de Vries, nonlinear Schrödinger,
and sine-Gordon equations, Heisenberg ferromagnet model, Toda lattices and many oth-
ers. The role of the nonlinear Fourier transform in the theory of integrable equations
is roughly analogous to the role of the linear Fourier transform and, more generally, the
Sturm-Liouville expansions in the theory of linear partial differential equations.
The transformation F used in this text can be thought of as a non-linearization of the
usual Fourier transformation. Let u : [0, 1] → R be a function. The nonlinear Fourier
transform F of u that we shall consider in this paper is of the form
F [u](n) = I +
(
0 F [u](n)
−F [u](−n) 0
)
+
∞∑
d=2
Ad[u](n),
where F is the linear Fourier transform (Fourier series) and u 7→ Ad[u] are the suitable
matrix-valued nonlinear operators.
The beta distribution is one of the oldest and most important probability distributions
with a broad spectrum of applications in different areas of probability and statistics, par-
ticularly in Bayesian statistical inference. It has been recently mentioned in virtually every
book on machine learning and related topics. The beta distribution Beta(x; a, b) with shape
parameters a and b is given by the probability density function
pβ(x; a, b) =
1
B(a+ 1, b+ 1)
xa(1− x)b, x ∈ [0, 1].
In this paper, we shall establish a link between the nonlinear Fourier transform and the
beta distribution. Let uc(x) ≡ u be a constant function. The transformation F is related to
a two-parameter family of polytopes D̂d(λ), where d ∈ N and λ ∈ [0, 1], given by
D̂d(λ) = {(x1, x2, . . . , xd); 1 ≥ x1 ≥ x2 . . . ≥ xd ≥ 0,
d∑
i=1
(−1)i−1xi = λ}
and their projections Dd(λ) in the hyperplane {(x1, x2, . . . , x(d−1), 0)} ⊂ Rd. For the
nonlinear Fourier transform F [uc](n) of the constant function uc ≡ u on [0, 1], we have
F [uc](n) = I +
∞∑
d=1
ud
∫ 1
0
Vol(Dd(λ))
(
e−2πiλn 0
0 e2πiλn
)
·
(
0 1
−1 0
)
dλ.
This formula is proved in Proposition 2.1 on page 7. We shall see that for every fixed d0,
the volumes of the family {Dd0(λ);λ ∈ [0, 1]} are given by the formula for the probability
density function of the beta distribution. Theorem 4.3 on page 16 gives the formula
Vol(Dd(λ)) =
1
d!
{
pβ(λ;
d
2 ,
d
2 + 1); d even
pβ(λ;
d+1
2 ,
d+1
2 ); d odd.
(1.1)
The probabilistic contents of the above formula will be described below.
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 3
The statement and the proof of Theorem 4.3 are obtained by considering a suitable
discretization FN of the nonlinear Fourier transform F . In the expression for FN [uc], the
role of the volumes of the polytopes Dd(λ) is assumed by the numbers
AQN (L, d) = ♯{(l1, l2, . . . , ld) ∈ N; l1 − l2 + l3 − . . .+ (−1)(d−1)ld = L},
where N − 1 ≥ l1 > l2 > . . . > ld ≥ 0. So, AQN (L, d) is the number of ordered
alternating partitions of L into d distinct parts not grater than N − 1.
The central result of the paper is the explicit formula for the numbers AQN (L, d). It is
given in Theorem 3.3 on page 10. We show that
AQN (L, d) =
{( L−1
⌊ d−12 ⌋
)(
N−L
⌊ d2 ⌋
)
; d even(
L
⌊ d−12 ⌋
)(
N−L−1
⌊ d2 ⌋
)
; d odd.
(1.2)
The relationship between the numbers AQN (L, d) and the nonlinear Fourier transform is
best described by the fact that the generating functions for the numbers AQN (L, d) are
in a natural way expressed in terms of the discrete nonlinear Fourier transform FN . This
is proved in Proposition 3.2 on page 9. We actually get separate generating functions for
odd and for even values of d. Understanding the structure of the numbers AQN (L, d) was
important in the construction of the inverse of FN in our recent paper [14].
Results (1.1) and (1.2) can be recast into probabilistic terms. Let our sample space
consist of all strictly decreasing d-tuples of integers
∆Dd (N) = {(l1, l2, . . . , ld); N − 1 ≥ l1 > l2 > . . . , ld ≥ 0)},
Let all the samples (l1, l2, . . . , ld) be equally probable and let
XAS [N, d] : ∆
D
d (N) → N
be the random variable which assigns to a randomly chosen point in ∆Dd (N) the alternating
sum,
XAS [N, d](l1, l2, . . . , ld) = l1 − l2 + l3 − . . .+ (−1)(d−1)ld.
We want to compute the probability P (XAS [N, d] = L) of the event that a randomly
chosen d-tuple has the alternating sum equal to L. We shall show that
AS[N, d](L) = P (XAS [N, d] = L) =

