THE BERNOULLI CLOCK PROBABILISTIC AND COMBINATORIAL INTERPRETATIONS OF THE BERNOULLI POLYNOMIALS BY CIRCULAR CONVOLUTION

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THE BERNOULLI CLOCK: PROBABILISTIC AND COMBINATORIAL
INTERPRETATIONS OF THE BERNOULLI POLYNOMIALS BY CIRCULAR
CONVOLUTION
YASSINE EL MAAZOUZ AND JIM PITMAN
Abstract.
The factorially normalized Bernoulli polynomials
bn
(
x
) =
Bn
(
x
)
/n
!are known to
be characterized by
b0
(
x
)=1and
bn
(
x
)for
n >
0is the anti-derivative of
bn−1
(
x
)subject to
R1
0bn
(
x
)
dx
= 0. We offer a related characterization:
b1
(
x
) =
x−
1
/
2and (
−
1)
n−1bn
(
x
)for
n >
0
is the
n
-fold circular convolution of
b1
(
x
)with itself. Equivalently, 1
−
2
nbn
(
x
)is the probability
density at
x∈
(0
,
1) of the fractional part of a sum of
n
independent random variables, each with
the beta(1
,
2) probability density 2(1
−x
)at
x∈
(0
,
1). This result has a novel combinatorial
analog, the Bernoulli clock: mark the hours of a 2
n
hour clock by a uniform random permutation
of the multiset
{
1
,
1
,
2
,
2
,...,n,n}
, meaning pick two different hours uniformly at random from
the 2
n
hours and mark them 1, then pick two different hours uniformly at random from the
remaining 2
n−
2hours and mark them 2, and so on. Starting from hour 0=2
n
, move clockwise
to the first hour marked 1, continue clockwise to the first hour marked 2, and so on, continuing
clockwise around the Bernoulli clock until the first of the two hours marked
n
is encountered,
at a random hour
In
between 1and 2
n
. We show that for each positive integer
n
, the event
(
In
= 1) has probability (1
−
2
nbn
(0))
/
(2
n
), where
n
!
bn
(0) =
Bn
(0) is the
n
th Bernoulli number.
For 1
≤k≤
2
n
, the difference
δn
(
k
) := 1
/
(2
n
)
−P
(
In
=
k
)is a polynomial function of
k
with
the surprising symmetry
δn
(2
n
+ 1
−k
)=(
−
1)
nδn
(
k
), which is a combinatorial analog of the
well known symmetry of Bernoulli polynomials bn(1 −x)=(−1)nbn(x).
Contents
1. Introduction 1
2. Circular convolution of polynomials 4
3. Probabilistic interpretation 6
4. Combinatorics of the Bernoulli clock 11
5. Generalized Bernoulli clock 15
6. Wrapping probability distributions on the circle 20
Appendix A. An elementary combinatorial proof of Theorem 1.1 21
Appendix B. Complement to the proof of Proposition 5.13 23
References 24
1. Introduction
The Bernoulli polynomials (
Bn
(
x
))
n≥0
are a special sequence of univariate polynomials with
rational coefficients. They are named after the Swiss mathematician Jakob Bernoulli (1654–1705),
who (in his Ars Conjectandi published posthumously in Basel 1713) found the sum of
m
th powers
of the first npositive integers using the instance x= 1 of the power sum formula
(1.1)
n−1
X
k=0
(x+k)m=Bm+1(x+n)−Bm+1(x)
m+ 1 ,(n= 1,2, . . . , m = 0,1,2, . . .).
Date: January 12, 2024.
2020 Mathematics Subject Classification. 60C05,05A15,05A05.
Key words and phrases. Bernoulli polynomials, Bernoulli clock, circular convolution, random permutations.
1
arXiv:2210.02027v2 [math.PR] 11 Jan 2024
2 YASSINE EL MAAZOUZ AND JIM PITMAN
The evaluations
Bm
:=
Bm
(0) and
Bm
(1) = (
−
1)
mBm
are known as the Bernoulli numbers, from
which the polynomials are recovered as
(1.2) Bn(x) =
n
X
k=0 n
kBn−kxk.
These polynomials have been well studied, starting from the early work of Faulhaber, Bernoulli,
Seki and Euler in the 17th and early 18th centuries. They can be defined in multiple ways. For
example, Euler defined the Bernoulli polynomials by their exponential generating function
(1.3) B(x, λ):=λeλx
eλ−1=∞
X
n=0
Bn(x)
n!λn(|λ|<2π).
