# A hydrodynamic analog of interaction-free measurement

Valeri Frumkin<EMAIL_ADDRESS>Department of Mathematics, Massachusetts
Institute of Technology.  John W. M. Bush<EMAIL_ADDRESS>Department of
Mathematics, Massachusetts Institute of Technology.

###### Abstract

Interaction-free measurement allows for quantum particles to detect objects
along paths they never traveled. As such, it represents one of the most
beguiling of quantum phenomena. Here, we present a classical analog of
interaction-free measurement using the hydrodynamic pilot-wave system, in
which a droplet self-propels across a vibrating fluid surface, guided by a
wave of its own making. We argue that existing rationalizations of
interaction-free quantum measurement in terms of particles being guided by
wave forms allow for a classical description manifest in our hydrodynamic
system, wherein the measurement is decidedly not interaction-free.

††preprint: APS/123-QED

Interaction-free measurement in quantum mechanics seemingly allows one to
obtain information about the quantum state of an object without its being
“disturbed” by the measurement process. Similar ideas, such as negative result
measurement, can be traced back to Renninger [1] and Dicke [2]. However, the
currently accepted notion of interaction-free measurement is due to Elitzur
and Vaidman [3], who argued that the presence of an object in an
interferometer could modify the interference of a photon, even without any
direct interaction between object and photon. The authors considered a Mach-
Zehnder interferometer with a bomb placed along one of its arms (figure 1).
The bomb has a highly sensitive trigger, so that any photon passing through it
will detonate the bomb; however, if the bomb malfunctions, the photon passes
undisturbed. Since a functioning or malfunctioning bomb is equivalent to its
being present or absent in the interferometer, we adopt the latter
description. The standard rationale for the effect is as follows [3]. As the
photon enters the first beam-splitter $B_{1}$, its wave function is split in
two, with each part traveling along one arm of the interferometer. In the
absence of a bomb, the two wave packets recombine at the second beam-splitter
$B_{2}$, in which case the photon will be detected at detector $D_{1}$ with
probability $1$. If the bomb is present, $50\%$ of the time the particle
follows path 1 and detonates the bomb; otherwise, it follows path 2. As it
does so, path 1 is blocked by the bomb, thus; interference between the two
wave packets at $B_{2}$ is prevented, and the particle is detected at $D_{1}$
or $D_{2}$ with equal probability. Thus, if a particle is detected at $D_{2}$,
it provides the experimenter information about the presence of a live bomb
positioned along a path that it never traveled.

Interaction-free measurement has been studied in various contexts, both
theoretically [4, 5, 6, 7, 8, 9, 10, 11, 12], and experimentally [13, 14, 15,
16, 17, 18, 7, 19, 20, 21]. However, there is still no widely accepted
consensus on the precise meaning of the term “interaction-free” [22]. Indeed,
the extent to which an experiment, like that proposed by Elitzur and Vaidman,
is interaction-free, depends on the particular interpretation of quantum
mechanics one chooses to adopt. Notably, the Copenhagen interpretation of
quantum mechanics provides no description of particles traveling along well
defined paths, so is at odds with the standard description of the effect. An
interpretation that does allow one to describe real particle paths is Bohmian
Mechanics (BM), also known as the de Broglie-Bohm pilot-wave theory [23, 24,
25]. According to BM, particles move under the influence of the wave function,
with a velocity equal to the quantum velocity of probability in the standard
quantum formalism. Both the particle and its pilot-wave are assumed to be real
physical objects. The Bohmian version of the Elitzur-Vaidman experiment, was
first presented by Hardy [8], who posited the following physical picture. At
the first beam splitter, the particle takes one of the two paths in the
interferometer, while its pilot-wave is split in two. One part of the pilot-
wave carries the particle along its path, while the other, so-called empty
wave, proceeds along the other path without the particle. Both components of
the pilot-wave recombine at $B_{2}$, producing the standard outcome. If a bomb
is present along the path that the particle didn’t take, it blocks the empty
wave, preventing it from interfering with the particle’s pilot-wave at
$B_{2}$, thus enabling the particle to arrive at detector $D_{2}$ with non-
zero probability. Note that according to both this physical picture and that
proposed by Elitzur and Vaidman [3], the experiment can be considered
“interaction-free” only in the sense that the particle does not interact with
the bomb: the wave form propagating along Path 1 interacts with and is altered
by the bomb without detonating it. We proceed by demonstrating that such
detection of an object by a particle that interacts with the particle only
through its associated waveform may likewise be achieved in a classical
system.