(
L−1
⌊ d−1
2
⌋)(
N−L
⌊ d
2
⌋ )
(Nd)
; d even
( L⌊ d−1
2
⌋)(
N−L−1
⌊ d
2
⌋ )
(Nd)
; d odd.
(1.3)
The random variable XAS is distributed according to the probability mass function AS[N, d]
defined by the right hand side of the above formula.
The question arises: Does this distribution have a sensible limit as N goes to infinity?
One possibility is to proceed as follows. Let λ ∈ [0, 1] be arbitrary. Let us choose a
sequence {LN}N∈N such that LN < N and limN→∞ LN/N = λ. We shall see that
lim
N→∞
P (XAS [N, d] = LN )N =
{
pβ(λ;
d
2 ,
d
2 + 1) ; d even
pβ(λ;
d+1
2 ,
d+1
2 ) ; d odd,
(1.4)
4 Ars Math. Contemp. 24 (2024) #P1.08
where pβ(λ; a, b) is the beta distribution with shape parameters a and b.
Let our sample space now be the ordered simplex ∆d ⊂ Rd of the dimension d, given
by
∆d = {(x1, x2, . . . , xd); 1 ≥ x1 ≥ x2 . . . ≥ xd ≥ 0},
and let all the samples (x1, x2, . . . , xd) be equally probable. This means that we assigned
on ∆d the uniform distribution v : ∆d → R given by v(x1, x2, . . . , xd) ≡ d!. Let the
random variable Xas[d] defined on ∆d be given by
Xas[d](x1, x2, . . . , xd) = x1 − x2 + x3 − . . .+ (−1)(d−1)xd.
Formula (1.4) shows that the cumulative probability distribution
Fas[d] : [0, 1] −→ R, λ → Fas[d](λ)
of the random variable Xas is given by
Fas[d](λ) = P (Xas[d] ≤ λ) =
{∫ λ
0
pβ(µ;
d
2 ,
d
2 + 1) dµ ; d even∫ λ
0
pβ(µ;
d+1
2 ,
d+1
2 ) dµ ; d odd.
This result can be recast in geometric terms. Taking into account that the d-dimensional
volume of the simplex ∆d is equal to 1d! , we see from the above that the (d−1)-dimensional
volume of the polytope Dd(λ) is indeed given by formula (1.1) explained in Theorem 4.3.
The equality (1.4) suggests a natural generalisation of the probability mass function of
XAS [N, d]. It can be defined by
PN (L; a, b) =
(
L−1
a
)(
N−L
b
)(
N
a+b+1
) ,
where L ∈ {1, 2, . . . , N} and are integers such that a + b < N . In Proposition 4.2 we
show that
lim
N→∞
PN (
LN
N
, a, b)N = pβ(λ; a, b) =
1
β(a+ 1, b+ 1)
λa(λ− 1)b.
So, the probability mass function PN (a, b) is a natural discretization of the continuous beta
distribution for arbitrary values of shape parameters. But, at the moment, a convincing
combinatorial or geometric description of PN (a, b) remains a task for the future.
Above, we have described a way how the beta distribution emerges as an appropriate
limit from a discrete and finite probability distribution. This result is reminiscent to the
relation between the Pólya-Eggenberger urn and the beta distribution. Pólya-Eggenberger
urn is an urn model with replacement and is tenable - one can draw the balls from the urn
infinitely many times. The limit of the quotient Wnn , where Wn is the number of white balls
drawn in n draws, is given by
lim
n→∞
P (
Wn
n
< λ) =
∫ λ
0
pβ(µ;
W0
s
,
B0
s
) dµ,
where W0 and B0 are the initial numbers of white and black balls in the urn and s is the
number of the balls added to the urn after drawing and returning a ball of the same colour.
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 5
Let s = 1. After a finite number n of draws, the probability of Wn = w and Bn = b is
equal to
P (Wn = w,Bn = b) =
(
w−1
W0−1
)(
b−1
B0−1
)(
n+τ0−1
τ0−1
) , (1.5)
where τ0 = W0 + B0. The proofs can be found in the comprehensive treatment of Pólya
urn models [9] by H. M. Mahmoud. The values in the (1.5) are related by w + b = W0 +
B0 + n = τ0 + n. Our formula (1.3) could therefore be tentatively understood as an
outcome of Pólya-Eggenberger process after roughly N + d steps. But the number of steps
in constructing an alternating sum l1−l2+. . .+(−1)d−1ld is d. That d is indeed the correct
number of steps in our process will become even clearer in the proofs of Theorem 3.3 and
Corollary 3.4. These proofs are different from the usual proof of formula (1.5). While
the number of steps in the limit of the Pólya-Eggenberger process is infinite, the number
of steps in our process remains d even after performing the limit. This comes naturally