Beyond evaluating power sums, the Bernoulli numbers and polynomials are useful in other
contexts and appear in many areas in mathematics, among which we mention number theory
[
2
,
4
,
28
,
1
], Lie theory [
6
,
36
,
7
,
27
], algebraic geometry and topology [
17
,
29
], probability
[25,26,19,20,39,34,5] and numerical approximation [38,33].
The factorially normalized Bernoulli polynomials
bn
(
x
) :=
Bn
(
x
)
/n
!can also be defined induc-
tively as follows (see [
30
, §9.5]). Beginning with
b0
(
x
) =
B0
(
x
) = 1, for each positive integer
n
, the
function x7→ bn(x)is the unique antiderivative of x7→ bn−1(x)that integrates to 0over [0,1]:
(1.4) b0(x)=1,d
dxbn(x) = bn−1(x)and Z1
0
bn(x) = 0 (n > 0).
So the first few polynomials bn(x)are
b0(x)=1, b1(x) = x−1/2,
b2(x) = 1
2!(x2−x−1/6), b3(x) = 1
3!(x3−3x2/2 + x/2).
As shown in [
30
, Theorem 9.7] starting from
(1.4)
, the functions
f
(
x
) =
bn
(
x
)with argument
x∈[0,1) are also characterized by the simple form of their Fourier transform
(1.5) b
f(k):=Z1
0
f(x)e−2πikxdx (k∈Z)
which is given by
(1.6) b
b0(k) = 1[k= 0],for k∈Z;
b
bn(0) = 0 and b
bn(k) = −1
(2πik)n,for n > 0and k̸= 0,
with the notation 1[
···
]equal to 1if [
···
]holds and 0otherwise. It follows from the Fourier
expansion of bn(x):
bn(x) = −2
(2π)n
∞
X
k=1
1
kncos 2kπx −nπ
2
that there exists a constant C > 0such that
(1.7) sup
0≤x≤1(2π)nbn(x) + 2 cos 2πx −nπ
2≤C2−nfor n≥2,
see [
22
]. So as
n↑ ∞
the polynomials
bn
(
x
)looks like shifted cosine functions. Besides
(1.3)
and
(1.4), several other characterizations of the Bernoulli polynomials are described in [23,10].
This article first draws attention to a simple characterization of the Bernoulli polynomials by
circular convolution and, more importantly, provides an interesting probabilistic and combinatorial
interpretation in terms of statistics of random permutations of a multiset.
THE BERNOULLI CLOCK 3
For a pair of functions
f
=
f
(
u
)and
g
=
g
(
u
), defined for
u
in [0
,
1) identified with the circle
group
T
:=
R/Z
= [0
,
1), with
f
and
g
integrable with respect to Lebesgue measure on
T
, their
circular convolution fgis the function
(1.8) (fg)(u) = ZT
f(v)g(u−v)dv for u∈T.
Here
u−v
is evaluated in the circle group
T
, that is modulo 1, and
dv
is the shift-invariant Lebesgue
measure on
T
with total measure 1. Iteration of this operation defines the
n
th convolution power
u7→ fn(u)for each positive integer n, each integrable f, and u∈T.
Theorem 1.1. The factorially normalized Bernoulli polynomials
bn
(
x
) =
Bn(x)
n!
are characterized
by:
(i) b0(x) = 1 and b1(x) = x−1/2,
(ii) for n > 0the n-fold circular convolution of b1(x)with itself is (−1)n−1bn(x); that is
(1.9) bn(x)=(−1)n−1bn
1(x).
In view of the identity
[
fg
=
b
fbg
, Theorem 1.1 follows from the classical Fourier evaluation
(1.6)
and uniqueness of the Fourier transform. A more elementary proof of Theorem 1.1, without
Fourier transforms, is provided in Section 2. So the Fourier evaluation
(1.6)
may be regarded as a
corollary of Theorem 1.1. That theorem can also be reformulated as follows:
Corollary 1.2. The following identities hold for circular convolution of factorially normalized
Bernoulli polynomials:
b0(x)b0(x) = b0(x)
b0(x)bn(x)=0,(n≥1),
bn(x)bm(x) = −bn+m(x),(n, m ≥1).
