# On the Chudakov effect for the most probable value of high-energy electron-
positron pair ionization loss in thin targets

S.V. Trofymenko<EMAIL_ADDRESS>National Science Center ‘Kharkiv
Institute of Physics and Technology’, 1 Akademichna st., 61108 Kharkiv,
Ukraine  Karazin Kharkiv National University, 4 Svobody sq., 61022 Kharkiv,
Ukraine

###### Abstract

The ionization loss of a high-energy electron-positron pair in thin targets is
considered. The analogue of the Landau distribution function is derived for
this loss under the condition when the Chudakov effect of the pair ionization
loss suppression is manifested. Expression for the most probable value of the
pair ionization loss $E_{MP}$ is obtained. It is shown that the magnitude of
Chudakov effect for $E_{MP}$ can be noticeably different from the magnitude of
this effect for the restricted mean value of the pair ionization loss.

###### keywords:

Electron-positron pair , ionization energy loss , straggling function ,
Chudakov effect

##  1 Introduction

The value of ionization energy loss of a fast particle traversing thin target
is stochastic. It is distributed according to the law firstly derived by
Landau [1]. Such a distribution, known as straggling function, is asymmetric
with respect to its single maximum corresponding to the most probable value of
the ionization loss $E_{MP}$. This value is one of the key parameters of the
straggling function. It is smaller than the average value $E_{AV}$ of the
particle ionization loss in the target, defined by the Bethe-Bloch formula.
$E_{MP}$ depends a bit differently on the target thickness $x$ than $E_{AV}$.
Namely, $E_{MP}\propto x(\ln x+a)$, where $a$ is some constant, and grows
faster than linearly with the increase of the target thickness, while,
naturally, $E_{AV}\propto x$. Straggling functions of various particles in
amorphous targets or disoriented crystals have been extensively studied both
theoretically [2, 3, 4, 5, 6] and experimentally [7, 8]. A series of studies
has been performed for the case of particle penetration through oriented
crystals [9, 10, 11] where the straggling functions have a more complicated
structure than in amorphous targets.

In the above works it was the ionization loss by single particles that was
investigated, i. e. the loss under the condition when the particles traverse
the target one at a time. As far as the quantity $E_{AV}$ is concerned, a
series of works has been devoted to the study of the influence of interference
effects on its value when several particles penetrate the target
simultaneously. Probably, the most well-known effect of this kind is the
Chudakov effect (or King-Perkins-Chudakov effect) [12, 13]. It is the
suppression of ionization energy loss of a high-energy electron-positron pair
in the vicinity of its creation point in substance compared to the sum of
independent electron and positron ionization energy losses. It occurs due to
mutual screening (destructive interference) of the electron’s and positron’s
proper Coulomb fields. Due to polarization of the substance the typical
transversal scale of these fields is $1/\omega_{p}$, where $\omega_{p}$ is the
plasma frequency111We use the system of units where the speed of light $c$ and
the Planck constant $\hbar$ are equal to unity.. The scale of the pair
divergence angle is $m/\mathcal{E}\ll 1$, where $m$ is the electron mass and
$\mathcal{E}$ is the pair energy. Thus, the electron’s and positron’s proper
fields can interfere within the distance $l\sim\mathcal{E}/m\omega_{p}$ from
the creation point, which is the region where the Chudakov effect is
manifested. The analogous effect has been also considered for a proton
cluster, bound by a common electron, which experiences a “Coulomb explosion”
after its entrance into a target [14, 15]. In this case, however, one deals
with constructive interference of the protons’ fields, which results in the
enhancement of the disintegrated cluster energy loss compared to the sum of
independent energy losses of separate protons.

The Chudakov effect has been studied experimentally with the use of cosmic ray
photons for pair production in emulsions [13, 16, 17], which showed a fairly
good agreement with the theories of this effect for the mean ionization loss
of the pair [12, 18, 19, 20, 21]. In [22, 23] there was reported the first
proof-of principle experiment of this kind, where the photons were generated
by high-energy electrons in an accelerator. It was noted that such an approach
allows for much more precise determination of the pair energy and achievement
of better statistics. In this experiment the pairs were created by multi-GeV
photons impinging on thin upstream golden foils, while the pair ionization
loss was registered in a thin downstream silicon detector with the active
layer thickness of 16 $\mu$m situated on some distance from the foils. In this
case it was the most probable value of the pair ionization loss $E_{MP}$ that
was measured. It was defined from the straggling functions of the pairs inside
the detector. The obtained results showed some discrepancy with the theories
[12, 18, 19, 20, 21] of the mean ionization loss of the pair. Namely, a more
significant suppression of the pair ionization loss, than the one expected
from these theories, was registered.

In the present Letter we obtain the straggling function of electron-positron
pairs in thin targets taking into account the discussed above interference
effect in the pair ionization loss and calculate the corresponding value of
the most probable ionization loss $E_{MP}$. The consideration is performed
both with the use of approach of [1], which allows obtaining analytical
expression for $E_{MP}$, and with the use of a more rigorous approach, which
provides more accurate results for straggling functions in very thin targets.
It is shown that in sufficiently thin targets the magnitude of Chudakov effect
is noticeably different for the quantities $E_{MP}$ and $E_{AV}$, and the
value $E_{MP}$ is stronger suppressed due to this effect than $E_{AV}$.
Therefore, it is essential to use the calculated value of $E_{MP}$, rather
than $E_{AV}$, for comparison with experimental results on the most probable
values of the pair ionization loss in thin detectors.

##  2 Targets of moderate thickness

In order to obtain the straggling function for $e^{+}e^{-}$ pairs in a target
of thickness $x$ we follow the approach developed in the seminal paper [1] for
single particles. Let $w(\varepsilon)d\varepsilon$ be the probability for the
pair to lose energy in the interval $(\varepsilon,\varepsilon+d\varepsilon)$
within the unit path. In this case the kinetic equation for the straggling
function of the pair is completely analogous to the one describing such a
function for single particles:

$\frac{df(x,\Delta)}{dx}=\int\limits_{0}^{\varepsilon_{\textrm{max}}}d\varepsilon
w(\varepsilon)[f(x,\Delta-\varepsilon)-f(x,\Delta)],$ (1)

where $\varepsilon_{\textrm{max}}$ is the maximum energy which the pair can
transfer to an atomic electron in a single collision, $\Delta$ is the energy
lost by the pair in the target. The straggling function $f(x,\Delta)$ is
different from zero only for positive $\Delta$. The solution of this equation
with the initial condition $f(0,\Delta)=\delta(\Delta)$ reads

$f(x,\Delta)=\frac{1}{2\pi
i}\int\limits_{-i\infty+a}^{+i\infty+a}dp\exp\bigg{\\{}p\Delta-x\int\limits_{0}^{\varepsilon_{\textrm{max}}}d\varepsilon
w(\varepsilon)(1-e^{-p\varepsilon})\bigg{\\}},$ (2)

where $a$ can be put equal zero and integration with respect to $p$ performed
along the imaginary axis. The integral with respect to $\varepsilon$ can be
split into two parts and presented as

$\int\limits_{0}^{\varepsilon_{1}}d\varepsilon
w(\varepsilon)(1-e^{-p\varepsilon})+\int\limits_{\varepsilon_{1}}^{\varepsilon_{\textrm{max}}}d\varepsilon
w(\varepsilon)(1-e^{-p\varepsilon}).$ (3)

The quantity $\varepsilon_{1}$ should satisfy the condition
$\varepsilon_{k}\ll\varepsilon_{1}\ll\varepsilon_{\textrm{max}}$, where
$\varepsilon_{k}$ are the binding energies of the atomic electrons. This
allows expanding the exponent in the first integral up to the term linear in
small parameter $p\varepsilon$. This integral becomes equal to the mean value
of the pair ionization loss per unit path due to collisions with energy
transfer not exceeding $\varepsilon_{1}$, multiplied by $p$. For this value we
apply the result from [18] and obtain

$\displaystyle\int\limits_{0}^{\varepsilon_{1}}d\varepsilon
w(\varepsilon)(1-e^{-p\varepsilon})\approx
p\int\limits_{0}^{\varepsilon_{1}}d\varepsilon\varepsilon w(\varepsilon)$ (4)
$\displaystyle=\frac{p\eta}{x}\bigg{\\{}\ln\frac{2m\varepsilon_{1}}{\omega_{p}^{2}}-2K_{0}(\omega_{p}s)\bigg{\\}},$

where $\eta=4\pi ne^{4}x/m$ with $n$ being the number of atomic electrons per
unit volume. In terms of conventional denominations applied in the papers on
straggling functions $\eta=2\xi$. Numerical estimations will be further made
for silicon targets. In this case $\eta/x\approx 0.0356~{}\textrm{keV}/\mu$m.
Here also $K_{0}$ is the Macdonald function and $s$ is the distance between
the electron and the positron in the direction orthogonal to the average
velocity of the pair. Following most of the papers on the theory of Chudakov
effect, we consider the simplest case when the electron and positron energies
are the same.

In order to calculate the second integral in (3) one needs the expression for
$w(\varepsilon)$ in the range of $\varepsilon$ where atomic electrons can be
considered as free (since $\varepsilon_{1}\gg\varepsilon_{k}$). It can be
expressed in terms of differential cross section of the energy transfer from
the pair to the electron as $w(\varepsilon)=nd\sigma/d\varepsilon$. Let us
first calculate the cross section $d\sigma/dq$ differential in the transverse
momentum transfer. Being an invariant quantity, it can be calculated in the
pair rest frame. Let us neglect the deflection of the high-energy particles
constituting the pair from the straight lines during the pair motion in the
medium. In this case in the discussed frame of reference one deals with a free
atomic electron scattering on the static potential of the pair in a polarized
medium222The electron-positron separation $s$, which is also an invariant, is
supposed to remain almost constant during the interaction time.. This
potential can be found by the Lorentz transformation of the corresponding
scalar and vector potentials in the laboratory frame. For a single particle of
the pair, which moves along the $z$-axis, they read [24]:

$\varphi_{L}=\frac{e}{\sqrt{\rho^{2}\gamma^{-2}+(z-vt)^{2}}}e^{-\omega_{p}\sqrt{\rho^{2}+\gamma^{2}(z-vt)^{2}}},~{}~{}{\bf
A}_{L}={\bf v}\varphi_{L},$ (5)

where $v=|{\bf v}|$ is the particle velocity, $\rho$ is the distance from the
$z$-axis to the observation point, $\gamma$ is the particle Lorentz-factor.
Let us neglect the divergence of the pair and consider the electron and
positron as moving in parallel directions. In this case the pair rest frame
coincides with the rest frame of each particle. The transformation of (5) to
the pair rest frame gives