from the source of our construction which is the nonlinear Fourier transform. The core
of our limiting construction is the replacement of the alternating sum of integers: first, by
alternating sums of rational numbers and then, in the limit, by the alternating sum of real
numbers. This also leads to the geometric expression of our results in terms of the volumes
of the polytopes Dd(λ), mentioned above.
There are other discretization of the beta distribution with useful properties. One pos-
sible approach was studied by A. Punzo in [11]. But, as far as the author is aware, this
discretization does not come from some combinatorial source and is given by a very differ-
ent formula.
The plan of the paper is the following. In section 2, we recall the AKNS-ZS type
of the nonlinear Fourier transform and prepare the necessary formulas. We establish the
connection between F and the family of polytopes {Dd(λ)}. We also introduce the dis-
cretization FN of F . In section 3, we describe and prove the main facts about our central
combinatorial problem: the evaluation of the numbers AQN (L, d), and the derivation of
the generating functions of the numbers AQN (L, d) in terms of FN . In section 4, we prove
proposition 4.2 and theorem 4.3 stated above. Section 5 contains graphs illustrating the re-
lation between the beta distribution and our discrete approximation. We conclude the paper
by mentioning some problems for further research.
2 Nonlinear Fourier transforms and its discretization
We review the definition of the nonlinear Fourier transform F which first appeared in the
work of M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur in [1] and [2] and more
or less simultaneously in the work of V. Zakharov and A. Shabat in [19]. They studied
and solved a certain class of integrable partial differential equations which are now called
AKNS-ZS equations. The acronym is also used to denote the nonlinear Fourier transform
which figures in the AKNS-ZS theory. In this section, we shall also introduce the Euler-
type discretization FN of F .
2.1 Nonlinear Fourier transform of AKNS-ZS type
We shall consider the nonlinear Fourier transform F which appears in the study of the
periodic AKNS-ZS problems. To every well-behaved function u(x) : [0, 1] → C it as-
signs the doubly infinite sequence {F [u](n)}n∈Z of SU(2) matrices, given by F [u](n) =
6 Ars Math. Contemp. 24 (2024) #P1.08
(−1)nΦ(x = 1, n), where Φ(x, n) is the solution of the linear initial value problem
Φx(x, n) = L(x, n) · Φ(x, n), Φ(0, n) = I. (2.1)
The coefficient matrix L(x, n) is given by
L(x, n) =
(
πi n u(x)
−u(x) −πi n
)
.
As we mentioned in the Introduction, we will see that F is of the form
F [u](n) = I +
(
0 F [u](n)
−F [u](−n] 0
)
+
∞∑
d=2
Ad[u](n).
The amount of literature on various aspects of the inverse scattering method is truly vast, so
we shall only mention a few works in which the Fourier analysis aspect is more pronounced.
The foundational work was done by Gardner, Greene, Kruskal and Miura in [8] and [7].
The transform, used in this paper was first constructed by Ablowitz, Kaup, Newell and
Segur, in [1, 2], and simultaneously by Zakharov and Shabat in [19]. Nonlinear Fourier
transforms of functions, defined on R and R+, were studied by I. Gelfan’d, A. Fokas and
B. Pelloni in [5, 6, 10], and in their other works. A version of transformation, closely
related to the one studied in this paper is described by T. Tao and C. Thiele in [15]. Some
aspects of the transformation, defined above, were studied in my papers [12, 13] and [14].
Definition of F , given above is the one that is usually found in the texts which study
the integrable ANKS-ZS equations. We shall rather represent F in a different gauge. Let
G(x, n) = diag(e−πinx, eπinx) be the (diagonal) matrix of our gauge transformation. In
the new gauge Φ is replaced by ΦG = G · Φ and ΦG is the solution of the initial-value
problem
ΦGx (x, n) = L
G(x, z) · ΦG(x, n), ΦG(0, n) = I. (2.2)