A classical pilot-wave system was discovered by Yves Couder in 2005 [26], and
consists of a millimetric droplet bouncing and self-propelling across the
surface of a vibrating liquid bath through an interaction with its own wave.
This system has provided the basis for a growing list of hydrodynamic quantum
analogs [27, 28], including single- and double-slit diffraction and single-
particle interference [29], quantization of orbital states [30, 31], the
emergence of wave-like statistics in corrals [32, 33], Friedel oscillations
[34], superradiance [35], and hydrodynamic spin lattices [36]. As the bouncing
droplets are the sources of their own pilot-wave field, the accompanying
physical picture is more closely aligned with the relativistic double-solution
pilot-wave theory proposed in the 1920s by Louis de Broglie [37] than with the
de Broglie-Bohm theory [28]. Nevertheless, the hydrodynamic pilot-wave system
has captured certain features predicted by BM, including surreal particle
trajectories [38], as arise in BM from the interference of empty waves with
the particle’s pilot-wave in an interferometer. As noted by Vaidman [39], the
phenomenon of surreal trajectories is related to that of interaction-free
measurement, in that both rely on interference effects between a particle-
carrying pilot-wave and an empty wave. Here, we build upon the experimental
setup used to realize surreal trajectories in pilot-wave hydrodynamics, in
order to demonstrate that when a particle is accompanied by a pilot-wave, an
analog of interaction-free measurement can be realized in a classical system.

Figure 1: A schematic of the Elitzur-Vaidman bomb experiment. A particle
emitted from a source $S$ passes through a beam-splitter $B_{1}$, at which
point its associated wave (a wave function [3] or a pilot-wave [8]) is split
in two. The wave is then recombined at a beam-splitter $B_{2}$, and the
particle continues toward the detectors. In the absence of a bomb, the
particle will be detected at $D_{1}$ $100\%$ of the time, while in its
presence, the particle will be detected at $D_{2}$ $25\%$ of the time. A
detection event at $D_{2}$ indicates the presence of a live bomb along a path
the particle never took. Red and blue dashed lines indicate possible paths
taken by a particle emitted from $S$ in the absence and presence of the bomb,
respectively.  Figure 2: Interaction-free measurement in pilot-wave
hydrodynamics: (a) A schematic illustration of the topography used in the
experiment. The upper half of the left barrier (orange) plays the role of a
“bomb”. (b) In a symmetric setup, the droplet enters the right or left channel
with equal probability. If the droplet goes to the left channel, it detonates
the “bomb”, and if it goes to the right, it is deflected away from the system
centerline, resulting in a “surreal” trajectory [38]. Thus, the droplet will
always be detected on the right side of the setup. Twenty such trajectories
are shown. (c) If the “bomb” is removed, the symmetry of the system is broken,
the droplet’s pilot-wave of the droplet does not interact with the bomb, and
“surreal” trajectories are suppressed. Thus, the droplet will always be
detected on the left side of the setup. The scale bar represents the Faraday
wavelength $\lambda_{F}=4.84$ mm

Our experimental system consists of a circular bath filled with a $7\pm 0.3$
mm deep layer of silicon oil with surface tension $\sigma=0.0209$ N/m,
viscosity $20$ cSt, and density $\rho=0.965\times
10^{-3}\,\,\,\text{kg}/\text{m}^{3}$. The height of the submerged topography
is $6\pm 0.2$ mm, so the oil depth in the overlaying shallow regions is
approximately $1$ mm. We vibrate the bath vertically using an electromagnetic
shaker with forcing $F(t)=\gamma\cos(2\pi ft)$, with peak vibrational
acceleration $\gamma=3.95$g and frequency $f=80$ Hz. The Faraday threshold, at
which the free surface destabilizes to a pattern of subharmonic Faraday waves,
was $\gamma_{F}=4.05$g. The electromagnetic shaker was connected to the bath
by a steel rod coupled with a linear air bearing, thus insuring spatially
uniform vibration [40]. Two accelerometers placed on opposite sides of the
bath allowed for control of the vibrational acceleration amplitude to within
$\pm 0.002$ g. We visualized the droplet trajectories and guiding pilot-wave
using a semi-reflective mirror angled at 45° relative to the bath, and a
charge-coupled device (CCD) camera mounted directly above the setup. We used a
diffuse-light lamp, positioned horizontally in front of the mirror, to
illuminate the bath, yielding images with bright regions corresponding to
horizontal regions of the bath surface, specifically extrema or saddle points.

The topographical configuration used in our experiments is depicted in figure
2(a). A walking droplet is launched towards a submerged rhombus that acts as a
beam-splitter, forcing the droplet along one of two adjoining channels with
equal probability. We consider the upper part of the left channel to be our
“bomb”, an object who’s presence we seek to detect without the droplet
interacting with it directly. When the “bomb” is present in the system (figure
2(b)), the setup is identical to that used to realize surreal trajectories
[38]. If the droplet takes the left channel, it hits the “bomb” and detonates
it. If, however, the droplet takes the right channel, its pilot-wave interacts
with its environment so as to deflect the droplet toward the right, along a
surreal trajectory. Thus, if the “bomb” is present, the droplet will either
detonate it, or be detected on the right side of the setup (see supplementary
video 1). When the “bomb” is absent (Figure 2(c)) and the droplet takes the
left path, there is nothing there to reflect it, so it continues along a
straight path to the left. If the droplet takes the right path, it is
reflected, then continues in a straight line towards the left. Thus in the
absence of the “bomb”, the droplet will always be detected on the left side of
the setup. After many realizations of the experiment in which the bomb was
present $50\%$ of the time, the experimenter has a $25\%$ chance of detecting
the droplet on the right side. Such a detection indicates that the “bomb” was
present in the left channel, even though the droplet took the right path and
so never interacted directly with it (see supplementary video 2).