$\varphi({\bf r})=\frac{e}{r}e^{-\omega_{p}r},~{}~{}{\bf A}=0,$ (6)

with $r$ being the distance from the particle. The transition amplitude
between the initial and final states of the incident atomic electron reads

$S_{fi}=-ie\int
d^{4}x\overline{\psi}_{f}(x)\varphi_{p}(x)\gamma_{0}\psi_{i}(x),$ (7)

where $\gamma_{0}$ is the gamma matrix, $\psi_{i,f}$ are the plane waves of
the initial and final electron states, $\varphi_{p}$ is the scalar potential
of the pair with the Fourier component

$\varphi_{p}({\bf q})=\frac{4\pi e}{q^{2}+\omega_{p}^{2}}\big{(}1-e^{-i{\bf
qs}}\big{)}.$ (8)

The vector ${\bf s}$ is directed from the electron (with the charge $e$) to
the positron. On the basis of (7) and (8) one obtains the required cross
section:

$\frac{d\sigma}{dq}=16\pi
e^{4}\frac{q}{\big{(}q^{2}+\omega_{p}^{2}\big{)}^{2}}\big{[}1-J_{0}(qs)\big{]},$
(9)

where $J_{0}$ is the Bessel function. Here we neglected the term proportional
to $q^{2}/p_{i}^{2}$, where $p_{i}$ is the electron initial momentum. It is
applicable due to the condition $\eta\ll\varepsilon_{\textrm{max}}$, which we
assume to be valid. The obtained cross section has the same form in the
laboratory frame due to invariance of $d\sigma/dq$ itself, as well as of all
the quantities which it includes (from the very beginning the value of
$\omega_{p}$ refers to the laboratory frame). Applying $\varepsilon=q^{2}/2m$
one finally obtains the cross section of the pair energy transfer to an atomic
electron at $\varepsilon\gg\varepsilon_{k}$ and the corresponding probability
$w(\varepsilon)$:

$w(\varepsilon)=\frac{\eta}{x}\frac{1-J_{0}\big{(}\sqrt{2m\varepsilon}s\big{)}}{\big{(}\varepsilon+\omega_{p}^{2}/2m\big{)}^{2}}.$
(10)

In the integrand of the second integral in (3) this expression for
$w(\varepsilon)$ can be considerably simplified. Indeed, the quantity
$\omega_{p}^{2}/2m$ is much smaller than 1 eV and can be neglected compared to
$\varepsilon\gg\varepsilon_{k}$. Moreover, for the mean ionization potential
of silicon $I=173$ eV, $\sqrt{2mI}\sim 7.5\cdot 10^{8}$ cm-1. Since
$\varepsilon\gg I$, the argument of $J_{0}$ is large even if $s$ is on the
order of interatomic distance (which is the minimum value of $s$ we will be
interested in). Thus, one can neglect $J_{0}$ in (10) as well. As a result,
the second integral in (3) can be calculated in the same way as in [1], since
the simplified expression for $w(\varepsilon)$ here becomes just twice as
large as the corresponding expression used in [1] for single particles.

Substituting the obtained result for (3) into (2) and integrating here along
the imaginary axis, one can present the distribution function in the form,
analogous to the one it has for single particles:

$f(x,\Delta)=\frac{1}{\pi\eta}\int\limits_{0}^{+\infty}dye^{-\pi
y/2}\cos(y\lambda+y\ln y),$ (11)

where

$\displaystyle\lambda=\frac{\Delta-\eta\ln(\eta/\varepsilon^{\prime})}{\eta}-1+\varGamma,~{}~{}\varGamma~{}\textrm{is
the Euler's constant},$ (12)
$\displaystyle\ln\varepsilon^{\prime}=\ln(\omega_{p}^{2}/2m)+2K_{0}(\omega_{p}s).$

It differs from the result [1] for single particles by the substitution
$\xi\to\eta=2\xi$ and a different expression for $\ln\varepsilon^{\prime}$.

As a function of $\lambda$, the integral in (11) has a maximum at
$\lambda\approx-0.223$. Thence, applying (12), one obtains the most probable
value of the ionization loss $\Delta$ of $e^{+}e^{-}$ pair in a thin target:

$E_{MP}(s)=\eta\bigg{\\{}\ln\frac{2m\eta}{\omega_{p}^{2}}-2K_{0}(\omega_{p}s)+0.2\bigg{\\}}.$
(13)

Substituting here $z=s/\vartheta$, where $\vartheta$ is the pair divergence
angle, one can express $E_{MP}$ in terms of distance $z$ from the pair
creation point. Note that expression (13) is valid for a pair of sufficiently
high energy ($\mathcal{E}\gtrsim 100$ MeV), for which the full value density
effect takes place in the ionization loss. In this case $E_{MP}$ does not
depend on the energy of the pair. The result (13) differs from the
corresponding value $E^{S}_{MP}$ for single particles by the substitution
$\xi\to\eta$ and the term with $K_{0}$. Due to the presence of $\eta$ in the
argument of the logarithm, for $s\gg\omega_{p}^{-1}$ the value (13) is larger
than $E^{S}_{MP}$ by a factor which exceeds 2. Thus, at large separations
between the electron and positron the most probable value of the pair
ionization loss slightly exceeds the sum of independent $E^{S}_{MP}$ values
for each particle of the pair (if the particles would have traversed the
target separately). This makes $E_{MP}$ different from the mean value $E_{AV}$
of the pair ionization loss, which for $s\gg\omega_{p}^{-1}$ becomes equal to
the sum of independent electron and positron mean ionization losses.

The fact that for large $s$ the value of $E_{MP}$ exceeds the doubled value of
$E^{S}_{MP}$ can be illustrated by the following consideration. Let the
straggling function for a single electron in a target of thickness $x$ be
$f(x,\Delta)$. The straggling function $f(2x,\Delta)$ in the target of
thickness $2x$ is defined by a convolution [6]

$f(2x,\Delta)=\int\limits_{0}^{\Delta}f(x,\epsilon)f(x,\Delta-\epsilon)d\epsilon,$
(14)

since the particle ionization losses in each layer of thickness $x$, which
comprise the target, are independent. As noted, the value of $E^{S}_{MP}$,
corresponding to this function, exceeds the doubled value of $E^{S}_{MP}$
corresponding to the function $f(x,\Delta)$. For $s\gg\omega_{p}^{-1}$ the
ionization losses of the electron and positron of the pair are independent.
Thus, for such $s$ the ionization loss of the pair in a target of thickness
$x$ is analogous to the ionization loss of a single electron333Presently we do
not distinguish between the struggling functions of high-energy electrons and
positrons, which is justified at least for $\Delta\sim E^{S}_{MP}$. in the
target of thickness $2x$, and the straggling functions in these cases are the
same. Therefore, for $s\gg\omega_{p}^{-1}$ the most probable value $E_{MP}$ of
the pair ionization loss in a target of thickness $x$ should be equal to the
corresponding value for a single electron in a target of thickness $2x$.
Hence, $E_{MP}$ should exceed the doubled value of the most probable
ionization loss $E^{S}_{MP}$ for a single electron in a target of thickness
$x$.

Concerning the mean value $E_{AV}$ of the pair ionization loss, with the
logarithmic accuracy it is defined as [18]

$E_{AV}(s)=\eta\bigg{\\{}\ln\frac{2m\varepsilon_{\textrm{max}}}{\omega_{p}^{2}}-2K_{0}(\omega_{p}s)\bigg{\\}}.$
(15)

For our estimations we take $\varepsilon_{\textrm{max}}$ equal to the electron
(or positron) energy $\gamma m$. In this case expression (15) differs from the
exact expression for the mean total value of the pair ionization loss by a
term on the order of unity in braces, which is small compared to the logarithm
at high energy of the pair. Further we will make numerical estimations of
$E_{AV}$ for the pair energy of $\mathcal{E}=1$ GeV.

For thin targets it might be more convenient to consider not the mean total
value of the ionization loss $E_{AV}$, but the restricted one $E_{AV(R)}$. It
is a part of the value $E_{AV}$ due to collisions with energy transfer not
exceeding some value $\varepsilon_{R}$, which is smaller than the maximum
energy transfer $\varepsilon_{\textrm{max}}$ defined by the collision
kinematics. It is caused by the fact that fast delta-electrons originating
from the collisions with a large energy transfer $\varepsilon>\varepsilon_{R}$
may leave the target without depositing a considerable amount of their energy
inside it. The energy transferred by the incident particle to such electrons
will not be registered in this case. The value of $E_{AV(R)}$ is defined by
the expression (15) with the substitution
$\varepsilon_{\textrm{max}}\to\varepsilon_{R}$. The value of $\varepsilon_{R}$
is usually chosen to be on the order of the energy at which the electron range
coincides with the target thickness.

In order to investigate the magnitude of Chudakov effect for $E_{MP}$ it is
convenient to consider the ratio of this quantity as a function of $s$ to its
value $E_{MP}(\infty)$ at $s\gg\omega_{p}^{-1}$:
$\alpha_{MP}=E_{MP}(s)/E_{MP}(\infty)$. The analogous ratios
$\alpha_{AV}=E_{AV}(s)/E_{AV}(\infty)$ and
$\alpha_{AV(R)}=E_{AV(R)}(s)/E_{AV(R)}(\infty)$ can be introduced for $E_{AV}$
and $E_{AV(R)}$ as well. Numerical estimations on the basis of (13) and (15)
show that generally $\alpha_{AV}>\alpha_{AV(R)}>\alpha_{MP}$ and the
difference between these quantities increases with the decrease of the target
thickness. The numerical results for $\alpha_{MP}$, $\alpha_{AV}$ and
$\alpha_{AV(R)}$ will be presented in the next section (see Fig. 2) where we
treat more accurately the case of very thin targets in which the difference in
the magnitude of Chudakov effect for $E_{MP}$ and $E_{AV}$ (or $E_{AV(R)}$) is
the most noticeable. Though, as will be noted, expression (13) nicely works
for calculation of $\alpha_{MP}$ (however, not $E_{MP}$ itself) even in
sufficiently thin targets.

##  3 Very thin targets

The consideration presented in the previous section, based on the method
applied in [1], does not properly take into account the fact that atomic
electrons are not free but exist in bound states in atoms. As shown (see [6]
and refs. therein), for single particles in silicon targets such a
consideration is valid for moderate target thickness $x\gtrsim 1$ mm. In
thinner targets this approach results in a shape of the straggling function
which is noticeably different from the one obtained experimentally. Though,
its maximum position, corresponding to $E_{MP}$, is not considerably shifted
from (13) in this case. In the present section we apply a more accurate
approach for calculation of a $e^{+}e^{-}$ pair straggling function, which is
valid also for targets of thickness much less than 1 mm, presently called very
thin targets.