The transformed coefficient matrix is then LG(x, n) = Gx ·G−1(x, n)+G(x, n) ·L(x, n) ·
G−1(x, n). Its explicit expression is
LG(x, n) =
(
0 e−2πinxu(x)
−e2πinxu(x) 0
)
. (2.3)
In the new gauge we set FG[u](n) = ΦG(x = 1, n). Since n ∈ Z, the equation
ΦG(1, n) = G(1, n) · Φ(1, n) gives F [u](n) = FG[u](n). The solution to the problem
(2.2) can be given in the form of the Dyson series.
ΦG(x, n) = I +
∞∑
d=1
∫
∆d(x)
LG(x1, n) · LG(x2, n) · · ·LG(xd, n) dx⃗, (2.4)
where ∆d(x) is the ordered simplex of dimension d with the edge length equal to x,
∆d(x) = {(x1, x2, . . . , xd) ∈ Rd;x ≥ x1 ≥ x2 ≥ . . . ≥ xd ≥ 0}.
Let us denote
E(x, n) =
(
eπixn 0
0 e−πixn
)
, J =
(
0 1
−1 0
)
, (2.5)
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 7
and let u(x) be real valued. Then we have LG(x, n) = u(x)E(−2x, n) · J. Matrices
E(x, n) and J do not commute. Instead, they obey the relation
E(x, n) · J = J · E(−x, n). (2.6)
Recall that D̂d(λ) denotes the polytope given by
D̂d(λ) = {(x1, x2, . . . xd) ∈ ∆d(1);
d∑
j=1
(−1)j−1xj = λ},
and Dd(λ) is its projection on the hyperplane xd = 0. These are the polytopes, mentioned
in the introduction. Denote
U(x1, x2, . . . , xd−1;λ) = u(x1) · · ·u(xd−1)u((−1)d−1(λ−
d−1∑
j=1
(−1)j−1xj)),
and let dλx⃗ be the measure on D̂d(λ) ⊂ Rd, inherited from the Euclidean measure on Rd.
Using (2.6) in the Dyson series and evaluating at x = 1 gives
F [u](n) = I +
∞∑
d=1
∫
∆d(1)
u(x1)u(x2) · · ·u(xd)E
(
−2(
d∑
j=1
(−1)j−1xj), n
)
· Jd dx⃗
which, upon setting x1 − x2 + . . .+ (−1)d−1xd = λ, can be rewritten as
F [u](n) = I +
∞∑
d=1
∫ 1
0
E(−2λ, n)
(∫
D̂d(λ)
u(x1)u(x2) · · ·u(xd) dλx⃗
)
· Jd 1√
d
dλ
= I +
∞∑
d=1
∫ 1
0
E(−2λ, n)
(∫
Dd(λ)
U(x1, . . . , xd−1;λ) dx1 · · · dxd−1
)
Jd dλ,
Inserting the constant function uc(x) ≡ u we immediately get the following proposition.
Proposition 2.1. In the case where uc(x) ≡ u is a constant function, we get
F [uc](n) = I +
∞∑
d=1
ud
∫ 1
0
Vol(Dd(λ))E(−2λ, n) · Jd dλ. (2.7)
2.2 Euler-type discretization of F
Many authors studied various discretizations of transformations similar to F , but usually
acting on the functions defined on R or R+, see e.g. [16, 17, 18]. Important are the dis-
cretizations that preserve the integrability of the AKNS-ZS systems. These are constructed
in well known works of M. Ablowitz and J. Ladik and also L. Faddeev and A. Yu Volkov,
see [3, 4]. A discrete nonlinear Fourier transform, similar to the one studied below, was
considered by Tao and Thiele in [15]. In the author’s paper [14] an algorithm for evaluat-
ing the inverse of the nonlinear Fourier transform, defined below, is constructed. (In [14]
a nonlinear Fourier transform of distributions of the form u(x) =
∑N
n=1 un δxn(x) is also
constructed, together with its inverse.)
8 Ars Math. Contemp. 24 (2024) #P1.08
We have obtained the nonlinear Fourier transform from an initial value problem for
a particular first-order linear differential equation. An obvious approach to construct a
discretization is to replace the differential equation with a suitable difference equation. Let
u⃗ = (u0, u1, . . . , uN−1) ∈ RN be a vector which plays a role of a function of a discrete
variable. Let the L-matrix be given by
LN (k, n) =
(
0 e−2πi
kn
N uk
−e2πi knN uk 0
)
.
Definition 2.2. Let k, n ∈ {0, 1, . . . , N − 1}. Discrete nonlinear Fourier transform FN [u⃗]
of u⃗ is defined by FN [u⃗](n) = ΦN (k = N − 1, n), where ΦN is the solution of the
difference initial value problem
ΦN (k + 1, n)− ΦN (k, n)
1
N
= LN (k, n) · ΦN (k, n), ΦN (0, n) = I.
Solving the above initial value problem and evaluating at k = N − 1 gives
FN [u⃗](n) =
0∏
k=N−1
(
I +
1
N
LN (k, n)
)
,
and this can be expanded into
FN [u⃗](n) = I +
N∑
d=1
1
Nd
∑
N−1≥l1>l2>...>ld≥0
LN (l1, n) · LN (l2, n) · · ·LN (ld, n). (2.8)
This expression is a discrete analogue of Dyson’s expansion (2.4).
Let us introduce the notation
Eδ(l, n) = E(l,
n
N
) =
(
eπil
n
N 0
0 e−πil
n
N
)
l, n ∈ {0, 1, . . . , N − 1}
where E is given by (2.5), and the subscript δ refers to the use in the discretized context.