Discussion: Our experiment demonstrates that if particles are accompanied by
guiding wave forms, the statistical behaviour that has led to the inference of
interaction-free measurement in quantum mechanics may be achieved in a
classical system. Notably, not all interpretations of quantum mechanics need
appeal to interaction-free measurement. In their original paper, Elitzur and
Vaidman write that according to the many-worlds interpretation [41] “since all
worlds take place in the physical universe, we cannot say that nothing has
touched the object. We get information about the object without touching it in
one world, but we pay the price of interacting with the object in the other
world” [3]. In BM, while the particle does not interact with the bomb, its
pilot-wave is blocked by it, so the measurement is not strictly interaction-
free. In the context of the Copenhagen interpretation of quantum mechanics,
before measurement, one can only talk about the particle’s wave function,
which is a global object that is determined by the configuration of the
interferometer as a whole. Thus, in the absence of a bomb, the wave function
can only collapse at detector $D_{1}$, while in its presence, the wave
function has a non-zero probability of collapsing at $D_{2}$. According to
this interpretation, the wave function is considered to be a mathematical
construction for computing probabilities rather than a physical field; thus,
one can argue that the effect is indeed interaction-free. However, a similar
argument can be applied quite generally in the Copenhagen picture, according
to which the measurement process is marked by the collapse of the wave
function, and the probability of a particular outcome is prescribed by the
configuration of the system as a whole. Thus, according to the Copenhagen
interpretation, a wide range of phenomena can be considered interaction-free,
rendering the notion superfluous.

It is important to note the differences between interaction-free measurement
in quantum mechanics, and in the classical analog presented here. First, the
quantum wave function is a global object that is determined by the entire
configuration of the experimental setup. As a result, interaction-free
measurement can be realized (at least in theory) with an interferometer that
has arms of arbitrary length, provided the length of the two arms differ by
$n\lambda$. In our system, the pilot-wave is a local object dressing the drop
that is dissipated under the effect of viscosity. Consequently, changing the
configuration of the setup can suppress the surreal trajectories and nullify
the effect. Second, in the BM version of the Elitzur-Vaidman experiment, the
bomb blocks the empty wave, thus eliminating interference. In our setup, the
situation is reversed: the presence of the “bomb” promotes interference of the
pilot-wave, resulting in a surreal trajectory. This configurational difference
does not alter the key statistical behavior common to the two systems.

All current attempts to rationalize interaction-free measurement rely on a
physical picture where localized wave-particle objects (e.g., traveling wave
packets, or Bohmian particles with their pilot-wave) travel along the arms of
the interferometer. Our study demonstrates that such localized wave-particle
descriptions of interaction-free measurement can also be achieved in a
classical system. We note that the validity of such descriptions can be tested
tested directly by realizing the Elitzur-Vaidman bomb experiment while
increasing the length of one of the interferometer arms by an amount
$n\lambda$, where $n$ is an integer and $\lambda$ the wavelength of the
photon. If one represents the photon as a wave packet that is split in two at
$B_{1}$, then for sufficiently large $n$, the two wave packets will not
recombine at $B_{2}$ since they travel paths of different length at the same
speed. Likewise, in the Bohmian picture suggested by Hardy [8], the
interference of the particle’s pilot wave with an empty wave at $B_{2}$ would
be eliminated. Thus, the photon would have a non-zero probability of being
detected at $D_{2}$ even in the absence of a bomb. If the effect does not
persist for arbitrary large $n$, it would validate these localized wave-
particle descriptions and, when considered in light of our results, suggest
that interaction-free measurement is a misnomer. If the effect does persist
for arbitrary large $n$ (as one might expect), this experiment would
invalidate the localized wave-packet description of the problem. One would
then be impelled to adopt a global view similar to that inherent in the
Copenhagen interpretation, specifically, that particles are guided by the non-
local quantum potential (as in the second-order formulation of Bohm’s theory
[23, 24, 25]), as is prescribed by the wave function. Since the wave function
is a global object, so too is the quantum potential; thus, the path taken by a
Bohmian particle is determined by the entire configuration of the
interferometer.

As was the case in the hydrodynamic analog of surreal trajectories, the
hydrodynamic pilot-wave system captures a feature of quantum mechanics
correctly predicted by Bohmian mechanics, without having to invoke the quantum
potential. The role of the quantum potential in guiding the particle in
Bohmian mechanics is played by the droplet’s guiding wave, which is generated
by the droplet but interacts with the system geometry in such a way as to
produce the surreal trajectories. In neither Bohmian mechanics nor our system
is the measurement really interaction free. In Bohmian mechanics, the quantum
potential changes form according to the presence or absence of the bomb; in
our system, the local, particle generated pilot-wave does likewise.