Let us obtain the expression for straggling function on the basis of a more
detailed consideration of the cross section of the incident $e^{+}e^{-}$ pair
energy transfer to an atomic electron. Just like for single incident
particles, this cross section can be divided into contributions from close and
distant collisions: $\sigma=\sigma_{c}+\sigma_{d}$. The distant collision part
$\sigma_{d}$ can be calculated with the use of equivalent photon method,
following the approach applied in [25, 26, 27] for consideration of $K$-shell
ionization. In this method the value of $\sigma_{d}$ per a single atomic
electron can be presented as

$\sigma_{d}=\frac{1}{Z}\sum_{k}\int\limits\frac{dN_{k}}{d\omega}\sigma^{k}_{ph}(\omega)d\omega,$
(16)

where $Z$ is the atomic number of the substance, $\sigma^{k}_{ph}(\omega)$ is
the photoionization cross section of the $k$-th atomic shell, $dN_{k}/d\omega$
is the spectral density of the equivalent photon number for the
electromagnetic field of the pair444Its value depends on the atomic shell
number [see formula (19)].. It can be found as

$\frac{dN_{k}}{d\omega}=(4\pi^{2}\omega)^{-1}\int|{\bf
E}_{\omega}|^{2}d^{2}\rho,$ (17)

where ${\bf E}_{\omega}$ is the Fourier component of electric field of the
pair. Let us again neglect the non-parallelism of the electron and positron
trajectories and direct the $z$-axis along the velocities of the particles. In
this case, provided the energy of the pair is sufficiently high, ${\bf
E}_{\omega}$ can be presented in the form of the following Fourier expansion
[28]:

${\bf E}_{\omega}=-\frac{ie}{\pi}e^{i\frac{\omega}{v}z}\int d^{2}q\frac{\bf
q}{q^{2}+\Omega^{2}}e^{i{\bf q}\boldsymbol{\rho}}\big{(}1-e^{-i{\bf
qs}}\big{)},$ (18)

where $\Omega^{2}=\omega^{2}/\gamma^{2}+\omega_{p}^{2}$. Substituting it to
(17) and following the procedure analogous to the one applied in [27] for a
single particle, one obtains

$\frac{dN_{k}}{d\omega}=\frac{2e^{2}}{\pi\omega}\Bigg{\\{}\ln\frac{2m\omega_{k}}{\Omega^{2}}-2K_{0}(\Omega
s)+\Omega sK_{1}(\Omega s)-1\Bigg{\\}}.$ (19)

Here we restricted the region of integration with respect to $q$, which can be
considered as the transferred momentum, by the value
$q_{0}=\sqrt{2m\omega_{k}}$. It is defined as the inverse of the minimum
impact parameter $\rho_{0}$ of the electron or positron with respect to the
atomic electron. This parameter is usually chosen to equal the Bohr radius of
the electron orbit at the considered atomic shell [25, 29] with the ionization
potential $\omega_{k}$.

The photoionization cross section of an atom with $Z$ electrons
$\sigma_{ph}=\sum\limits_{k}\sigma^{k}_{ph}$ can be expressed in terms of
oscillator strength distribution function $f(\varepsilon)$ as [30]

$\sigma_{ph}(\omega)=\frac{2\pi^{2}e^{2}Z\hbar}{mc}f(\varepsilon),$ (20)

where, for convenience, we restored the quantities $c$ and $\hbar$. The
function $f(\varepsilon)$ can be approximated by a set of $\delta$-functions
corresponding to different atomic shells [5] as
$f(\varepsilon)=\sum\limits_{k}F_{k}\delta(\varepsilon-\varepsilon_{k})$,
where $\varepsilon=\hbar\omega$, $\varepsilon_{k}=\hbar\omega_{k}$ and
$F_{k}=Z_{k}/Z$. Here $Z_{k}$ is the number of electrons at the $k$-th shell.
Substituting (19) and (20) to (16) and considering $\omega$ as the energy
$\varepsilon$ transferred during the collision, at
$\gamma\gg\omega_{k}/\omega_{p}$ for each $k$, one obtains the distant
collision contribution to the differential cross section of $e^{+}e^{-}$ pair
energy transfer to an atomic electron:

$\displaystyle\frac{d\sigma_{d}}{d\varepsilon}=$ $\displaystyle\frac{4\pi
e^{4}}{m}\sum\limits_{k}\frac{F_{k}}{\varepsilon_{k}}\Bigg{\\{}\ln\frac{2m\varepsilon_{k}}{\omega_{p}^{2}}-2K_{0}(\omega_{p}s)+\omega_{p}sK_{1}(\omega_{p}s)$
(21) $\displaystyle-1\Bigg{\\}}\delta(\varepsilon-\varepsilon_{k}).$

The close collision part $d\sigma_{c}/d\varepsilon$ is defined by the
expression (10) divided by $n$, since in such collisions the atomic electron
is considered as free. Following [8, 5], we will apply the approximation in
which the atomic electron is considered as free from the very ionization edge
$\varepsilon=\varepsilon_{k}$, while for $\varepsilon<\varepsilon_{k}$ the
cross section $d\sigma_{c}/d\varepsilon$ is set equal zero. In [8] it was
demonstrated that such an approximation for $d\sigma_{c}/d\varepsilon$,
together with the model for $d\sigma_{d}/d\varepsilon$ as a set of
$\delta$-functions, nicely work for the description of experimental results on
single particle straggling functions in thin silicon detectors555Though, since
in this model a single effective excitation energy $\varepsilon_{k}$ is
applied for the $k$-th atomic shell in (21), the model might be expected to be
valid in the case when the average number of collisions experienced by the
particle in the target is not very small. For silicon targets this results in
the appropriate target thickness of more than several microns..

In the present case it is also possible to neglect the second term in the
denominator of (10) and the term with $J_{0}$ in the nominator. Concerning the
term with $J_{0}$, this might seem inapplicable, since presently we consider
the cross section at energies $\varepsilon>\varepsilon_{k}$, but not only
$\varepsilon\gg\varepsilon_{k}$, as we did in the previous section. However,
taking into account this term for $\varepsilon\sim\varepsilon_{k}$ would
result in violation of self-consistency of our model since in this region of
$\varepsilon$, as noted, the cross section $d\sigma_{c}/d\varepsilon$ is
applied in the form, typical for free atomic electrons, which it has at
$\varepsilon\gg\varepsilon_{k}$. Thus, $d\sigma_{c}/d\varepsilon$ reads:

$\frac{d\sigma_{c}}{d\varepsilon}=\frac{4\pi
e^{4}}{m}\sum\limits_{k}\frac{F_{k}}{\varepsilon^{2}}\theta(\varepsilon-\varepsilon_{k}),$
(22)

where $\theta$ is the Heaviside step function. Expressions (21) and (22) lead
to the following formula for the mean value of the pair ionization loss in the
target:

$\displaystyle E_{AV}(s)$
$\displaystyle=nx\int\limits_{0}^{\varepsilon_{\textrm{max}}}\frac{d\sigma}{d\varepsilon}d\varepsilon=\eta\bigg{\\{}\ln\frac{2m\varepsilon_{\textrm{max}}}{\omega_{p}^{2}}-2K_{0}(\omega_{p}s)$
(23) $\displaystyle+\omega_{p}sK_{1}(\omega_{p}s)-1\bigg{\\}}.$

It differs from the result (15) by the term
$[\omega_{p}sK_{1}(\omega_{p}s)-1]$ in braces. For $s<\omega_{p}^{-1}$, where
the Chudakov effect is the most significant, this term is close to zero. For
$s\gg\omega_{p}^{-1}$ it tends to $-1$ and remains small compared to the
logarithmic term in (23). Thus, the result for $E_{AV}$, obtained with the use
of the equivalent photon method, is very close to the result of [18] and the
derived expression for $d\sigma/d\varepsilon$ can be considered as a valid
approximation of the exact interaction cross section. Nevertheless, in order
to be completely compatible with our consideration in the previous section,
where the expression (15) for $E_{AV}$ was applied, we will neglect the small
term $[\omega_{p}sK_{1}(\omega_{p}s)-1]$ in (21), which results in neglecting
the same term in (23), and apply the following expression for the probability
of the pair energy loss $w(\varepsilon)=nd\sigma/d\omega$:

$\displaystyle w(\varepsilon)=$
$\displaystyle\frac{\eta}{x}\sum\limits_{k}\Bigg{[}\frac{F_{k}}{\varepsilon_{k}}\Bigg{\\{}\ln\frac{2m\varepsilon_{k}}{\omega_{p}^{2}}-2K_{0}(\omega_{p}s)\Bigg{\\}}\delta(\varepsilon-\varepsilon_{k})$
(24)
$\displaystyle+\frac{F_{k}}{\varepsilon^{2}}\theta(\varepsilon-\varepsilon_{k})\Bigg{]}.$

It differs from the corresponding expression for single particles (for
$\gamma\gg\varepsilon_{k}/\omega_{p}$ for each $k$), derived in [8], by the
substitution $\xi\to\eta$ and the term with $K_{0}$ in braces.

In order to obtain the expression for $f(x,\Delta)$ it is necessary to
substitute (24) into (2) and calculate the corresponding integral. Following
the procedure applied in [5] for single particles, one obtains

$f(x,\Delta)=\frac{1}{\pi\eta}\int\limits_{0}^{+\infty}dy\exp\Bigg{\\{}-\frac{\pi
y}{2}+g(y,\eta)\Bigg{\\}}\cos\Bigg{[}\frac{y\Delta}{\eta}+h(y,\eta)\Bigg{]},$
(25)

where

$\displaystyle
g(y,\eta)=\sum_{k}\bigg{\\{}yF_{k}\textrm{Si}\big{(}y\varepsilon_{k}/\eta\big{)}-N_{k}\big{[}1-\cos\big{(}y\varepsilon_{k}/\eta\big{)}\big{]}\bigg{\\}},$
(26) $\displaystyle
h(y,\eta)=\sum_{k}\bigg{\\{}yF_{k}\textrm{Ci}\big{(}y\varepsilon_{k}/\eta\big{)}-N_{k}\sin\big{(}y\varepsilon_{k}/\eta\big{)}\bigg{\\}},$

Si and Ci are the sine and cosine integral functions, defined in such way that
at $t\to+\infty$: $\textrm{Si}(t)=\pi/2$, $\textrm{Ci}(t)=0$. The function
(25) has the same structure as the one derived in [5] for single particles,
but differs from it by the substitution $\xi\to\eta$ and the expression for
the mean number of collisions $N_{k}$ with the atomic electrons at $k$-th
shell, which is presently

$N_{k}=\eta\frac{F_{k}}{\varepsilon_{k}}\Bigg{\\{}\ln\frac{2m\varepsilon_{k}}{\omega_{p}^{2}}-2K_{0}(\omega_{p}s)+1\Bigg{\\}}.$
(27)

At large target thickness $x$, when $\eta\gg\varepsilon_{k}$ for each $k$, the
function (25) coincides with (11).