The coefficient matrix LN can be written in the form
LN (l, n) = ul Eδ(−2l, n) · J,
with J defined in (2.5). By means of relation (2.6), we can collect all the copies of J in
(2.8) on the right. Let u⃗c = (u, . . . , u) be a constant vector. We get
FN [u⃗c](n) = I+
N∑
d=1
(
u
N
)d
∑
N−1≥l1>l2>...>ld≥0
Eδ
(
−2(l1− l2+ . . .+(−1)d−1ld), n
)
·Jd.
If we denote L = l1 − l2 + . . .+ (−1)d−1ld, we can finally write
FN [u⃗c](n) = I +
N−1∑
d=1
(
u
N
)d
N−1∑
L=0
Eδ(−2L, n)
∑
(l1,...,ld)∈D̂discd,N (L)
Jd, (2.9)
where
D̂discd,N (L) = {(l1, . . . , ld) ∈ Nd; N − 1 ≥ l1 > . . . > ld ≥ 0,
d∑
j=1
(−1)j−1lj = L}.
(2.10)
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 9
3 Ordered alternating partitions with distinct parts
In this section we introduce the central combinatorial object of the paper, namely the num-
bers AQN (L, d). We establish the connection between the family {AQN (L, d)} and the
discrete nonlinear Fourier transform FN . The transformation FN yields the generating
functions for {AQN (L, d)} separately for even and odd values of d. The main result of the
section is the statement and proof of the explicit formula for the numbers AQN (L, d) and
the evaluation of the probability distribution of these numbers.
Definition 3.1. Let
∆DN,d = {(l1, l2, . . . , ld) ∈ Nd;N − 1 ≥ l1 > l2 > . . . > ld ≥ 0}
be the discrete ordered simplex. Denote by AQN (L, d) the numbers which count the or-
dered alternating partitions of L ∈ N into d distinct parts not greater than N − 1,
AQN (L, d) = ♯{(l1, l2, . . . , ld) ∈ ∆DN,d; l1 − l2 + l3 − . . . (−1)d−1ld = L}. (3.1)
In other words, AQN (L, d) is the number of solutions of the equation
l1 − l2 + l3 − . . .+ (−1)d−1ld = L
where (l1, l2, . . . , ld) is an element of the simplex ∆DN,d.
The next proposition shows that FN [uc](n) can, roughly speaking, be understood as
the discrete linear Fourier transform of the generating polynomial of the finite sequence
{AQN (L, d)}Nd=1.
Let us denote by Fev[uc](n) the upper left entry and by Fodd[uc](n) the upper right
entry of the 2× 2 matrix FN [uc](n).
Proposition 3.2. The power series expansion of FN [uc] around u = 0 is given by
FN [uc](n) = I +
N∑
d=1
(
u
N
)d
N−1∑
L=0
AQN (L, d)Eδ(−2L, n) · Jd. (3.2)
For every L ∈ {0, 1, . . . , N − 1}, the generating polynomials of the numbers
{AQN (L, 2k)}k=1,...,⌊N2 ⌋ and {AQN (L, 2k − 1)}k=1,...,⌊N+12 ⌋
are given by the equations
⌊N2 ⌋∑
k=1
(−1)k( u
N
)2kAQN (L, 2k) =
N−1∑
n=0
e2πi
Ln
N · Fev[uc](n) (3.3)
⌊N+12 ⌋∑
k=1
(−1)k+1( u
N
)2k−1AQN (L, 2k − 1) =
N−1∑
n=0
e−2πi
Ln
N · Fodd[uc](n). (3.4)
10 Ars Math. Contemp. 24 (2024) #P1.08
Proof. Recall formula (2.9)
FN [u⃗c](n) = I +
N−1∑
d=1
(
u
N
)d
N−1∑
L=0
Eδ(−2L, n)
∑
(l1,...,ld)∈D̂discd,N (L)
Jd.
The last sum in the formula yields the constant matrix Jd multiplied by the integer
♯D̂discd,N (L). The number AQN (L, d) is by its definition the number of elements in D̂
disc
d,N (L),
so we have ∑
(l1,...,ld)∈D̂discd,N (L)
Jd = AQN (L, d) · Jd.
Let us now take into account
J2k = (−1)k · I =
(
(−1)k 0
0 (−1)k
)
and
J2k−1 = (−1)k+1 · J =
(
0 (−1)2k−1
−(−1)2k−1 0
)
,
and consider the diagonal and anti-diagonal parts of FN separately. From 3.2, we get two
equations, one for each parity of k:
Fev[uc](n) =
⌊N2 ⌋∑
k=1
(−1)k( u
N
)2k
N−1∑
L=0
e−2πi
Ln
N ·AQN (L, 2k)
Fodd[uc](n) =
⌊N+12 ⌋∑
k=1
(−1)k+1( u
N
)2k−1
N−1∑
L=0
e2πi
Ln
N AQN (L, 2k − 1).
Now, we perform the inverse discrete linear Fourier transforms on both of the above equa-
tions and get the expressions (3.3) and (3.4).
We now state and prove the explicit formula for the function AQN (L, d).
Theorem 3.3. For any N ∈ N, d ≤ N and L ∈ {0, . . . , N − 1}, we have
AQN (L, d) =
{( L−1
⌊ d−12 ⌋
)(
N−L
⌊ d2 ⌋
)
; d even(
L
⌊ d−12 ⌋
)(
N−L−1
⌊ d2 ⌋
)
; d odd.
(3.5)
Above we use the definition of the binomial symbol for which
(
a
b
)
= 0 for negative a.
Proof. Let us define
ÂQN (L, d) = ♯{(l1, . . . , ld);N ≥ l1 > . . . > ld ≥ 1, and
d∑
j=1
(−1)j−1lj = L}.
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 11