In particular, for positive integers
n
and
m
, this evaluation of (
bnbm
)(1) yields an identity
which appears in [32, p. 31]:
(1.10) (−1)mZ1
0
bn(u)bm(u)du =Z1
0
bn(u)bm(1 −u)du =−bn+m(1).
Here the first equality is due to the well known reflection symmetry of the Bernoulli polynomials
(1.11) (−1)mbm(u) = bm(1 −u) (m≥0)
which is the identity of coefficients of
λm
in the elementary identity of Eulerian generating functions
(1.12) B(u, −λ) = (−λ)e−λu
e−λ−1=λeλ(1−u)
eλ−1=B(1 −u, λ).
The rest of this article is organized as follows. Section 2gives an elementary proof for Theorem 1.1,
and discusses circular convolution of polynomials. In Section 3we highlight the fact that 1
−
2
nbn
(
x
)
is the probability density at
x∈
(0
,
1) of the fractional part of a sum of
n
independent random
variables, each with the beta(1
,
2) probability density 2(1
−x
)at
x∈
(0
,
1). Because the minimum
of two independent uniform [0
,
1] variables has this beta(1
,
2) probability density the circular
convolution of
n
independent beta(1
,
2) variables is closely related to a continuous model we call
the Bernoulli clock : Spray the circle
T
= [0
,
1) of circumference 1with 2
n
i.i.d uniform positions
U1, U′
1, . . . , Un, U′
nwith order statistics
U1:2n<··· < U2n:2n.
Starting from the origin 0, move clockwise to the first of position of the pair (
U1, U′
1
), continue
clockwise to the first position of the pair (
U2, U′
2
), and so on, continuing clockwise around the
circle until the first of the two positions (
Un, U′
n
)is encountered at a random index 1
≤In≤
2
n
4 YASSINE EL MAAZOUZ AND JIM PITMAN
(i.e. we stop at
UIn:2n
) after having made a random number 0
≤Dn≤n−
1turns around the
circle. Then for each positive integer n, the event (In= 1) has probability
P(In= 1) = 1−2nbn(0)
2n
where n!bn(0) = Bn(0) is the nth Bernoulli number. For 1≤k≤2n, the difference
δk:2n:= 1
2n−P(In=k)
is a polynomial function of
k
, which is closely related to
bn
(
x
). In particular, this difference has
the surprising symmetry
δ2n+1−k:2n= (−1)nδk:2n,for 1≤k≤2n
which is a combinatorial analog of the reflection symmetry (1.11) for the Bernoulli polynomials.
Stripping down the clock model, the random variables
In
and
Dn
are two statistics of permuta-
tions of the multiset
(1.13) 12···n2:= {1,1,2,2, . . . , n, n}.
Section 4discusses the combinatorics behind the distributions of
In
and
Dn
. In Section 5we
generalize the Bernoulli clock model to offer a new perspective on the work of Horton and Kurn [
18
]
and the more recent work of Clifton et al [
8
]. In particular, we provide a probabilistic interpretation
for the permutation counting problem in [
18
] and prove Conjectures 4.1 and 4.2 of [
8
]. Moreover,
we explicitly compute the mean function on [0
,
1] of a renewal process with i.i.d. beta(1
, m
)-jumps.
The expression of this mean function is given in terms of the complex roots of the exponential
polynomial
Em
(
x
)
:
= 1 +
x/
1! +
···
+
xm/m
!, and its derivatives at 0are precisely the moments of
these roots, as studied in [40].
The circular convolution identities for Bernoulli polynomials are closely related to the decom-
position of a real valued random variable
X
into its integer part
⌊X⌋ ∈ Z
and its fractional part
X◦∈T:= R/Z= [0,1):
(1.14) X=⌊X⌋+X◦.
If
γ1
is a random variable with standard exponential distribution, then for each positive real
λ
we
have the expansion
(1.15) d
duP((γ1/λ)◦≤u) = λe−λu
1−e−λ=B(u, −λ) = X
n≥0
bn(u)(−λ)n.