Figure 1: Straggling functions of a high-energy $e^{+}e^{-}$ pair for two
values of the target thickness $x$. Solid lines – calculation on the basis of
(25), dashed lines – on the basis of (11). Separation between the particles in
the pair is $s=0.5\omega_{p}^{-1}$.

For numerical estimations we apply the following effective values of
$\varepsilon_{k}$ for K, L and M atomic shells in silicon targets [8]:
$\varepsilon_{K}=4033$ eV, $\varepsilon_{L}=241$ eV and $\varepsilon_{M}=17$
eV. Here also $F_{K}=2/14$, $F_{L}=8/14$ and $F_{M}=4/14$. Fig. 1 shows the
examples of straggling functions of $e^{+}e^{-}$ pairs in silicon targets of
various thickness calculated on the basis of (11) and (25). The example of a
target with $x=16~{}\mu$m is chosen since a silicon detector of this thickness
has been applied in the experiment [22, 23] discussed above. We see that, like
in the case of single particles, in thin targets the distribution (25) is
broader than (11). For not very small $x$ the maximum positions of these
distributions do not noticeably differ, which allows applying the expression
(13) in this case as well. This difference, however, becomes a bit larger with
the decrease of $x$. At small $x$ a hump appears in the distribution (25) on
the right of its maximum, which is associated with the small probability of
collision with K-shell electrons in this case (for details see [5]).

Fig. 2 demonstrates the dependence of ratios $\alpha_{MP}$, $\alpha_{AV}$ and
$\alpha_{AV(R)}$ on $s$ for the target thickness of $16~{}\mu$m. The quantity
$\alpha_{MP}$ is calculated numerically with the use of expression (25). It
turns out that the obtained dependence for $\alpha_{MP}$ in this case almost
coincides with the one, calculated on the basis of (13) for this thickness.
Thus, one concludes that although expression (13) does not provide an accurate
result for the absolute value of $E_{MP}$ in sufficiently thin targets, it is
still applicable here for calculation of the relative suppression of $E_{MP}$
compared to its value at $s\gg\omega_{p}^{-1}$. Following [22], for the
considered target thickness we apply the value $\varepsilon_{R}=100$ keV for
calculation of $E_{AV(R)}$ and estimation of the quantity $\alpha_{AV(R)}$.
The figure shows that the magnitude of Chudakov effect (i. e. the magnitude of
the pair ionization loss suppression) is different for $E_{MP}$ and $E_{AV}$,
as well as for $E_{MP}$ and $E_{AV(R)}$. This difference grows with the
decrease of separation $s$ between the particles in the pair. For the smallest
separations we presently consider, which are on the order of interatomic
distance $a_{0}\sim 0.1$ nm, the relative difference
$(\alpha_{AV}-\alpha_{MP})/\alpha_{MP}$ reaches about 85% for the chosen
target thickness, while the value of
$(\alpha_{AV(R)}-\alpha_{MP})/\alpha_{MP}$ is around 45%.

Figure 2:  Dependence of the relative value of $E_{MP}$ (thick solid line),
$E_{AV}$ (dot-dashed line) and $E_{AV(R)}$ (thin solid line) on separation
between the electron and positron in a silicon target of 16 $\mu$m thickness;
$\alpha_{AV}$ is estimated for the pair energy of $\mathcal{E}=1$ GeV.

Fig. 2 shows that the value $E_{MP}$ is stronger suppressed due to Chudakov
effect than $E_{AV(R)}$. As noted, in the experiment [22, 23] there was
registered a stronger suppression of the pair ionization loss, than it was
expected from the available theories for $E_{AV(R)}$. Thus, the application of
the theory, developed in the present letter, to the analysis of measurements
[22, 23] may improve (at least partially) the coincidence between the
experimental results and the theoretical predictions. Though, as shown in
[23], a comprehensive analysis of the discussed experiment requires taking
into account such factors as angular and energy distribution of particles in
the pair, their multiple scattering and uncertainty in the position of the
pair creation point, which were quite noticeable under the conditions of this
experiment. Such an analysis is beyond the scope of the present letter and
could be performed elsewhere. For some discussion of this experiment see also
[28].

##  4 Conclusion

In this letter the straggling function of a high-energy electron-positron pair
is calculated under the condition of manifestation of the Chudakov effect in
the pair ionization loss. The expression for the most probable value of the
pair ionization loss $E_{MP}$ is derived. The consideration is made both with
the use of approach, which neglects the effects of atomic electron binding,
valid in the targets of moderate thickness, and with the use of a more
rigorous approach, which allows taking into account such effects and is valid
in much thinner targets as well. It is shown that in sufficiently thin targets
the value of $E_{MP}$ can be noticeably stronger suppressed due to Chudakov
effect than the total $E_{AV}$ or restricted $E_{AV(R)}$ mean value of the
pair ionization loss. The approach, developed in Sec. 3, could be also applied
for the study of straggling functions of larger groups of high-energy
particles in very thin detectors in the case when interference effects are
significant in the particle energy loss.

## Acknowledgments

The work was partially supported by the National Academy of Sciences of
Ukraine (budget program “Support for the Development of Priority Areas of
Scientific Research” (6541230); projects 0121U111556 and 0121U111839).

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# Characterization of the Bernoulli polynomials via the Raabe functional
equation

Bakir FARHI
Laboratoire de Mathématiques appliquées
Faculté des Sciences Exactes
Université de Bejaia, 06000 Bejaia, Algeria
<EMAIL_ADDRESS>
http://farhi.bakir.free.fr/

###### Abstract

The purpose of the present paper is to show that in certain classes of real
(or complex) functions, the Bernoulli polynomials are essentially the only
ones satisfying the Raabe functional equation. For the class of the real
$1$-periodic functions which are expandable as Fourier series, we point out
new solutions of the Raabe functional equation, not relating to the Bernoulli
polynomials. Furthermore, we will give for the considered classes various
proofs, making the mathematical content of the paper quite rich.

MSC 2020: Primary 11B68; Secondary 39B22, 39B32, 30D05.
Keywords: The Raabe multiplicative theorem, Bernoulli polynomials, functional
equations for real or complex functions.

##  1 Introduction and Notation

Throughout this paper, we let ${\mathbb{N}}$ denote the set of positive
integers and ${\mathbb{N}}_{0}:={\mathbb{N}}\cup\\{0\\}$ the set of
nonnegative integers. For $x\in{\mathbb{R}}$, we let $\lfloor{x}\rfloor$ and
$\langle{x}\rangle$ respectively denote the integer part and the fractional
part of $x$. Next, we let $B_{n}(X)$ ($n\in{\mathbb{N}}_{0}$) denote the
Bernoulli polynomials which can be defined through their exponential
generating function:

$\frac{te^{Xt}}{e^{t}-1}=\sum_{n=0}^{+\infty}B_{n}(X)\frac{t^{n}}{n!}.$ (1.1)

The Bernoulli polynomials, together with their values at $X=0$ (called the
Bernoulli numbers), have many important properties and appear in several areas
of mathematics, including number theory, real analysis, special functions,
combinatorics, orthogonal polynomials, and so on. We refer to the book of
Nielsen [5] for an elementary (but old) presentation and to the lecture notes
of Kouba [4] for a modern presentation. It is known for example that for every
$n\in{\mathbb{N}}_{0}$ the Bernoulli polynomial $B_{n}(X)$ is monic of degree
$n$ and satisfies the following polynomial identities ((see [4])):

$\displaystyle B_{n}^{\prime}(X)$ $\displaystyle=nB_{n-1}(X),$ (1.2)
$\displaystyle B_{n}(X+1)-B_{n}(X)$ $\displaystyle=nX^{n-1}.$ (1.3)

It is also known that the Fourier series expansion of the Bernoulli polynomial
$B_{n}(X)$ ($n\in{\mathbb{N}}$) in the interval $[0,1)$ is given by (see [4]):

$B_{n}(x)=\frac{-2\cdot
n!}{(2\pi)^{n}}\sum_{k=1}^{+\infty}\frac{\cos\left(2\pi
kx-\frac{n\pi}{2}\right)}{k^{n}}$ (1.4)

(which is valid for all $x\in[0,1)$ if $n\geq 2$ and all $x\in(0,1)$ if
$n=1$). By analogy with (1.4), let us define

$\overline{B}_{n}(x)=\frac{-2\cdot
n!}{(2\pi)^{n}}\sum_{k=1}^{+\infty}\frac{\sin\left(2\pi
kx-\frac{n\pi}{2}\right)}{k^{n}}$ (1.5)

(for all $(n,x)\in{\mathbb{N}}\times[0,1)\setminus\\{(1,0)\\}$).

In [6], Raabe showed that for every $n\in{\mathbb{N}}_{0}$ and every
$a\in{\mathbb{N}}$, the Bernoulli polynomial $B_{n}(X)$ satisfies the
remarkable functional equation (in $f$):

$f(ax)=a^{n-1}\left(f(x)+f\left(x+\frac{1}{a}\right)+f\left(x+\frac{2}{a}\right)+\dots+f\left(x+\frac{a-1}{a}\right)\right).$
(${\mathbb{E}}_{n,a}$)

(Remark that this equation is trivial for $a=1$). Nielsen [5] remarked that
for every fixed integer $a\geq 2$, the Bernoulli polynomial $B_{n}(X)$ is the
only monic polynomial satisfying $({\mathbb{E}}_{n,a})$; a simple proof of
this result will be given later in this paper. Carlitz [2] generalized the
Raabe theorem by showing that for all $n\in{\mathbb{N}}_{0}$ and all
$a,b\in{\mathbb{N}}$, we have:

$a^{n-1}\sum_{k=0}^{a-1}B_{n}\left(\frac{x}{a}+\frac{bk}{a}\right)=b^{n-1}\sum_{\ell=0}^{b-1}B_{n}\left(\frac{x}{b}+\frac{a\ell}{b}\right).$
(1.6)

Notice that (1.6) is symmetric in $a$ and $b$ and that the Raabe theorem can
be derived from it by taking $b=1$ and replacing $x$ by $ax$. In [3], Carlitz
showed in addition that for every $n\in{\mathbb{N}}_{0}$, the unique monic
polynomial satisfying (1.6) for two different values $a$ and $b$ is the
Bernoulli polynomial $B_{n}(X)$. Related results concerning not only the
Bernoulli polynomials but also the Euler polynomials are given in [1, 2, 3].