                         

                         d                       l1

                                                                  

                      d-1

                                                    l2

                      d-2

                                                                   l3

                                                                                                     

                                                         l4



                                                                    l5                       



                          3                                 l6



                          2                                   l7 

  

                          1                                  l8

                                   



                                    a1    a2   a3      L      b4       b3     b2         b1      N    
Figure 1: Zigzag path interpretation of an element of ÂQN (L, d) with d = 8.
We claim that for ÂQN (L, d) we have
ÂQN (L, d) =
(
L− 1
⌊d−12 ⌋
)(
N − L
⌊d2⌋
)
. (3.6)
Suppose that d = 2k is even. Let us consider the partial sums of the alternating sum
ÂQN (L, d) = (l1 − l2) + (l3 − l4) + . . .+ (ld−1 − ld) = L, (3.7)
namely:
a1 = (l1 − l2)
a2 = (l1 − l2) + (l3 − l4)
...
...
ak−1 = (l1 − l2) + (l3 − l4) + (l5 − l6) + . . .+ (ld−3 − ld−2).
Let us also introduce the integers bm, given by
b1 = (N − l1)
b2 = (N − l1) + (l2 − l3)
...
...
bk = (N − l1) + (l2 − l3) + (l4 − l5) + . . .+ (ld−2 − ld−1)
From the above definitions we see that
l1 = N − b1
l2m = (N − bm)− am
l2m−1 = (N − bm)− am−1.
12 Ars Math. Contemp. 24 (2024) #P1.08
We shall now turn the situation around. Let
1 ≤ α1 < α2 < . . . , < αk−1 ≤ L− 1 (3.8)
be an arbitrary ordered subset of {1, 2, 3 . . . , L− 1} and let
0 ≤ β1 < β2 < . . . < βk ≤ N − L− 1 (3.9)
be an arbitrary ordered subset of {0, 1, 2, . . . , N − L− 1}. Let us define
λ1 = N − β1
λ2m = (N − βm)− αm, m = 1, 2, . . . , k − 1
λ2m−1 = (N − βm)− αm−1, m = 2, 3, . . . , k
From (3.8) and (3.9) we see that
N ≥ λ1 > λ2 > λ3 > . . . > λd−1 > 1
and
λ1 − λ2 + λ3 − . . .+ λd−1 ≥ L+ 1.
Therefore there exists precisely one λd such that
(λ1 − λ2 + λ3 − . . .+ λd−1)− λd = L
From the construction we also see that λd < λd−1.
We have shown that for every choice of a pair (3.8) and (3.9) of subsets of
{1, 2, 3 . . . , L− 1} and {0, 1, 2, . . . , N − L− 1},
respectively, there exists precisely one solution {λ1, λ2, . . . , λd} of the equation (3.7).
Since the number of such pairs is equal to(
L− 1
k − 1
)(
N − L
k
)
=
(
L− 1
⌊d−12 ⌋
)(
N − L
⌊d2⌋
)
,
our proposition is proved for even d. The proof for odd d is only a slight variation of the
above and we shall omit it.
Proof by induction. Our formula can be proved by induction on N . For N = 2, formula
(3.6) can be checked by hand. If we divide the alternating sums from ÂQN (L, d) into
those, for which l1 = N and those for which l1 < N , we get the recursion relation
ÂQN (L, d) = ÂQN−1(L, d) + ÂQN−1(N − L, d− 1).
By the induction hypothesis, the above equation becomes
ÂQN (L, d) =
(
L− 1
⌊d−12 ⌋
)(
N − L− 1
⌊d2⌋
)
+
(
N − L− 1
⌊d−22 ⌋
)(
L− 1
⌊d−12 ⌋
)
=
(
L− 1
⌊d−12 ⌋
)(
N − L
⌊d2⌋
)
,
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 13
and this proves (3.6). The second equality above comes from the recurrence relation of the
Pascal triangle.
Finally, we observe that
AQN (L, d) = ÂQN (L, d), for d even, and AQN (L, d) = ÂQN (L+1, d), for d odd.
These relations, together with formula (3.6), prove the proposition.
Two of the central results of this paper are corollaries of the above theorem.
Corollary 3.4. Let the random variable
XAS [N, d] : ∆
D
d (N) −→ R
defined on the discrete ordered simplex
∆Dd (N) = {(l1, l2, . . . , ld); N − 1 ≥ l1 > l2 > . . . > ld ≥ 0}
be given by
XAS [N, d](l1, l2, . . . , ld) = l1 − l2 + l3 − . . .+ (−1)(d−1)ld.
Then its probability mass function is
P (XAS [N, d] = L) =