Here the first two equations hold for all real
λ̸
= 0 and
u∈
[0
,
1), but the final equality holds with
a convergent power series only for 0
<|λ|<
2
π
. Section 6presents a generalization of formula
(1.15)
with the standard exponential variable
γ1
replaced by the gamma distributed sum
γr
of
r
independent copies of
γ1
, for a positive integer
r
. This provides an elementary probabilistic
interpretation and proof of a formula due to Erdélyi, Magnus, Oberhettinger, and Tricomi [
15
,
Section 1.11, page 30] relating the Hurwitz-Lerch zeta function (first studied in [24])
(1.16) Φ(z, s, u) = X
m≥0
zm
(u+m)s
to Bernoulli polynomials. Moreover, the expansion
(6.4)
in Proposition 6.1 quantifies how the
distribution of the fractional part of a
γr,λ
random variable approaches the uniform distribution on
the circle in terms of Bernoulli polynomials, where the latter are viewed as signed measures on the
circle.
2. Circular convolution of polynomials
Theorem 1.1 follows easily by induction on
n
from the characterization
(1.4)
of the Bernoulli
polynomials, and the action of circular convolution by the function
(2.1) −b1(u)=1/2−u,
as described by the following lemma.
THE BERNOULLI CLOCK 5
Lemma 2.1. For each Riemann integrable function
f
with domain [0
,
1), the circular convolution
h=f(−b1)is continuous on T, implying h(0) = h(1−). Moreover,
(2.2) Z1
0
h(u)du = 0
and at each u∈(0,1) at which fis continuous, his differentiable with
(2.3) d
duh(u) = f(u)−Z1
0
f(v)dv.
In particular, if
f
is bounded and continuous on (0
,
1), then
h
=
f
(
−b1
)is the unique continuous
function hon Tsubject to (2.2)with derivative (2.3)at every u∈(0,1).
Proof. According to the definition of circular convolution (1.8),
(fg)(u) = Zu
0
f(v)g(u−v)dv +Z1
u
f(v)g(1 + u−v)dv.
In particular, for g(u) = −b1(u), and a generic integrable function f,
(f(−b1))(u) = Zu
0
f(v)(v−u+ 1/2)dv +Z1
u
f(v)(v−u−1/2)dv
=1
2Zu
0
f(v)dv −Z1
u
f(v)dv−uZ1
0
f(v)dv +Z1
0
vf(v)dv.
Differentiate this identity with respect to
u
, to see that
h
:=
f
(
−b1
)has the derivative displayed
in
(2.3)
at every
u∈
(0
,
1) at which
f
is continuous, by the fundamental theorem of calculus. Also,
this identity shows
h
is continuous on (0
,
1) with
h
(0) =
h
(0+) =
h
(1
−
), hence
h
is continous with
respect to the topology of the circle
T
. This
h
has integral 0by associativity of circular convolution:
h
1 =
f
(
−b1
)
1 =
f
0=0. Assuming further that
f
is bounded and continuous on (0
,
1),
the uniqueness of his obvious. □
The reformulation of Theorem 1.1 in Corollary 1.2 displays how simple it is to convolve Bernoulli
polynomials on the circle. On the other hand, convolving monomials is less pleasant, as the
following calculations show.
Lemma 2.2. For real parameters n > 0and m > −1,
(2.4) xmxn=xnxm=n
m+ 1xn−1xm+1 +xn−xm+1
m+ 1 .
Proof. Integrate by parts to obtain
xnxm=Zx
0
un(x−u)mdu +Z1
x
un(1 + x−u)mdu
=n
m+ 1 Zx
0
un−1(x−u)m+1du +n
m+ 1 Z1
x
un−1(1 + x−u)mdu +xn−xm+1
m+ 1
and hence (2.4). □
Proposition 2.3 (Convolving monomials).For each positive integer n
(2.5) 1xn=xn1 = 1
n+ 1,
and for all positive integers mand n
(2.6) xmxn=xnxm=n!m!
(n+m+ 1)! +
n−1
X
k=0
n!
(n−k)!(m+ 1)k+1
(xn−k−xm+k+1)
and with the Pochhammer notation (m+ 1)k+1 := (m+ 1) . . . (m+k+ 1). In particular
xxn=x−xn+1
n+ 1 +1
(n+ 1)(n+ 2).
摘要：

THEBERNOULLICLOCK:PROBABILISTICANDCOMBINATORIALINTERPRETATIONSOFTHEBERNOULLIPOLYNOMIALSBYCIRCULARCONVOLUTIONYASSINEELMAAZOUZANDJIMPITMANAbstract.ThefactoriallynormalizedBernoullipolynomialsbn(x)=Bn(x)/n!areknowntobecharacterizedbyb0(x)=1andbn(x)forn>0istheanti-derivativeofbn−1(x)subjecttoR10bn(x)dx=...

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