In This paper, given $n\in{\mathbb{N}}_{0}$ and $a\geq 2$ an integer, we show
that the Bernoulli polynomial $B_{n}(X)$ is essentially the only solution (up
to a multiplicative constant) of Equation $({\mathbb{E}}_{n,a})$ in certain
classes of real (or complex) functions. The considered classes of functions
are, in this order, the complex polynomials, the complex analytic functions,
$\mathscr{C}^{n}({\mathbb{R}},{\mathbb{R}})$,
$\mathscr{C}^{0}({\mathbb{R}},{\mathbb{R}})$, and the real $1$-periodic
functions which are expandable as Fourier series. Exceptionally for the last
class, the set of solutions of $({\mathbb{E}}_{n,a})$ contains new solutions
not (directly) related to the Bernoulli polynomial $B_{n}(X)$ (see Theorem 3.8
and Corollary 3.10). Further, the methods used to solve Equation
$({\mathbb{E}}_{n,a})$ in each of the considered classes of functions are
various, which makes the mathematical content of the paper quite rich.

##  2 Some useful lemmas about Equation $({\mathbb{E}}_{n,a})$

The first lemma below is obvious and needs no proof.

######  Lemma 2.1.

Let $n\in{\mathbb{N}}_{0}$ and $a\in{\mathbb{N}}$. Then the set of functions
$f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ which are solutions of Equation
$({\mathbb{E}}_{n,a})$ constitutes an ${\mathbb{R}}$-vector space. Similarly,
the set of functions $f:{\mathbb{C}}\rightarrow{\mathbb{C}}$ which are
solutions of Equation $({\mathbb{E}}_{n,a})$ constitutes a
${\mathbb{C}}$-vector space. $\square$

The next two lemmas are crucial for proving our main results (see §3); they
provide important properties for the spaces of solutions of Equations
$({\mathbb{E}}_{n,a})$.

######  Lemma 2.2.

Let $n,a\in{\mathbb{N}}$. If a differentiable real (or complex) function $f$
is a solution of Equation $({\mathbb{E}}_{n,a})$ then its derivative
$f^{\prime}$ is a solution of Equation $({\mathbb{E}}_{n-1,a})$.

###### Proof.

Let $f$ be a differentiable real (or complex) function. Suppose that $f$ is a
solution of Equation $({\mathbb{E}}_{n,a})$; that is

$f(ax)=a^{n-1}\left[f(x)+f\left(x+\frac{1}{a}\right)+f\left(x+\frac{2}{a}\right)+\dots+f\left(x+\frac{a-1}{a}\right)\right].$

By differentiating this last identity, we get (after simplifying and
rearranging):

$f^{\prime}(ax)=a^{n-2}\left[f^{\prime}(x)+f^{\prime}\left(x+\frac{1}{a}\right)+f^{\prime}\left(x+\frac{2}{a}\right)+\dots+f^{\prime}\left(x+\frac{a-1}{a}\right)\right],$

implying that $f^{\prime}$ is a solution of Equation $({\mathbb{E}}_{n-1,a})$,
as required. ∎

######  Lemma 2.3.

Let $n\in{\mathbb{N}}_{0}$ and $a,b\in{\mathbb{N}}$. If a real (or complex)
function $f$ is both a solution of Equation $({\mathbb{E}}_{n,a})$ and
Equation $({\mathbb{E}}_{n,b})$ then it is a solution of Equation
$({\mathbb{E}}_{n,ab})$.

###### Proof.

Suppose that a real (resp. complex) function $f$ is both a solution of
Equation $({\mathbb{E}}_{n,a})$ and Equation $({\mathbb{E}}_{n,b})$. Then we
have for all $x\in{\mathbb{R}}$ (resp. $x\in{\mathbb{C}}$):

$f(abx)=f(a\cdot bx)=a^{n-1}\sum_{k=0}^{a-1}f\left(bx+\frac{k}{a}\right).$
(2.1)

Next, since $f$ satisfies Equation $({\mathbb{E}}_{n,b})$ then we have for all
$x\in{\mathbb{R}}$ (resp. $x\in{\mathbb{C}}$) and all
$k\in\\{0,1,\dots,a-1\\}$:

$\displaystyle f\left(bx+\frac{k}{a}\right)$
$\displaystyle=f\left(b\cdot\left(x+\frac{k}{ab}\right)\right)$
$\displaystyle=b^{n-1}\sum_{\ell=0}^{b-1}f\left(x+\frac{k}{ab}+\frac{\ell}{b}\right)$
$\displaystyle=b^{n-1}\sum_{\ell=0}^{b-1}f\left(x+\frac{a\ell+k}{ab}\right).$

By inserting this (for $0\leq k\leq a-1$) into (2.1), we get:

$\displaystyle f(abx)$
$\displaystyle=a^{n-1}\sum_{k=0}^{a-1}b^{n-1}\sum_{\ell=0}^{b-1}f\left(x+\frac{a\ell+k}{ab}\right)$
$\displaystyle=(ab)^{n-1}\sum_{\begin{subarray}{c}0\leq k\leq a-1\\\
0\leq\ell\leq b-1\end{subarray}}f\left(x+\frac{a\ell+k}{ab}\right).$

But since the sequence of integers ${(a\ell+k)}_{\begin{subarray}{c}0\leq
k\leq a-1\\\ 0\leq\ell\leq b-1\end{subarray}}$ is nothing else the sequence
$0,1,\dots,ab-1$ (up to an order), we conclude to

$f(abx)=(ab)^{n-1}\sum_{r=0}^{ab-1}f\left(x+\frac{r}{ab}\right),$

for all $x\in{\mathbb{R}}$ (resp. $x\in{\mathbb{C}}$). This shows that $f$ is
a solution of Equation $({\mathbb{E}}_{n,ab})$, as required. ∎

######  Remark 2.4.

Let $n\in{\mathbb{N}}_{0}$ be fixed. According to Lemma 2.3, for a function
$f$ to be a solution of all Equations $({\mathbb{E}}_{n,a})$
($a\in{\mathbb{N}}$), it suffices that $f$ is a solution of all Equations
$({\mathbb{E}}_{n,p})$ ($p$ prime).

##  3 The main results

The first result below is already pointed out by Nielsen [5]. Here, we give
for it an alternative nice proof.

######  Theorem 3.1.

Let $n\in{\mathbb{N}}_{0}$ and $a\geq 2$ an integer. Then the solutions of
Equation $({\mathbb{E}}_{n,a})$ in ${\mathbb{C}}[X]$ are the polynomials
$\lambda B_{n}(X)$ ($\lambda\in{\mathbb{C}}$).

###### Proof.

By the Raabe multiplication theorem, we know that the polynomials $\lambda
B_{n}(X)$ ($\lambda\in{\mathbb{C}}$) are all solutions of Equation
$({\mathbb{E}}_{n,a})$. To prove the converse, we argue by induction on
$n\in{\mathbb{N}}_{0}$.
• For $n=0$. Let $P\in{\mathbb{C}}[X]$ be a solution of Equation
$({\mathbb{E}}_{0,a})$; that is

$P(aX)=\frac{1}{a}\left[P(X)+P\left(X+\frac{1}{a}\right)+P\left(X+\frac{2}{a}\right)+\dots+P\left(X+\frac{a-1}{a}\right)\right].$

If $P=0_{{\mathbb{C}}[X]}$ then we are done (since
$0_{{\mathbb{C}}[X]}=0B_{0}(X)$). Suppose now that $P\neq 0_{{\mathbb{C}}[X]}$
and let $d\in{\mathbb{N}}_{0}$ denote the degree of $P$ and
$\lambda\in{\mathbb{C}}^{*}$ denote the leading coefficient of $P$. By
comparing the leading coefficients of the two sides of the last polynomial
identity, we get

$\lambda a^{d}=\lambda,$

giving $d=0$. This means that $P$ is constant, that is $P(X)=\lambda=\lambda
B_{0}(X)$, as required.
• Let $n\in{\mathbb{N}}$. Suppose that every solution in ${\mathbb{C}}[X]$ of
Equation $({\mathbb{E}}_{n-1,a})$ has the form $\lambda B_{n-1}(X)$
($\lambda\in{\mathbb{C}}$) and let us show that every solution in
${\mathbb{C}}[X]$ of Equation $({\mathbb{E}}_{n,a})$ has the form $\lambda
B_{n}(X)$ ($\lambda\in{\mathbb{C}}$). Let $P\in{\mathbb{C}}[X]$ be a solution
of Equation $({\mathbb{E}}_{n,a})$; so (according to Lemma 2.2) its derivative
$P^{\prime}\in{\mathbb{C}}[X]$ is a solution of Equation
$({\mathbb{E}}_{n-1,a})$. Thus, according to our induction hypothesis, there
exists $\lambda\in{\mathbb{C}}$ such that $P^{\prime}(X)=\lambda B_{n-1}(X)$.
It follows by integrating and taking into account (1.2) that:
$P(X)=\frac{\lambda}{n}B_{n}(X)+c$ (for some $c\in{\mathbb{C}}$). So, it
remains to show that $c=0$. Since $P(X)$ and $B_{n}(X)$ are both solutions of
Equation $({\mathbb{E}}_{n,a})$ then their linear combination
$\left(P(X)-\frac{\lambda}{n}B_{n}(X)\right)=c$ is also a solution of
$({\mathbb{E}}_{n,a})$ (according to Lemma 2.1), meaning that:

$c=a^{n-1}\left(\underbrace{c+c+\dots+c}_{a\text{ times}}\right)=a^{n}c,$

implying that $c=0$, as required. This completes this induction and achieves
the proof. ∎

######  Theorem 3.2.

Let $n\in{\mathbb{N}}_{0}$ and $a\geq 2$ be an integer. Then the only complex
analytic functions on ${\mathbb{C}}$ which are solutions of Equation
$({\mathbb{E}}_{n,a})$ are the polynomials $\lambda B_{n}(X)$
($\lambda\in{\mathbb{C}}$).