Finally, given the numerous applications proposed for interaction-free
measurement [9, 42, 43, 44], it is noteworthy that none of these requires the
non-local features of the wave function for their implementation, only its
interference properties. It is thus worth considering whether some of these
applications might be implemented in the hydrodynamic pilot-wave system. A
classical macroscopic implementation of these ideas could potentially allow
one to bypass the difficulty of decoherence inherent in all quantum systems,
and so conceivably serve as a physical platform for quantum-inspired classical
computing.

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## Acknowledgments

### Funding

The authors gratefully acknowledge the financial support of the National
Science Foundation grant CMMI-2154151.

### Competing interests

The authors declare no competing interests.

### Data and materials availability

All data are available in the main text or the supplementary materials.


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# Some Neumann-Bessel series
and the Laplacian on polygons

Luca Guido Molinari  L. G. Molinari: Physics Department Aldo Pontremoli,
Università degli Studi di Milano and I.N.F.N. sez. Milano, Via Celoria 16,
20133 Milano, Italy<EMAIL_ADDRESS>
(Date: 9 nov 2020)

###### Abstract.

Several sums of Neumann series with Bessel and trigonometric functions are
evaluated, as finite sums of trigonometric functions. They arise from a
generalization of the Neumann expansion of the eigenstates of the Laplacian in
regular polygons.

###### Key words and phrases:

Bessel function, Neumann series, Laplace equation in polygon

###### 2010 Mathematics Subject Classification:

Primary 33C10, Secondary 35J05

## Introduction

The ground state of the Laplace equation in a regular polygon with Dirichlet
boundary conditions at the $n$ sides, has a natural expression as a Neumann
series of Bessel and trigonometric functions,

$\psi_{n}(r,\theta)=J_{0}(\lambda_{n}r)+2{\sum}_{k=1}^{\infty}h_{k,n}J_{kn}(\lambda_{n}r)\cos(kn\theta),$

with coefficients $h_{k,n}$ to be found and eigenvalue $-\lambda_{n}^{2}$ that
scales with the area. For the equilateral triangle and the square, the
solutions are known as sums of few trigonometric functions of the coordinates
$x=r\cos\theta$ and $y=r\sin\theta$. Such solutions have a corresponding
Neumann expression [7]. For the square of area $\pi$:

(1) $\displaystyle
J_{0}(\sqrt{2\pi}r)+2{\sum}_{k=1}^{\infty}J_{4k}(\sqrt{2\pi}r)\cos(4k\theta)=\tfrac{1}{2}\cos(x\sqrt{2\pi})+\tfrac{1}{2}\cos(y\sqrt{2\pi})$

The triangle requires some work to establish the equivalence:

(2) $\displaystyle
J_{0}(\lambda_{3}r)+2{\sum}_{k=1}^{\infty}\frac{\cos(k\pi/2-\pi/6)}{\cos(\pi/6)}J_{3k}(\lambda_{3}r)\cos(3k\theta)=\tfrac{2}{3\sqrt{3}}\sin(\tfrac{4\pi}{3R_{3}}x+\tfrac{2\pi}{3})$
$\displaystyle-\tfrac{2}{3\sqrt{3}}[\sin[\tfrac{2\pi}{3R_{3}}(x+y\sqrt{3})-\tfrac{2\pi}{3}]-\tfrac{2}{3\sqrt{3}}\sin[\tfrac{2\pi}{3R_{3}}(x-y\sqrt{3})-\tfrac{2\pi}{3}]$

where, for area $\pi$, $\lambda^{2}_{3}=\frac{4\pi}{\sqrt{3}}$ and
$R_{3}=\tfrac{2}{3}\sqrt{\pi\sqrt{3}}$. In [7] I also obtained a sum that
generalizes the integrable cases $n=3,4$:

(3) $\displaystyle f_{n}(x,y)=$ $\displaystyle
J_{0}(r)+2{\sum}_{k=1}^{\infty}\frac{\cos[nk\tfrac{3\pi}{2}-\tfrac{\pi}{2n}]}{\cos(\tfrac{\pi}{2n})}J_{nk}(r)\cos(nk\theta)$
$\displaystyle=$
$\displaystyle\frac{1}{n}{\sum}_{\ell=0}^{n-1}\frac{\cos[r\cos(\theta+\tfrac{2\pi}{n}\ell)+\tfrac{\pi}{2n}]}{\cos(\tfrac{\pi}{2n})}$
$\displaystyle=$
$\displaystyle\frac{1}{n}{\sum}_{\ell=0}^{n-1}\frac{\cos[x\cos(\tfrac{2\pi}{n}\ell)-y\sin(\tfrac{2\pi}{n}\ell)+\tfrac{\pi}{2n}]}{\cos(\tfrac{\pi}{2n})}$

