# The Circular Polarization Reverse in Magnetar Bursts

Tomoki Wada Jiro Shimoda  Institute for Cosmic Ray Research, The University of
Tokyo, Kashiwa, Chiba, Japan

###### Abstract

Magnetars, whose magnetic fields are strongest in the neutron stars, show
diverse bursting activities. The shape of the magnetar deviates from an
exactly pure spherical shape because of its strong magnetic field, and it can
be a triaxial body. We study the free precession of the triaxial magnetar
focusing on the unstable situation, in which the main rotation axis turns
upside-down. This flip is the so-called Dzhanibekov effect. We find that
during the flip, the Euler force can suddenly disturb the force balance on the
magnetar surface, which may result in a crack at the surface. We suggest that
the crack can trigger bursting activities of the magnetar, such as X-ray short
bursts and flares with sufficiently large baryon fractions. In the burst with
a large baryon fraction, right- or left-handed circularly polarized photons
are selectively scattered, depending on the magnetic field direction. The
association between the flip and the X-ray bursts will be confirmed by the
observations of X-ray circular polarization. We also discuss the association
of fast radio bursts, ultrahigh-energy cosmic rays, and gravitational waves.

Magnetars — X-ray bursts —X-ray transient sources

##  1 Introduction

Magnetars are neutron stars whose photospheric luminosities are higher than
their spin-down luminosities (Duncan & Thompson, 1992). Almost all of the
magnetars have strong magnetic field $\sim 10^{14}$–$10^{15}\,{\rm G}$ and
exhibit energetic short bursts, outbursts, and flares (Kaspi & Beloborodov,
2017; Enoto et al., 2019, for reviews). These bursts are thought to be
triggered by a release of its magnetic energy that would occur with some
activity in its crust, such as starquakes (Thompson & Duncan, 1995, 1996). The
magnetars are also considered to be the origin of at least some fast radio
bursts (FRBs) because FRB 20200428A is detected from a Galactic magnetar, SGR
1935+2154 (Bochenek et al., 2020; CHIME/FRB Collaboration et al., 2020a;
Mereghetti et al., 2020; Li et al., 2021; Ridnaia et al., 2021; Tavani et al.,
2021).

The shape of the magnetars can deviate from a pure sphere. The strong Lorentz
force, the elastic force, the near-zone radiative field, or the multipolar
magnetic field can cause the magnetar shape to deviate from an exactly pure
spherical shape. If the magnetar composes a binary, the accretion along the
magnetic field from the companion also causes the deviations (Fujisawa et al.,
2022). Her X-1 is an X-ray pulsar, which is an accreting neutron star around a
main-sequence star, HZ Her. In addition to the spin period of the neutron
star, $1.24\,{\rm s}$, and the orbital period, $1.7\,{\rm days}$, a long-term
35-day X-ray flux modulation is observed (Tananbaum et al., 1972). The 35-day
modulation can be explained by a free precession of a triaxial body whose
moments of inertia are all different (Shakura et al., 1998; Kolesnikov et al.,
2022; Heyl et al., 2023). The ratio of the spin period to the precession
period suggests that the ellipticity of the Her X-1 is of order $10^{-6}$
(Kolesnikov et al., 2022). The continuous gravitational wave observation set
an upper limit on the ellipticity of neutron stars (Abbott et al., 2022). They
report the upper limits of the ellipticity, $\sim 10^{-6}$ at the frequency of
$\sim 10^{3}\,{\rm Hz}$, and $\gtrsim 1$ at $\lesssim 10\,{\rm Hz}$. Thus, the
shape of a magnetar can be triaxial.

The rotation around the second principal axis of a triaxial body is not always
stable (e.g., Ashbaugh et al., 1991; Mardešić et al., 2020). If the rotation
around the second principal axis is dominant initially, a reversal of the
orientation of the second principal axis (hereafter referred to as a flip)
occurs during a very short part of the precession period. This is the so-
called Dzhanibekov effect (e.g., Shakura et al., 1998, for neutron stars) and
is different from the precession explaining the superperiodicity of the Her
X-1 (Kolesnikov et al., 2022; Heyl et al., 2023). The connection between the
flip and the magnetar bursts has not been discussed.