###### Proof.

Since the polynomials $\lambda B_{n}(X)$ ($\lambda\in{\mathbb{C}}$) are all
solutions of Equation $({\mathbb{E}}_{n,a})$ (by the Raabe theorem) then we
have just to show that any complex analytic function on ${\mathbb{C}}$ which
satisfies $({\mathbb{E}}_{n,a})$ has the form $\lambda B_{n}(X)$
($\lambda\in{\mathbb{C}}$). We shall show this fact in two steps.
• 1st step. (The case $n=0$). Let $f$ be a complex analytic function on
${\mathbb{C}}$ which satisfies Equation $({\mathbb{E}}_{0,a})$, that is

$f(az)=\frac{1}{a}\left[f(z)+f\left(z+\frac{1}{a}\right)+f\left(z+\frac{2}{a}\right)+\dots+f\left(z+\frac{a-1}{a}\right)\right]~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
z\in{\mathbb{C}}),$ (3.1)

and let us show that $f$ is necessarily constant. Since $f$ is analytic (so
continuous) on ${\mathbb{C}}$ then it is in particular continuous on the
closed ball $\overline{B}(0,2):=\\{u\in{\mathbb{C}}:|u|\leq 2\\}$ of
${\mathbb{C}}$. Next, since $\overline{B}(0,2)$ is compact (it is closed and
bounded) then $f$ is bounded on $\overline{B}(0,2)$. So, let us define

$M:=\sup_{\begin{subarray}{c}u\in{\mathbb{C}}\\\ |u|\leq
2\end{subarray}}\left|{f(u)}\right|\in[0,+\infty).$

By leaning on (3.1), we are going to show that $f$ is even bounded on
${\mathbb{C}}$ by $M$, which implies (according to Liouville’s theorem) that
$f$ is constant. Let $z\in{\mathbb{C}}$ arbitrary. Consider a positive integer
$k$ such that $a^{k}\geq|z|$ (such a $k$ exists since
$\lim_{m\rightarrow+\infty}a^{m}=+\infty$) and set $z_{0}:=\frac{z}{a^{k}}$.
So we have $|z_{0}|\leq 1$. Next, since $f$ satisfies Equation
$({\mathbb{E}}_{0,a})$ then (by iterating the result of Lemma 2.3) it
satisfies all Equations $({\mathbb{E}}_{0,a^{r}})$ ($r\in{\mathbb{N}}$); in
particular $f$ satisfies Equation $({\mathbb{E}}_{0,a^{k}})$. Thus we have

$f(z)=f\left(a^{k}z_{0}\right)=\frac{1}{a^{k}}\sum_{\ell=0}^{a^{k}-1}f\left(z_{0}+\frac{\ell}{a^{k}}\right).$

It follows by using the triangle inequality in ${\mathbb{C}}$ that:

$\left|{f(z)}\right|\leq\frac{1}{a^{k}}\sum_{0\leq\ell\leq
a^{k}-1}\left|{f\left(z_{0}+\frac{\ell}{a^{k}}\right)}\right|.$

But for every $\ell\in\\{0,1,\dots,a^{k}-1\\}$, we have that
$\left|{z_{0}+\frac{\ell}{a^{k}}}\right|\leq\left|{z_{0}}\right|+\frac{\ell}{a^{k}}\leq
1+1=2$, implying that
$\left|{f\left(z_{0}+\frac{\ell}{a^{k}}\right)}\right|\leq M$. It follows from
this fact that:

$\left|{f(z)}\right|\leq\frac{1}{a^{k}}\sum_{0\leq\ell\leq a^{k}-1}M=M.$

Consequently, $f$ is bounded on ${\mathbb{C}}$; thus (by Liouville’s theorem)
it is constant, as required.
• 2nd step. (The general case). Let $f$ be a complex analytic function on
${\mathbb{C}}$ which satisfies Equation $({\mathbb{E}}_{n,a})$. By iterating
the result of Lemma 2.2, we derive that the $n$th derivative $f^{(n)}$ of $f$
is a solution of Equation $({\mathbb{E}}_{0,a})$, implying (according to the
result of the first step above) that $f^{(n)}$ is constant. Thus $f$ is a
polynomial of degree $\leq n$, which implies (according to Theorem 3.1) that
$f$ has the required form $f(X)=\lambda B_{n}(X)$ ($\lambda\in{\mathbb{C}}$).
This completes the proof of the theorem. ∎

######  Theorem 3.3.

Let $a\geq 2$ be an integer. Then the only solutions of Equation
$({\mathbb{E}}_{0,a})$ in $\mathscr{C}^{0}({\mathbb{R}},{\mathbb{R}})$ are the
constant functions.

###### Proof.

It is obvious that the constant functions are solutions of Equation
$({\mathbb{E}}_{0,a})$. Conversely, let
$f\in\mathscr{C}^{0}({\mathbb{R}},{\mathbb{R}})$ be a solution of
$({\mathbb{E}}_{0,a})$ and let us prove that $f$ is constant. According to
Lemma 2.3, $f$ is a solution of all Equations $({\mathbb{E}}_{0,a^{k}})$
($k\in{\mathbb{N}}$); that is for all $k\in{\mathbb{N}}$ and all
$x\in{\mathbb{R}}$, we have

$f\left(a^{k}x\right)=\frac{1}{a^{k}}\left[f(x)+f\left(x+\frac{1}{a^{k}}\right)+f\left(x+\frac{2}{a^{k}}\right)+\dots+f\left(x+\frac{a^{k}-1}{a^{k}}\right)\right].$
(3.2)

We remark that the right-hand side of (3.2) is nothing else the left Riemann
sum of $f$ with respect to the regular subdivision $x_{0}=x$,
$x_{1}=x+\frac{1}{a^{k}}$, $x_{2}=x+\frac{2}{a^{k}}$, …, $x_{a^{k}}=x+1$ of
the interval $[x,x+1]$. Since $f\in\mathscr{C}^{0}({\mathbb{R}},{\mathbb{R}})$
then it is Riemann integrable on every bounded closed interval of
${\mathbb{R}}$; in particular, $f$ is Riemann integrable on $[x,x+1]$ for
every $x\in{\mathbb{R}}$. So we derive from (3.2) that we have for all
$x\in{\mathbb{R}}$:

$\lim_{k\rightarrow+\infty}f\left(a^{k}x\right)=\int_{x}^{x+1}f(t)\,dt.$ (3.3)

Now, by using (3.3), we have for all $x\in{\mathbb{R}}$ and all
$\ell\in{\mathbb{N}}$:

$\displaystyle\int_{x}^{x+1}f(t)\,dt$
$\displaystyle=\lim_{k\rightarrow+\infty}f\left(a^{k}x\right)$
$\displaystyle=\lim_{k\rightarrow+\infty}f\left(a^{k-\ell}x\right)$
$\displaystyle=\lim_{k\rightarrow+\infty}f\left(a^{k}\left(\frac{x}{a^{\ell}}\right)\right)$
$\displaystyle=\int_{\frac{x}{a^{\ell}}}^{\frac{x}{a^{\ell}}+1}f(t)\,dt~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left(\text{by
applying \eqref{eq7} for }\frac{x}{a^{\ell}}\text{ instead of }x\right).$

By tending $\ell$ to infinity and taking into account the continuity of $f$,
it follows that we have for all $x\in{\mathbb{R}}$:

$\int_{x}^{x+1}f(t)\,dt=\int_{0}^{1}f(t)\,dt.$ (3.4)

Then by differentiating, we get

$f(x+1)-f(x)=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall x\in{\mathbb{R}}),$

meaning that $f$ is $1$-periodic. Next, set

$\displaystyle\sigma$ $\displaystyle:=\int_{0}^{1}f(t)\,dt$ and $\displaystyle
E$
$\displaystyle:=\left\\{\frac{r}{a^{s}-1};~{}r\in{\mathbb{Z}},s\in{\mathbb{N}}\right\\}\subset{\mathbb{R}}.$

Our strategy now consists in showing that $f$ is constant on $E$ (equal to
$\sigma$) and that $E$ is dense in ${\mathbb{R}}$, concluding that $f$ is
constant on ${\mathbb{R}}$ (since it is continuous on ${\mathbb{R}}$).
According to (3.3) and (3.4), we have for all $x\in{\mathbb{R}}$:

$\lim_{k\rightarrow+\infty}f\left(a^{k}x\right)=\sigma.$ (3.5)

Let us first show that $f$ is constant on $E$; more precisely that
$f(x)=\sigma$, $\forall x\in E$. Let $x\in E$ arbitrary. So we can write
$x=\frac{r}{a^{s}-1}$ for some $r\in{\mathbb{Z}}$ and some $s\in{\mathbb{N}}$.
We remark that for all $k\in{\mathbb{N}}$, we have
$(a^{ks}-1)x\in{\mathbb{Z}}$ (since the integer $((a^{ks}-1)$ is a multiple of
the integer $(a^{s}-1)$); that is $a^{ks}x-x\in{\mathbb{Z}}$. But since $f$ is
$1$-periodic (as proved above), we derive that

$f(x)=f\left(a^{ks}x\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
k\in{\mathbb{N}}).$

It follows by tending $k$ to $+\infty$ and using (3.5) that:

$f(x)=\sigma.$

Thus $f$ is constant (equal to $\sigma$) on $E$, as claimed it to be. Now, let
us show that $E$ is dense in ${\mathbb{R}}$. So we have to show that for all
$u\in{\mathbb{R}}$, there exists a sequence ${(u_{n})}_{n\in{\mathbb{N}}}$ of
$E$ which converges to $u$. Let $u\in{\mathbb{R}}$ arbitrary. For all
$n\in{\mathbb{N}}$, define

$r_{n}:=\left\lfloor\left(a^{n}-1\right)u\right\rfloor~{}~{}~{}~{}\text{and}~{}~{}~{}~{}u_{n}:=\frac{r_{n}}{a^{n}-1}.$

It is clear that ${(u_{n})}_{n}$ is a sequence of $E$. In addition, we have
for all $n\in{\mathbb{N}}$:

$\displaystyle u-u_{n}$ $\displaystyle=u-\frac{r_{n}}{a^{n}-1}$
$\displaystyle=\frac{\left(a^{n}-1\right)u-\left\lfloor\left(a^{n}-1\right)u\right\rfloor}{a^{n}-1}$
$\displaystyle=\frac{\left\langle\left(a^{n}-1\right)u\right\rangle}{a^{n}-1}.$

Thus

$0\leq u-u_{n}<\frac{1}{a^{n}-1}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
n\in{\mathbb{N}}),$

implying that ${(u_{n})}_{n}$ converges to $u$ (since
$\frac{1}{a^{n}-1}\rightarrow 0$ as $n\rightarrow+\infty$). Hence $E$ is dense
in ${\mathbb{R}}$, as claimed it to be. Finally, the three facts: “$f$ is
continuous on ${\mathbb{R}}$”, “$f$ is constant on $E$”, and “$E$ is dense in
${\mathbb{R}}$” conclude that $f$ is constant on ${\mathbb{R}}$, as required.
The proof is complete. ∎

######  Remark 3.4.

There exist solutions of Equation $({\mathbb{E}}_{0,a})$ ($a\in{\mathbb{N}}$,
$a\geq 2$) which are not constant. We may think, for example, of the real
function $1\mathrm{I}_{{\mathbb{Q}}}$ (the indicator function of the set of
rational numbers ${\mathbb{Q}}$).