For $n\to\infty$ the Riemann sum in the second line is
$\int_{0}^{2\pi}\frac{dt}{2\pi}\cos(r\cos t)=J_{0}(r)$; for $n=2$ it is
$f_{2}(x,y)=\cos x$. For $n=6$:

(4) $\displaystyle
f_{6}(x,y)=J_{0}(r)+2\sum_{k=1}^{\infty}(-1)^{k}J_{6k}(r)\cos(6k\theta)=\tfrac{1}{3}\cos
x+\tfrac{2}{3}\cos(\tfrac{1}{2}x)\cos(\tfrac{\sqrt{3}}{2}y)$

The functions $f_{n}$ are eigenfunctions of the Laplace operator with
eigenvalue $-1$ but, for $n>4$, they no longer vanish on the boundary of a
$n$-polygon.

Figure 1.  Left: contour plots of $f_{6}$ and $f_{7}$ in eq.(3). Right: the
separatrix line $f_{6}(x,y)=-1/3$ (Kagomé lattice, the eq. can be written as
$0=\cos(x/2)[\cos(x/2)+\cos(\sqrt{3}y/2)]$. Sites are the doule zeros.), and
$f_{7}(x,y)=-1.9633...$ (Mathematica)

I only remark that the level curves $f_{n}(x,y)=C_{n}$ are closed around the
origin (where $f_{n}=1$) up to a separatrix with $n$ self-intersections, with
values $C_{5}=-0.334909$, $C_{6}=-1/3$, $C_{7}=0.19633$, etc. The lines are
shown in Fig.1.

The Laplacian in polygons has a long history. The ground state beyond the
square cannot be finite sums of trigonometric functions, and has been
investigated analytically and numerically in $1/n$ expansion (see for example
[7, 4, 3]).

In this paper I generalize the identity (3), and obtain a number of new
formulas for Neumann series whose sums contain a finite number of terms. For
certain values of the parameters, they are identities that are found in the
tables by Gradshteyn and Ryzhik [2], Prudnikov, Brychkov and Marichev [8], a
recent paper by Al-Jarrah, Dempsey and Glasser [1], and two old papers by
Takizawa and Kobayasi [9, 5]. In the last ones, the Neumann series appear as
correlation functions for the heat flow in coupled harmonic oscillators.

## The summation formula

The source equation of various sums in this paper is:

(5)
$\displaystyle\boxed{\sum_{k=-\infty}^{+\infty}J_{kn+p}(z)e^{ikny}=\frac{1}{n}\sum_{\ell=0}^{n-1}e^{iz\sin(y+\tfrac{2\pi}{n}\ell)-ip(y+\tfrac{2\pi}{n}\ell)}}$

For $y=0$ and even $n$ it is eq.1 in [9]. Sums of this sort are tabulated only
for $n=1,2$ in [8].

###### Proof.

The result follows from the Fourier integral of a Bessel function of integer
order. For fixed $z\in\mathbb{C}$,
$\sum_{k=-\infty}^{\infty}e^{ik\theta}J_{kn+p}(z)$ is uniformly convergent in
$\theta$ by the bound $|J_{\pm m}(z)|\leq C|z/2|^{m}/m!$ (Nielsen, see §3.13
in [10]).

$\displaystyle\sum_{k=-\infty}^{\infty}e^{ikny}J_{kn+p}(z)=\sum_{k=-\infty}^{\infty}e^{ikny}\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{iz\sin\theta-i(kn+p)\theta}$
$\displaystyle=\sum_{k=-\infty}^{\infty}e^{ikny}\sum_{j=0}^{n-1}\int_{\frac{2\pi}{n}j}^{\frac{2\pi}{n}(j+1)}\frac{d\theta}{2\pi}e^{iz\sin\theta-
ip\theta}e^{-ikn\theta}$

The sums are exchanged:
$=\sum_{j=0}^{n-1}\sum_{k=-\infty}^{\infty}e^{ikny}\int_{0}^{\frac{2\pi}{n}}\frac{d\theta}{2\pi}e^{iz\sin(\theta+\frac{2\pi}{n}j)-ip(\theta+\frac{2\pi}{n}j)}e^{-ikn\theta}$.
The functions $\sqrt{n/2\pi}\,e^{ikny}$ are a complete orthonormal basis in
L${}^{2}(0,2\pi/n)$. The infinite sum is the Fourier representation of
$\exp[iz\sin(y+\frac{2\pi}{n}j)-ip(y+\frac{2\pi}{n}j)]$. ∎

### 1

The case $p=0$ and $y=\frac{\pi}{2}+\alpha$ is an extension with angle
$\alpha$ of the equations 19 and 20 in [1], where $\alpha=0$. With
$J_{-m}(z)=(-)^{m}J_{m}(z)$:

(6) $\displaystyle
J_{0}(z)+2\sum_{k=1}^{+\infty}e^{ikn\frac{\pi}{2}}J_{kn}(z)\cos(kn\alpha)=\frac{1}{n}\sum_{\ell=0}^{n-1}e^{iz\cos(\alpha+\tfrac{2\pi}{n}\ell)}$

For $n=1$, separation of even and odd parity parts in $z$ gives the Jacobi
expansions (eqs. 5.7.10.4 and 5 in [8]):

(7) $\displaystyle
J_{0}(z)+2\sum_{k=1}^{\infty}(-)^{k}J_{2k}(z)\cos(2k\alpha)=\cos(z\cos\alpha)$
(8)
$\displaystyle\sum_{k=0}^{\infty}(-)^{k}J_{2k+1}(z)\cos[(2k+1)\alpha]=\tfrac{1}{2}\sin(z\cos\alpha)$

If $\alpha$ is replaced by $\alpha+\pi/2$ they are eqs. 8.514.5 and 6 in [2]
and 10.4, 10.5 in [6]:

(9) $\displaystyle
J_{0}(z)+2\sum_{k=1}^{\infty}J_{2k}(z)\cos(2k\alpha)=\cos(z\sin\alpha)$ (10)
$\displaystyle\sum_{k=0}^{\infty}J_{2k+1}(z)\sin[(2k+1)\alpha]=\tfrac{1}{2}\sin(z\sin\alpha)$

### 1.1

For $n$ replaced by $2n$, eq.(6) is:

(11) $\displaystyle
J_{0}(z)+2\sum_{k=1}^{\infty}(-)^{kn}J_{2kn}(z)\cos(2kn\alpha)=\frac{1}{2n}\sum_{\ell=0}^{2n-1}\cos[z\cos(\alpha+\tfrac{\pi}{n}\ell)]$

Since terms $\ell$ and $n+\ell$ are the same, the sum is replaced by
$2\sum_{\ell=0}^{n-1}$. The value $y=\frac{\pi}{2}$ yields eq.(23) in [1].
For $n=1$ the derivative of (11) in $\alpha=\frac{\pi}{4}$ is:

(12) $\displaystyle
2J_{2}(z)-6J_{6}(z)+10J_{10}(z)-14J_{14}(z)+...=z\tfrac{\sqrt{2}}{4}\sin(z\tfrac{\sqrt{2}}{2})$

For $n=2$ eq.(11) gives

(13) $\displaystyle
J_{0}(z)+2\sum_{k=1}^{\infty}J_{4k}(z)\cos(4k\alpha)=\frac{1}{2}[\cos(z\sin\alpha)+\cos(z\cos\alpha)]$

The values $\alpha=0,\frac{\pi}{4}$ give eqs. 5.7.1.19. Case $n=3$, $\alpha=0$
gives eq. 5.7.1.21 in [8].
The derivative of (13) with $n=4$ is:

(14)
$\displaystyle\sum_{k=1}^{\infty}kJ_{4k}(z)\sin(4k\alpha)=\frac{z}{16}[\sin(z\sin\alpha)\cos\alpha-\sin(z\cos\alpha)\sin\alpha]$

The expansion in small $\alpha$ gives:

(15) $\displaystyle\sum_{k=1}^{\infty}k^{2}J_{4k}(z)=\frac{z}{64}(z-\sin z)$

### 1.2

For $n$ replaced by $2n+1$, eq.(6) is:

$\displaystyle
J_{0}(z)+2\sum_{k=1}^{\infty}e^{i(2n+1)k\frac{\pi}{2}}J_{(2n+1)k}(z)\cos[(2n+1)k\alpha]=\frac{1}{2n+1}\sum_{\ell=0}^{2n}e^{iz\cos(\alpha+\tfrac{2\pi}{2n+1}\ell)}$

The even-parity and odd-parity parts in the exchange $z\to-z$ are:

$\displaystyle
J_{0}(z)+2\sum_{k=1}^{\infty}(-)^{k}J_{(4n+2)k}(z)\cos[(4n+2)k\alpha]=\frac{1}{2n+1}\sum_{\ell=0}^{2n}\cos[z\cos(\alpha+\tfrac{2\pi}{2n+1}\ell)]$
$\displaystyle
2\sum_{k=0}^{\infty}(-)^{n+k}J_{(2n+1)(2k+1)}(z)\cos[(2n+1)(2k+1)\alpha]=\frac{1}{2n+1}\sum_{\ell=0}^{2n}\sin[z\cos(\alpha+\tfrac{2\pi}{2n+1}\ell)]$

Examples of the second equation are

(16)
$\displaystyle\sum_{k=0}^{\infty}(-)^{k}J_{6k+3}(z)\cos[(6k+3)\alpha]=-\frac{1}{6}\sum_{\ell=0}^{2}\sin[z\cos(\alpha+\tfrac{2\pi}{3}\ell)]$
(17)
$\displaystyle\sum_{k=0}^{\infty}(-)^{k}J_{10k+5}(z)\cos[(10k+5)\alpha]=\frac{1}{10}\sum_{\ell=0}^{4}\sin[z\cos(\alpha+\tfrac{2\pi}{5}\ell)]$