In this paper, we study the connection between the magnetar bursts and the
flips of the triaxial magnetars. In Sectioin 2, we review the equation of
motion of a triaxial body and show that the triaxial body can flip (so-called
Dzhanibekov effect). In Section 3.1, we show that the Euler force on the
surface of the magnetar is stronger during the flip than in the other phase,
and suggest that the force can trigger some magnetar bursts. In Section 3.2,
we consider the X-ray bursts associated with the flip, and suggest that the
flip of the magnetar can be identified by the circular polarization of X-ray
short bursts. In Section 3.3, we discuss the possibility of the FRBs
associated with the flip. In Section 3.4, we discuss multi-messenger
astrophysical events associated with the flip. Section 4 is devoted to the
conclusion. We use $Q_{,x}=Q/10^{x}$ in cgs units.

##  2 Triaxial Free Precession and Flip of a Magnetar

Figure 1:  Definition of Euler angles (top panel). The axes $X,\,Y,$ and $Z$
denote an inertial frame. The axes $x_{1},\,x_{2}$ and $x_{3}$ denote the body
axes. Typical values of the Euler angles that appear in this paper (bottom
panels). The configuration in bottom left panel corresponds to
$\cos\theta\simeq 0$ and $\cos\psi\simeq 1$, and this condition is satisfied
for $0\lesssim t\lesssim 10^{6}P$ and $5\times 10^{6}P\lesssim t\lesssim
6\times 10^{6}P$ in Figure 2. The configuration in the bottom right panel
corresponds to $\cos\theta\simeq 0$ and $\cos\psi\simeq-1$, and this condition
is satisfied for $2\times 10^{6}P\lesssim t\lesssim 4\times 10^{6}P$ in Figure
2. During these configurations, the magnetar mainly rotates around the $x_{2}$
axis (i.e., $\Omega_{2}\gg\Omega_{1},\Omega_{3}$) while the sign of
$\Omega_{2}$ is different.

We consider the motion of a triaxial magnetar (Landau & Lifshitz, 1969). We
assume that the magnetar is triaxial and the spin axis does not coincide with
any principal axes of inertia initially. We also assume that the magnetar
rotates as a rigid body with a constant moment of inertia. The three
eigenvalues of the moment of inertia tensor, $I_{i}\,(i=1,2,3)$, are defined
to be $I_{3}>I_{2}>I_{1}$. The energy and angular momentum conservation laws
can be written as follows.

$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{3}I_{i}\Omega_{i}{}^{2},$ (1)
$\displaystyle{\mathcal{M}}^{2}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}I_{i}{}^{2}\Omega_{i}{}^{2},$ (2)

where $E$, and ${\mathcal{M}}$ are the kinetic energy of the rotation and the
total angular momentum of the magnetar, and $\Omega_{i}~{}({\rm i=1,2,3})$ are
the angular velocities around each principal axis of inertia, $x_{1}$,
$x_{2}$, and $x_{3}$ (see top panel of Figure 1). If the magnetar, whose mass
is $M_{\rm NS}$ and radius is $R_{\rm NS}$, is a uniform sphere, the
eigenvalues of the momentum of the inertia tensor is
$I_{1}=I_{2}=I_{3}=(2/5)M_{\rm NS}R_{\rm NS}{}^{2}$. In this paper, we vary
$I_{1}$ and $I_{3}$ slightly from $(2/5)M_{\rm NS}R_{\rm NS}{}^{2}$ and set
$I_{3}>I_{2}>I_{1}$. The differences of $I_{1},\,I_{2}$, and $I_{3}$ will be
discussed below.

The top panel of Figure 1 shows the definitions of the Euler angles,
$\theta,\,\psi$, and $\phi$. The inertial frame is denoted by $(X,Y,Z)$. The
rotation of the body around the $Z$ axis and the $x_{3}$ axis is denoted by
the angle $\phi$ and $\psi$. $\theta$ is the angle between the $Z$ axis and
the $x_{3}$ axis. For $\theta=0,\,\psi=0,$ and $\phi=0$, the $x_{1}$, $x_{2}$,
and $x_{3}$ axes respectively coincide with the $X,\,Y,$ and $Z$ axes.