We now derive from Theorem 3.3 the following more general result:

######  Corollary 3.5.

Let $n\in{\mathbb{N}}_{0}$ and $a\geq 2$ be an integer. Then the only
solutions of Equation $({\mathbb{E}}_{n,a})$ in
$\mathscr{C}^{n}({\mathbb{R}},{\mathbb{R}})$ are the polynomials $\lambda
B_{n}(X)$ ($\lambda\in{\mathbb{R}}$).

###### Proof.

Let $f\in\mathscr{C}^{n}({\mathbb{R}},{\mathbb{R}})$ be a solution of Equation
$({\mathbb{E}}_{n,a})$. Then (by iterating the result of Lemma 2.2) the $n$th
derivative $f^{(n)}$ of $f$ (which belongs to
$\mathscr{C}^{0}({\mathbb{R}},{\mathbb{R}})$ since $f$ belongs to
$\mathscr{C}^{n}({\mathbb{R}},{\mathbb{R}})$) is a solution of Equation
$({\mathbb{E}}_{0,a})$. It follows (according to Theorem 3.3) that $f^{(n)}$
is constant. Thus $f$ is a polynomial, which implies (according to Theorem
3.1) that $f$ has the required form $f(X)=\lambda B_{n}(X)$
($\lambda\in{\mathbb{R}}$). This achieves the proof. ∎

We are now interested in the solutions of Equations $({\mathbb{E}}_{n,a})$ in
$\mathscr{C}^{0}({\mathbb{R}},{\mathbb{R}})$. We have the following theorem:

######  Theorem 3.6.

Let $n\in{\mathbb{N}}_{0}$ and $a\geq 2$ be an integer. Let
$f\in\mathscr{C}^{0}({\mathbb{R}},{\mathbb{R}})$ such that the limit
$\lim_{h\rightarrow 0}\frac{\int_{h}^{h+1}f(t)\,dt}{h^{n}}$ exists (possibly
infinite). Then $f$ is a solution of Equation $({\mathbb{E}}_{n,a})$ if and
only if it has the form

$f(X)=\lambda B_{n}(X)+\tau(X),$ (3.6)

where $\lambda\in{\mathbb{R}}$ and $\tau$ is a continuous real $1$-periodic
function which is a solution of $({\mathbb{E}}_{n,a})$.

###### Proof.

It is immediate that the functions having the form in (3.6) are solutions of
$({\mathbb{E}}_{n,a})$. Conversely, let
$f\in\mathscr{C}^{0}({\mathbb{R}},{\mathbb{R}})$ be a solution of
$({\mathbb{E}}_{n,a})$ and let us show that it has the form in (3.6).
According to Lemma 2.3, $f$ is a solution of all Equations
$({\mathbb{E}}_{n,a^{k}})$ ($k\in{\mathbb{N}}$); that is for all
$k\in{\mathbb{N}}$ and all $x\in{\mathbb{R}}$, we have

$f\left(a^{k}x\right)=a^{k(n-1)}\left[f(x)+f\left(x+\frac{1}{a^{k}}\right)+f\left(x+\frac{2}{a^{k}}\right)+\dots+f\left(x+\frac{a^{k}-1}{a^{k}}\right)\right];$

or equivalently

$\frac{f\left(a^{k}x\right)}{a^{kn}}=\frac{1}{a^{k}}\sum_{0\leq\ell<a^{k}}f\left(x+\frac{\ell}{a^{k}}\right).$
(3.7)

We remark that the right-hand side of (3.7) is exactly the left Riemann sum of
$f$ with respect to the regular subdivision $x_{0}=x$,
$x_{1}=x+\frac{1}{a^{k}}$, $x_{2}=x+\frac{2}{a^{k}}$, …, $x_{a^{k}}=x+1$ of
the interval $[x,x+1]$. Since $f\in\mathscr{C}^{0}({\mathbb{R}},{\mathbb{R}})$
then it is Riemann integrable on every bounded closed interval of
${\mathbb{R}}$; in particular, $f$ is Riemann integrable on $[x,x+1]$ for
every $x\in{\mathbb{R}}$. So by tending $k$ to $+\infty$ in (3.7), we derive
that we have for all $x\in{\mathbb{R}}$:

$\lim_{k\rightarrow+\infty}\frac{f\left(a^{k}x\right)}{a^{kn}}=\int_{x}^{x+1}f(t)\,dt.$
(3.8)

Now, by using (3.8), we have for all $x\in{\mathbb{R}}^{*}$ and all
$\ell\in{\mathbb{N}}$:

$\displaystyle\int_{x}^{x+1}f(t)\,dt$
$\displaystyle=\lim_{k\rightarrow+\infty}\frac{f\left(a^{k}x\right)}{a^{kn}}$
$\displaystyle=\lim_{k\rightarrow+\infty}\frac{f\left(a^{k-\ell}x\right)}{a^{(k-\ell)n}}$
$\displaystyle=a^{\ell
n}\lim_{k\rightarrow+\infty}\frac{f\left(a^{k}\left(\frac{x}{a^{\ell}}\right)\right)}{a^{kn}}$

$\displaystyle=a^{\ell
n}\int_{\frac{x}{a^{\ell}}}^{\frac{x}{a^{\ell}}+1}f(t)\,dt~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\text{by
applying \eqref{eq12} for }\frac{x}{a^{\ell}}\text{ instead of }x)$
$\displaystyle=x^{n}\left(\frac{a^{\ell}}{x}\right)^{n}\int_{\frac{x}{a^{\ell}}}^{\frac{x}{a^{\ell}}+1}f(t)\,dt.$

Then, by tending $\ell$ to infinity and setting $h=\frac{x}{a^{\ell}}$ in the
last expression, we get for all $x\in{\mathbb{R}}^{*}$:

$\int_{x}^{x+1}f(t)\,dt=x^{n}\lim_{h\rightarrow
0}\frac{\int_{h}^{h+1}f(t)\,dt}{h^{n}}.$

By specializing in particular $x$ to $1$ in this last equality, we obtain that

$\lim_{h\rightarrow
0}\frac{\int_{h}^{h+1}f(t)\,dt}{h^{n}}=\int_{1}^{2}f(t)\,dt.$

So, setting

$\sigma:=\int_{1}^{2}f(t)\,dt,$

we derive that we have for all $x\in{\mathbb{R}}^{*}$:

$\int_{x}^{x+1}f(t)\,dt=\sigma x^{n}.$ (3.9)

But in view of the continuity of $f$, (3.9) is still true for $x=0$, so it is
true for all $x\in{\mathbb{R}}$. Then, by differentiating (3.9), we get for
all $x\in{\mathbb{R}}$:

$f(x+1)-f(x)=n\sigma x^{n-1};$

that is (according to (1.3)):

$f(x+1)-f(x)=\sigma\left(B_{n}(x+1)-B_{n}(x)\right).$

So, by defining a function $\tau$ as:

$\tau(x):=f(x)-\sigma B_{n}(x)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
x\in{\mathbb{R}}),$

we have that

$\tau(x+1)=\tau(x)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall x\in{\mathbb{R}}),$

showing that $\tau$ is $1$-periodic. In addition, $\tau$ is continuous and it
is a solution of Equation $({\mathbb{E}}_{n,a})$ (since it is a linear
combination of $f$ and $B_{n}$, which are both continuous and solutions of
Equation $({\mathbb{E}}_{n,a})$). Consequently, $f$ has the required form
$f(X)=\sigma B_{n}(X)+\tau(X)$. This completes the proof. ∎

######  Remarks 3.7.

  1. 1.

The condition of the existence of the limit $\lim_{h\rightarrow
0}\frac{\int_{h}^{h+1}f(t)\,dt}{h^{n}}$ occurring in Theorem 3.6 is
automatically verified for $n=0$ (since $f$ is continuous). This is why this
condition doesn’t appear in Theorem 3.3.

  2. 2.

An alternative condition which can replace the condition “$\lim_{h\rightarrow
0}\frac{\int_{h}^{h+1}f(t)\,dt}{h^{n}}$ exists” in Theorem 3.6 is
“$\lim_{y\rightarrow+\infty}\frac{f(y)}{y^{n}}$ exists”. Remark that this new
condition is not automatically verified for $n=0$ (contrary to the first one).

  3. 3.

We can easily show that the function $\tau$ involved in Theorem 3.6 is
$o(x^{n})$ (as $x\rightarrow+\infty$). For $n=0$, this information implies
that $\tau$ is a zero function (this is why $\tau$ doesn’t appear in Theorem
3.3); but for $n\geq 1$, the fact $\tau(x)=o(x^{n})$ is obvious (since $\tau$
is continuous and $1$-periodic, so bounded).