The first equation with $\alpha=\pi$ is eq.22 in [1].
Both sums are eigenfunctions of the Laplacian with eigenvalue $\lambda=-1$
(see Fig. 2). The sum (17), with $z=r$, $x=r\cos\alpha$ and $y=r\sin\alpha$,
is

(18) $\displaystyle f(x,y)=\sin
x-2\sin(x\cos\tfrac{\pi}{5})\cos(y\sin\tfrac{\pi}{5})+2\sin(x\cos\tfrac{2\pi}{5})\cos(y\sin\tfrac{2\pi}{5})$

Figure 2. Contour plots of the sums (16) and (18). The first is the ground
state of the equilateral triangle (no nodal lines) and an excited state of the
hexagon. The second function is zero on a line close to the first zero of
$J_{5}(r)$.

### 1.3

In (5) with $p=0$, multiply by $\exp(i\beta)$ ($\beta$ real) and take the real
part. The left hand side becomes:

$\displaystyle J_{0}(x)\cos\beta+{\sum}_{k=1}^{\infty}J_{kn}(x)\;{\rm
Re}[e^{i\beta}(e^{ikny}+e^{-ikn(y+\pi)})]$
$\displaystyle=J_{0}(x)\cos\beta+2{\sum}_{k=1}^{\infty}\cos(\beta-
kn\tfrac{\pi}{2})J_{kn}(x)\cos[kn(y+\tfrac{\pi}{2})]$

The identity (3) is obtained, when $\beta=\tfrac{\pi}{2n}$ and
$y+\tfrac{\pi}{2}=\theta+\pi$.

### 2

Parseval’s identity is applied to (5):

$\displaystyle\sum_{k\in\mathbb{Z}}J^{2}_{kn+p}(x)$
$\displaystyle=\frac{1}{n^{2}}\sum_{k\ell}e^{ip\tfrac{2\pi}{n}(k-\ell)}\int_{0}^{2\pi}\frac{dy}{2\pi}e^{ix\sin(y-\tfrac{\pi}{n}(k-\ell))-ix\sin(y+\tfrac{\pi}{n}(k-\ell))}$
$\displaystyle=\frac{1}{n^{2}}\sum_{k\ell}e^{ip\tfrac{2\pi}{n}(k-\ell)}\int_{0}^{2\pi}\frac{dy}{2\pi}e^{-i2x\cos
y\sin(\tfrac{\pi}{n}(k-\ell))}$
$\displaystyle=\frac{1}{n^{2}}\sum_{k\ell}e^{ip\tfrac{2\pi}{n}(k-\ell)}J_{0}[2x\sin(\tfrac{\pi}{n}(k-\ell))]$
$\displaystyle=\frac{1}{n}+\frac{2}{n^{2}}\sum_{k=1}^{n-1}k\cos(\tfrac{2\pi}{n}kp)J_{0}(2x\sin\tfrac{\pi
k}{n})$

The last sum is unchanged if $k$ is replaced by $n-k$:

(19)
$\displaystyle\sum_{k\in\mathbb{Z}}J^{2}_{kn+p}(x)=\frac{1}{n}+\frac{1}{n}\sum_{k=1}^{n-1}\cos(\tfrac{2\pi}{n}kp)J_{0}(2x\sin\tfrac{\pi
k}{n})$

The left-hand side is
$J_{p}(x)^{2}+\sum_{k=1}^{\infty}J^{2}_{kn+p}(x)+J^{2}_{kn-p}(z)$.
For the special case $p=0$ and $n\to 2n$ in (19), the sum is amenable to eq.29
in [1]:

(20) $\displaystyle
J_{0}^{2}(x)+2\sum_{k=1}^{\infty}J_{2kn}^{2}(x)=\frac{1}{2n}+\frac{1}{2n}J_{0}(2x)+\frac{1}{n}\sum_{k=1}^{n-1}J_{0}(2x\cos\tfrac{\pi}{2n}k)$

### 2.1

If $n\to 2n$ and $p=n$ in (19), with simple steps one obtains:

(21)
$\displaystyle\sum_{k=0}^{\infty}J^{2}_{(2k+1)n}(x)=\frac{1}{4n}+\frac{(-)^{n}}{4n}J_{0}(2x)+\frac{1}{2n}\sum_{\ell=1}^{n-1}(-1)^{\ell}J_{0}(2x\sin\tfrac{\pi\ell}{2n})$

### 3

In eq.(5) the variable $y$ is shifted to $y+2t$. The equation is multiplied by
$e^{iz^{\prime}\sin y-iqy}$ and integrated in $y$:

$\displaystyle\sum_{k=-\infty}^{+\infty}J_{p+kn}(z)$ $\displaystyle
J_{q-kn}(z^{\prime})e^{i(kn+p)2t}$ (22) $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{\ell=0}^{n-1}e^{-ip\tfrac{2\pi}{n}\ell}\int_{0}^{2\pi}\frac{dy}{2\pi}e^{iz\sin(y+2t+\tfrac{2\pi}{n}\ell)+iz^{\prime}\sin
y-i(p+q)y}$

In the integral, the shift $y$ to $y-t-\frac{\pi}{n}\ell$ changes the exponent
to

$i(z+z^{\prime})\sin y\cos(t+\tfrac{\pi}{n}\ell)+i(z-z^{\prime})\cos
y\sin(t+\tfrac{\pi}{n}\ell)-i(p+q)(y-t-\tfrac{\pi}{n}\ell)$

### 3.1

With $z=z^{\prime}$ we obtain eq.1 in [5]:

(23)
$\displaystyle\sum_{k=-\infty}^{+\infty}J_{p+kn}(z)J_{q-kn}(z)e^{2iknt}=\frac{1}{n}\sum_{\ell=0}^{n-1}e^{-i(p-q)(t+\tfrac{\pi}{n}\ell)}J_{p+q}[2z\cos(t+\tfrac{\pi}{n}\ell)]$

For $n=1,2$ it is (with a shift of the index $k$ in the first identity and
renaming of parameter):

(24)
$\displaystyle\sum_{k=-\infty}^{+\infty}J_{k}(z)J_{p-k}(z)e^{2ikt}=e^{ipt}J_{p}(2z\cos
t)$ (25)
$\displaystyle\sum_{k=-\infty}^{+\infty}J_{p+2k}(z)J_{q-2k}(z)e^{4ikt}=\tfrac{1}{2}e^{-i(p-q)t}[J_{p+q}(2z\cos
t)+i^{p-q}J_{p+q}(2z\sin t)]$

The first one is eq.8.530 [2]. The second one, for
$t=\frac{\pi}{4},\frac{\pi}{2}$, becomes:

(26)
$\displaystyle\sum_{k=-\infty}^{+\infty}(-)^{k}J_{p+2k}(z)J_{q-2k}(z)=J_{p+q}(z\sqrt{2})\cos[(p-q)\tfrac{\pi}{4}]$
(27)
$\displaystyle\sum_{k=-\infty}^{+\infty}J_{p+2k}(z)J_{q-2k}(z)=\tfrac{1}{2}J_{p+q}(2z)$

For $p=q$ the first one is eq. 5.7.11.25 [8].

### 3.2

Eq.(22) with $p=q=0$ and $t=0$ is:

$\displaystyle
J_{0}(z)J_{0}(z^{\prime})+2\sum_{k=1}^{\infty}(-)^{kn}J_{kn}(z)J_{kn}(z^{\prime})=\frac{1}{n}\sum_{\ell=0}^{n-1}\int_{0}^{2\pi}\frac{dy}{2\pi}e^{iz\sin(y+\tfrac{2\pi}{n}\ell)+iz^{\prime}\sin
y}$

For $n=1,2$ they are eqs. 5.7.11.1 and 5.7.11.18 in [8] and, for
$z=z^{\prime}$: eqs. 31, 32 in [1]. A new example is:

(28)
$\displaystyle\sum_{k=1}^{\infty}J_{4k}(x)J_{4k}(y)=\tfrac{1}{8}[J_{0}(x+y)+J_{0}(x-y)-4J_{0}(x)J_{0}(y)+2J_{0}(\sqrt{x^{2}+y^{2}})]$

### 4

Multiplication of (5) by $\exp(-ay)$ ($a>0$) with $p=0$ and $n=1$, and
integration on $\mathbb{R}^{+}$ give:

$\displaystyle\frac{1}{a}J_{0}(z)+\sum_{k=1}^{\infty}J_{2k}(z)\frac{2a}{a^{2}+4k^{2}}+\sum_{k=0}^{\infty}J_{2k+1}(z)\frac{2i(2k+1)}{a^{2}+(2k+1)^{2}}=\int_{0}^{\infty}dy\,e^{iz\sin
y-ay}$

The integral in the right-hand side is done by series expansion, with
eqs.3.895.1 and 3.895.4 [2]. The even and odd terms are:

(29)
$\displaystyle\frac{1}{a}J_{0}(z)+\sum_{k=1}^{\infty}J_{2k}(z)\frac{2a}{a^{2}+4k^{2}}=\sum_{k=0}^{\infty}(-)^{k}\frac{z^{2k}}{a(a^{2}+4)...(a^{2}+4k^{2})}$
(30)
$\displaystyle\sum_{k=0}^{\infty}J_{2k+1}(z)\frac{2(2k+1)}{a^{2}+(2k+1)^{2}}=\sum_{k=0}^{\infty}(-)^{k}\frac{z^{2k+1}}{(a^{2}+1)(a^{2}+9)...(a^{2}+(2k+1)^{2})}$

More and more identities can be obtained by derivation, or integration with
functions. Here I limited myself to simple and, hopefully, useful examples.

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