(
L−1
⌊ d−1
2
⌋)(
N−L
⌊ d
2
⌋ )
(Nd)
; d even
( L⌊ d−1
2
⌋)(
N−L−1
⌊ d
2
⌋ )
(Nd)
; d odd.
Proof. The number of the favourable events is given by Theorem 3.3, proved above. To
evaluate the number of all outcomes it helps to consider Figure 1. We see that the number
of all outcomes is equal to the number of the subsets which are composed of all the integer
points ai, all the points bi and the point L. These are precisely all the subsets with d
elements in the set {1, 2, . . . , N}. Their number is of course
(
N
d
)
. This proves our corollary.
Inserting the formula (3.5) in the expressions (3.3) and (3.4) yields the following corol-
lary:
Corollary 3.5. The power series of FN [uc](n) around u = 0 is given by
FN [uc](n) = I +
⌊N2 ⌋∑
k=1
(−1)k( u
N
)2k
N−1∑
L=0
(
L− 1
k − 1
)(
N − L
k
)
· Eδ(−2L, n)
+
⌊N+12 ⌋∑
k=0
(−1)k( u
N
)2k+1
N−1∑
L=0
(
L
k
)(
N − L− 1
k
)
· Eδ(−2L, n) · J.
14 Ars Math. Contemp. 24 (2024) #P1.08
4 Beta distribution and polytopes Dd(λ)
In this section we prove our second theorem which is the expression of the volumes of
polytopes Dd(λ) in terms of the probability density function of the beta distribution. We
obtain this result by taking a suitable limit of the probability mass functions of the random
variables XAS [N, d].
4.1 Discretization of beta distribution
The subset D̂discd,N (L), given by (2.10) of the discrete ordered simplex
∆discd,N = {(l1, l2, . . . , ld) ∈ (N ∪ {0})d;N − 1 ≥ l1 > l2 > . . . > ld ≥ 0}
with the edge of size N is given by one equation. The size AQN (L, d) of D̂discd,N (L) is
therefore of the order Nd−1.
Lemma 4.1. Let λ be a real number in [0, 1] and let {LN}N∈N be a sequence of positive
integers such that LN < N and limN→∞ LNN = λ. Then we have
lim
N→∞
AQN (LN , d)(
N
d
) N = {pβ(λ; d2 , d2 + 1) ; d even
pβ(λ;
d+1
2 ,
d+1
2 ) ; d odd.
(4.1)
where
pβ(λ; a, b) =
1
B(a+ 1, b+ 1)
λa(1− λ)b
is the probability density function of the beta distribution Beta(λ; a, b).
Proof. We shall prove the formula only for even d. The proof for odd d is essentially
the same. Consider first the numerator of the quotient under the limit. For d = 2m,
formula (3.5) gives
AQN (LN , d) =
(
LN − 1
m− 1
)(
N − LN
m
)
.
This expression can be expanded into
AQN (LN , d) =
1
(m− 1)!m!
m−2∏
k=0
((LN − 1)− k)
m−1∏
k=0
((N − LN )− k). (4.2)
Consider the first product above. It is a polynomial of degree m− 1 in the variable (LN −
1) = (N LNN − 1). Expanding this polynomial gives
(N
LN
N
− 1)m−1 +
m−2∑
k=1
(N
LN
N
− 1)k · n(k) = (N LN
N
− 1)m−1 +O(N LN
N
)m−2
For large N we can replace LNN by λ. Taking into account also the second product, (4.2)
gives
AQN (LN , d) =
1
(m− 1)!m!
(
(N
LN
N
− 1)m−1 +R1
)(
(N −N LN
N
)m +R2
)
=
1
(m− 1)!m!
(
Nm+m−1(
LN
N
)m−1(1− LN
N
)m +R3
)
(4.3)
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 15
where
R1 = R3 = O(
1
(Nλ)m−2
) and R2 = O(
1
(Nλ)m−1
).
For the denominator
(
N
d
)
we have(
N
d
)
=
1
d!
(
N(N − 1) · · · (N − (d− 1))
)
=
Nd
d!
+O( 1
N (d−2)
). (4.4)
Because d− 1 = m+ (m− 1) and limN→∞ LNN = λ formulas (4.3) and (4.4) yield
lim
N→∞
AQN (LN , d)(
N
d
) N = d!
(m− 1)!m!
λm−1λm.
The definition of the Euler beta function for positive integers gives d!(m−1)!m! =
1
B(m,m+1) ,
and this proves formula (4.1) for even d.
The above calculation suggests the definition of a discrete version BetaN (a, b) of beta
distribution for arbitrary choice of the shape parameters. Let a, b and N be integers. Let
the probability mass function of BetaN (a, b) be defined by
PN (L; a, b) =
(
L−1
a
)(
N−L
b
)(
N
a+b+1
)
for L ∈ {1, 2, . . . , N}.
Proposition 4.2. Let λ be an arbitrary real number in the unit interval [0, 1] and let
{LN}N∈N be a sequence, such that LN < N and limN→∞ LNN = λ. Then
lim
N→∞
PN (
LN
N
, a, b)N = pβ(λ; a, b) =
1
β(a+ 1, b+ 1)
λa(λ− 1)b
Proof. The proof is an obvious adaptation of the proof of Lemma 4.1. We only have to
replace the particular values m − 1 and m of the shape parameters by an arbitrary pair a
and b of positive integers. Then the same calculations as those performed in the proof of
Lemma 4.1 yield the proof of the proposition.
4.2 Volumes of polytopes Dd(λ)
Recall formula (2.7):
F [uc](n) = I +
∞∑
d=1
ud
∫ 1
0
Vol(Dd(λ))
(
e−2πiλn 0
0 −e2πiλn
)
·
(
0 1
−1 0
)d
dλ.
We have the following theorem.
16 Ars Math. Contemp. 24 (2024) #P1.08
Theorem 4.3. For every dimension d, the volumes of polytopes Dd(λ) are essentially dis-
tributed according to the beta distribution with the shape parameters (d2 ,
d
2 + 1), if d is
even, and (d+12 ,
d+1
2 ), if d is odd. More concretely, we have the following expression:
Vol(Dd(λ)) =
1
d!