The statement of Theorem 3.6 above incites us to solve Equation
$({\mathbb{E}}_{n,a})$ ($n,a\in{\mathbb{N}}_{0}$, $a\geq 2$) in the set of
real $1$-periodic functions. In what follows, we propose more precisely to
solve that equation in the set of real $1$-periodic functions which are
expandable as Fourier series. In this direction, we have obtained the
following result:

######  Theorem 3.8.

Let $n\in{\mathbb{N}}$ and $a\geq 2$ be an integer. Let also $f$ be a real
$1$-periodic function which is expandable as a Fourier series. Then $f$ is a
solution of Equation $({\mathbb{E}}_{n,a})$ if and only if it has the form:

$f(x)=\sum_{k=1}^{+\infty}\left(u_{k}\frac{\cos(2\pi
kx)}{k^{n}}+v_{k}\frac{\sin(2\pi kx)}{k^{n}}\right),$

where ${(u_{k})}_{k\geq 1}$ and ${(v_{k})}_{k\geq 1}$ are real sequences which
satisfy the relations:

$\left\\{\begin{array}[]{rcl}u_{ak}&=&u_{k},\\\
v_{ak}&=&v_{k}\end{array}\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
k\in{\mathbb{N}}).$

###### Proof.

For simplicity, we use the exponential form of Fourier series. Let

$f(x)=\sum_{k\in{\mathbb{Z}}}\alpha_{k}e^{2\pi ikx}$

be the Fourier series expansion of $f$, where
${(\alpha_{k})}_{k\in{\mathbb{Z}}}$ is a sequence of complex numbers. Using
the elementary formula

$e^{2\pi ikx}+e^{2\pi ik\left(x+\frac{1}{a}\right)}+e^{2\pi
ik\left(x+\frac{2}{a}\right)}+\dots+e^{2\pi
ik\left(x+\frac{a-1}{a}\right)}=\begin{cases}ae^{2\pi ikx}&\text{if }k\equiv
0\ (\mathrm{mod}\ a)\\\ 0&\text{else}\end{cases}$

(valid for all $k\in{\mathbb{Z}}$ and all $x\in{\mathbb{R}}$), we find that we
have for all $x\in{\mathbb{R}}$:

$\displaystyle
f(x)+f\left(x+\frac{1}{a}\right)+f\left(x+\frac{2}{a}\right)+\dots+f\left(x+\frac{a-1}{a}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}k\in{\mathbb{Z}}\\\ k\equiv 0\
(\mathrm{mod}\ a)\end{subarray}}\alpha_{k}ae^{2\pi ikx}$
$\displaystyle=\sum_{\ell\in{\mathbb{Z}}}a\alpha_{a\ell}e^{2\pi ia\ell
x}~{}(\text{by putting }k=a\ell).$

Thus (for all $x\in{\mathbb{R}}$):

$a^{n-1}\left(f(x)+f\left(x+\frac{1}{a}\right)+f\left(x+\frac{2}{a}\right)+\dots+f\left(x+\frac{a-1}{a}\right)\right)=\sum_{\ell\in{\mathbb{Z}}}a^{n}\alpha_{a\ell}e^{2\pi
ia\ell x}.$ (3.10)

On the other hand, we have for all $x\in{\mathbb{R}}$:

$f(ax)=\sum_{k\in{\mathbb{Z}}}\alpha_{k}e^{2\pi iakx}.$ (3.11)

By comparing between (3.10) and (3.11), we derive that:

$f\text{ is a solution of
}({\mathbb{E}}_{n,a})\Longleftrightarrow\alpha_{ak}=\frac{\alpha_{k}}{a^{n}}~{}~{}~{}~{}(\forall
k\in{\mathbb{Z}}).$

For $k=0$, the relation $\alpha_{ak}=\frac{\alpha_{k}}{a^{n}}$ gives
$\alpha_{0}=0$. Next, for $k\in{\mathbb{Z}}^{*}$, by setting
$\alpha_{k}=\frac{\beta_{k}}{k^{n}}$ (so
${(\beta_{k})}_{k\in{\mathbb{Z}}^{*}}$ is a sequence of complex numbers), we
derive that:

$f\text{ is a solution of
}({\mathbb{E}}_{n,a})\Longleftrightarrow\alpha_{0}=0\text{ and
}\beta_{ak}=\beta_{k}~{}(\forall k\in{\mathbb{Z}}^{*}).$

Finally, by setting for all $k\in{\mathbb{N}}$:

$u_{k}:=2\,\Re(\beta_{k})~{}~{}\text{and}~{}~{}v_{k}:=-2\,\Im(\beta_{k})$

and taking into account that $f$ is real, the Fourier series expansion of $f$
becomes

$\displaystyle f(x)$ $\displaystyle=\sum_{k\in{\mathbb{Z}}}\alpha_{k}e^{2\pi
ikx}$
$\displaystyle=\alpha_{0}+\sum_{k\in{\mathbb{Z}}^{*}}\beta_{k}\frac{e^{2\pi
ikx}}{k^{n}}$
$\displaystyle=\alpha_{0}+\sum_{k=1}^{+\infty}\left(u_{k}\frac{\cos(2\pi
kx)}{k^{n}}+v_{k}\frac{\sin(2\pi kx)}{k^{n}}\right)$

and we have

$f\text{ is a solution of
}({\mathbb{E}}_{n,a})\Longleftrightarrow\alpha_{0}=0,u_{ak}=u_{k}~{}(\forall
k\in{\mathbb{N}}),\text{ and }v_{ak}=v_{k}~{}(\forall k\in{\mathbb{N}}).$

The theorem is proved. ∎

######  Examples 3.9.

Let $n$ be a fixed positive integer which we take $\geq 2$ if necessary.
Leaning on Theorem 3.8, let us give some examples of real functions which are
solutions of some Equations $({\mathbb{E}}_{n,a})$ without being solutions of
others. Denoting by $s_{2}(k)$ the sum of the binary digits of a given
positive integer $k$, it is clear that we have $s_{2}(2k)=s_{2}(k)$ for every
$k\in{\mathbb{N}}$. So, according to Theorem 3.8, the real function defined
by:

$f_{1}(x):=\sum_{k=1}^{+\infty}\frac{s_{2}(k)}{k^{n}}\cos(2\pi
kx)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall x\in{\mathbb{R}})$

is a solution of Equation $({\mathbb{E}}_{n,2})$ but it is not a solution of
Equation $({\mathbb{E}}_{n,3})$ for example (since $s_{2}(3k)\neq s_{2}(k)$ in
general). Next, denoting by $\mathrm{odd}(k)$ the odd part of a given positive
integer $k$ (i.e., the greatest odd positive divisor of $k$), the real
function defined by:

$f_{2}(x):=\sum_{k=1}^{+\infty}\frac{\mathrm{odd}(k)}{k^{n}}\cos(2\pi
kx)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall x\in{\mathbb{R}})$

is a solution of Equation $({\mathbb{E}}_{n,2})$ (since we have obviously
$\mathrm{odd}(2k)=\mathrm{odd}(k)$ for any $k\in{\mathbb{N}}$), but it is not
a solution of any other Equation $({\mathbb{E}}_{n,a})$ if $a$ is not a power
of $2$. Finally, denoting by ${(\mu_{k})}_{k\in{\mathbb{N}}}$ the sequence
defined by:

$\mu_{k}:=\begin{cases}1&\text{if }k\text{ is a power of }2\\\
0&\text{else}\end{cases}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
k\in{\mathbb{N}})$

(which clearly satisfies the condition $\mu_{2k}=\mu_{k}$, for all
$k\in{\mathbb{N}}$), we obtain that the real function given by:

$f_{3}(x):=\sum_{k=1}^{+\infty}\frac{\mu_{k}}{k^{n}}\cos(2\pi
kx)=\sum_{\ell=0}^{+\infty}\frac{\cos\left(2^{\ell+1}\pi x\right)}{2^{\ell
n}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall x\in{\mathbb{R}})$

is a solution of Equation $({\mathbb{E}}_{n,2})$ but it is not a solution of
any other Equation $({\mathbb{E}}_{n,a})$ for $a$ not a power of $2$. Remark
also that $f_{3}\in\mathscr{C}^{n-1}({\mathbb{R}},{\mathbb{R}})$ but
$f_{3}\not\in\mathscr{C}^{n}({\mathbb{R}},{\mathbb{R}})$.

Now, for a given $n\in{\mathbb{N}}$, we are going to characterize, among the
real $1$-periodic functions expandable as Fourier series, those which are
solutions of all Equations $({\mathbb{E}}_{n,a})$ ($a\in{\mathbb{N}}$). We
have the following corollary:

######  Corollary 3.10.

Let $n\in{\mathbb{N}}$ and $f$ be a real $1$-periodic function which is
expandable as a Fourier series. Then $f$ is a solution of all Equations
$({\mathbb{E}}_{n,a})$ ($a\geq 2$ an integer) if and only if it has the form:

$f(x)=\alpha\sum_{k=1}^{+\infty}\frac{\cos(2\pi
kx)}{k^{n}}+\beta\sum_{k=1}^{+\infty}\frac{\sin(2\pi kx)}{k^{n}}$ (3.12)

(where $\alpha,\beta\in{\mathbb{R}}$), or equivalently the form:

$f(x)=\lambda B_{n}(x)+\mu\overline{B}_{n}(x)~{}~{}~{}~{}~{}~{}(\text{valid
for all $x\in[0,1)$ if $n\geq 2$ and all $x\in(0,1)$ if $n=1$})$

(where $\lambda,\mu\in{\mathbb{R}}$).

###### Proof.

By Theorem 3.8, it is clear that the functions of the form in (3.12) are
solutions of all Equations $({\mathbb{E}}_{n,a})$ ($a\geq 2$). Conversely,
suppose that $f$ is a solution of all Equations $({\mathbb{E}}_{n,a})$ ($a\geq
2$) and let us show that $f$ has the form in (3.12). By Theorem 3.8, $f$ would
have the form:

$f(x)=\sum_{k=1}^{+\infty}\left(u_{k}\frac{\cos(2\pi
kx)}{k^{n}}+v_{k}\frac{\sin(2\pi kx)}{k^{n}}\right),$

where ${(u_{k})}_{k\geq 1}$ and ${(v_{k})}_{k\geq 1}$ are real sequences
satisfying the relations

$\left\\{\begin{array}[]{rcl}u_{ak}&=&u_{k}\\\
v_{ak}&=&v_{k}\end{array}\right.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
k,a\in{\mathbb{N}}).$

In particular (by specializing $k$ to $1$), we find that for any
$a\in{\mathbb{N}}$: $u_{a}=u_{1}$ and $v_{a}=v_{1}$. In other words, the
sequences ${(u_{k})}_{k}$ and ${(v_{k})}_{k}$ are constant. Consequently, we
have

$f(x)=u_{1}\sum_{k=1}^{+\infty}\frac{\cos(2\pi
kx)}{k^{n}}+v_{1}\sum_{k=1}^{+\infty}\frac{\sin(2\pi kx)}{k^{n}},$

which is the required form of $f$. The last part of the corollary is an
immediate consequence of Formulas (1.4) and (1.5) and some elementary
trigonometric formulas. This completes the proof of the corollary. ∎

######  Example 3.11.

The Fourier series expansion of the real $1$-periodic function
$x\mapsto\log\left(2|\sin(\pi x)|\right)$ is well known and given by:

$\log\left(2\left|\sin(\pi
x)\right|\right)=-\sum_{k=1}^{+\infty}\frac{\cos(2\pi
kx)}{k}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
x\in{\mathbb{R}}\setminus{\mathbb{Z}}).$

So, according to Corollary 3.10, this “elementary” function is a solution of
all Equations $({\mathbb{E}}_{1,a})$ ($a\in{\mathbb{N}}$).

## References

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