1
B( d2 ,
d
2+1)
λ
d
2−1(1− λ) d2 = pβ(λ; d2 ,
d
2 + 1); d even
1
B( d+12 ,
d+1
2 )
λ
d−1
2 (1− λ) d−12 = pβ(λ; d+12 ,
d+1
2 ); d odd,
(4.5)
where pβ(λ; a, b) is the probability density function of the distribution with shape parame-
ters a and b.
Proof. Recall the set D̂discd,N (L), given by (2.10). Rescaling it by the factor 1/N gives the
set
D̂discd (
L
N
) = {( l1
N
,
l2
N
. . . ,
ld
N
);
N − 1
N
≥ l1
N
> . . . >
ld
N
≥ 0,
d∑
j=1
(−1)j−1lj = L}
which contains the same number of points as D̂discd,N (L), but lies in the polytope D̂d(
L
N ).
Let Ddiscd (
L
N ) denote the orthogonal projection of D̂
disc
d (
L
N ) on the hyperplane
{(x1, . . . , xd−1, 0)} ⊂ Rd.
The number ♯Ddiscd (
L
N ) of points in D
disc
d (
L
N ) is clearly equal to the number of points in
D̂discd (
L
N ). The value
1
Nd−1
♯Ddiscd (
L
N ) is approximately equal to the volume Vol(Dd(
L
N ))
of the projection Dd( LN ) of D̂d(
L
N ) on the hyperplane xd = 0 in R
d. So, on the one hand,
the number ♯Ddiscd (
L
N ) is equal to AQN (L, d), while on the other, the value
1
Nd−1
♯Ddiscd (
L
N )
is an approximation of Vol(Dd(L)). Let now {λN}N∈N be a sequence of rationals LNN
converging to λ ∈ [0, 1]. We have
lim
N→∞
1
Nd−1
AQN (NLN , d) = Vol(Dd(λ)).
In the proof of Lemma 4.1 we have seen that(
N
d
)
=
Nd
d!
+O( 1
N (d−2)
),
so (
N
d
)
N
=
N (d−1)
d!
+O( 1
N (d−3)
).
Therefore
lim
N→∞
1
Nd−1
AQN (NLN , d) = lim
N→∞
AQN (LN , d)(
N
d
) N.
This equality, together with Lemma 4.1, proves our theorem.
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 17
5 Quantitative comparisons
In this section we shall investigate by experimental means the comparison between the
probability density function of Beta(l; a, b) distribution and its approximations, given by
the probability mass functions PN (l; a, b). For the sake of brevity we shall concentrate
on the shape parameters (a, b) = (m − 1,m) which appear in connection with the non-
linear Fourier transform. It is now clear that absolute value of the difference pβ(l; a, b) −
PN (l; a, b) decreases for every l = LNN as N increases. But the quality of the approx-
imation also depends crucially on the choice of the shape parameters. We shall see that,
roughly speaking, the value |pβ(l; a, b)−PN (l; a, b)| at a fixed N , increases with increasing
of a+ b. Explicit formula for this difference can be deduced from formulas (4.1) and (4.2),
but it is quite complicated. The images will provide a better illustration of the relations
between pβ(l; a, b) and PN (l; a, b).
The two images in Figure 2 show the comparison between pβ(l; 21, 22) and PN (l; 21, 22)
for N = 200 and N = 1000.
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
6
(a) N = 200, a = 21, b = 22
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
(b) N = 1000, a = 21, b = 22
Figure 2: Comparison of graphs.
Figure 3 shows that for any choice of the shape parameters the difference PN (l; a, b)−
pβ(l; a, b)) has three local extrema. For the cases, related to the number of alternating
partitions of integers where a = b − 1 or a = b, the maximum is located roughly at the
center of the interval [0, 1].
0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.2
0.4
0.6
0.8
(a) N = 200, a = 21, b = 22
0.2 0.4 0.6 0.8 1.0
-0.1
0.1
0.2
(b) N = 1000, a = 21, b = 22
Figure 3: The shape of the difference.
18 Ars Math. Contemp. 24 (2024) #P1.08
The two images in Figure 4 illustrate the dependence of the difference pβ(l; a, b) −
PN (l; a, b) on the size of the shape parameters. Again we consider (a, b) = (a, a+ 1). We
see, that at a fixed N the difference increases with increasing of the shape parameter a.
(a) N = 100, a ∈ [3, 20] (b) N = 1000, a ∈ [3, 20]
Figure 4: Dependence of the difference on the size of the shape parameter.
Even if the shape parameters a and b are very different, the corresponding graphs are of
similar shapes to the above. The only difference is that, in case where the shape parameters
a and b are significantly different the peaks of the graphs are shifted away from the center.
This is clear from the following fact. Suppose that a is considerably larger than b. Then
the left zero of limit function pβ(l; a, b) = 1B(a+1,b+1) l
a(1 − l)b is of higher degree than
the right one. The function is therefore flatter and closer to zero in the vicinity of 0 and
the peak of the graph is pushed towards the right. Qualitatively the shape of the difference
does not change.
6 Conclusions and outlook
In the paper we arrived at the construction of a discrete probability distribution with proba-
bility mass function PN (l; a, b) which converges to the probability density function
pβ(l; a, b) as N → ∞. The result is precisely stated in Proposition 4.2. Crucial in the
construction is the connection of PN (l; a, a) and PN (l; a−1, a) to the following combina-
torial problem: find the number AQN (L, d) of alternating ordered partitions of the positive
integer L < N into d distinct parts, not greater than N − 1. The number AQN (L, d) can
also be represented by the number of the zig-zag paths, drawn in Figure (1). This combina-
torial problem naturally appeared in the context of the discretization FN of the nonlinear
Fourier transform F , described in Section 2. The essential connection between the num-
bers AQN (L, d) and FN is given by Proposition 3.2 where we show that the inverse linear
Fourier transform of the entries of FN yields the generating polynomials of the numbers
AQN (L, d).
The formula for distribution PN (l; a, b) can also be interpreted as the distribution de-
scribing the Pólya-Eggenberger urn, but this interpretation is different from ours. We have
the connection of PN (l; a, b) to the combinatorial problem and the nonlinear Fourier trans-
form only for the shape parameters of the form (a, b) = (a, a) or (a, b) = (a − 1, a).
The natural question arises: can we find a combinatorial problem whose relation with
PN (l; a, b) for an arbitrary choice of a and b would be analogous to the relation be-
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 19
tween PN (l; a − 1, a) and PN (l; a, a) and the problem of alternating ordered partitions
AQN (L, d)? Does there exist a meaningful generalisation Fa,b of the nonlinear Fourier
transform F , whose relation with pβ(x; a, b) would be analogous to the relation between
F and pβ(x; a, a) and pβ(x; a − 1, a), described in theorem 4.3. These are the natural
problems for further investigation, based on this paper. Finding answers to these questions
would importantly improve understanding the nonlinear Fourier transform and its structure.
In this paper, we considered the nonlinear Fourier transform F [u] evaluated on the
simplest of functions, namely, the constant function u ≡ c. An obvious direction of further
research is to try to extend the approach used in this paper, to the context, where F [u] is
evaluated on some more interesting class of functions u.
ORCID iDs
Pavle Saksida https://orcid.org/0000-0003-3093-9863
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