The equations of motion (Euler’s equations for free rotation) are

$\displaystyle I_{1}\dot{\Omega}_{1}$ $\displaystyle=$
$\displaystyle(I_{2}-I_{3})\Omega_{2}\Omega_{3},$ (3) $\displaystyle
I_{2}\dot{\Omega}_{2}$ $\displaystyle=$
$\displaystyle(I_{3}-I_{1})\Omega_{3}\Omega_{1},$ (4) $\displaystyle
I_{3}\dot{\Omega}_{3}$ $\displaystyle=$
$\displaystyle(I_{1}-I_{2})\Omega_{1}\Omega_{2}.$ (5)

For simplicity, we assume that $(I_{3}-I_{2})/I_{2}=(I_{2}-I_{1})/I_{2}$ and
set $I_{2}=(2/5)M_{\rm NS}R_{\rm
NS}^{2},\,I_{1}=I_{2}(1-\delta),\,I_{3}=I_{2}(1+\delta)$, where $\delta$ is a
dimensionless parameter describing the non-sphericity. $\delta$ is much
smaller than unity and typically $\delta\sim 10^{-6}$– $10^{-11}$ (Kolesnikov
et al., 2022 for the observational values and Zanazzi & Lai, 2020 for a
summary of the theoretical values) while these values depend on the origin of
the non-sphericity. There are some possible origins of the non-sphericity, the
internal magnetic field of the magnetar, the elasticity of the magnetar
surface, the magnetic field exterior to the magnetar, the effect of
magnetosphere plasma, and so on (see Zanazzi & Lai, 2020).

In this paper, we consider the case that $\Omega_{2}\gg\Omega_{1},\Omega_{3}$
at the initial condition. Then, the equation of motion for $\Omega_{1}$ is
rewritten as

$\displaystyle\ddot{\Omega}_{1}\simeq\frac{(I_{3}-I_{2})(I_{2}-I_{1})}{I_{1}I_{3}}\Omega_{2}^{2}\Omega_{1}\simeq\delta^{2}\Omega_{2}^{2}\Omega_{1},$
(6)

and the rotation around the $x_{1}$ axis is unstable. In a similar way, we can
find that the rotation around the $x_{3}$ axis is also unstable under the same
initial condition. The timescale of this instability is

$\displaystyle T_{\rm
ins}\simeq\sqrt{\frac{I_{1}I_{3}}{(I_{3}-I_{2})(I_{2}-I_{1})\Omega_{2}^{2}}}\sim\delta^{-1}\frac{P}{2\pi},$
(7)

where $P$ is the spin period and we used $\Omega_{2}\sim(2\pi)/P$. According
to the detailed analysis in Appendix A, $\Omega_{i}\,(i=1,2,3)$ are periodic
functions of time, $t$. The Euler angles $\theta$ and $\psi$ are also periodic
function of time and the period, $T$, is

$\displaystyle T$ $\displaystyle\sim$ $\displaystyle
4K\delta^{-1}\Omega_{2,0}{}^{-1}\sim 0.2\,{\rm
yr}\,K_{1}\delta_{-6}{}^{-1}P_{0},$ (8)

where $K$ is the complete elliptic integral of the first kind given in
Appendix A.

Figure 2:  Time evolution of the Euler angles, $\theta,\psi$, during one
period $T$. The $x_{2}$ axis flips ($\cos\psi$ suddenly changes from $1$ to
$-1$, or from $-1$ to $1$) around $t\sim 1.6\times 10^{6}P$ and $t\sim
4.7\times 10^{6}P$. The configurations before and after the flip are shown in
the bottom panels of Figure 1.

Figure 2 shows the time evolution of the Euler angles (see the top panel of
Figure 1 for the definition of the Euler angles, $\phi$, $\theta$, and
$\psi$). In this figure, we adopt $\delta=10^{-6}$ and set the initial angular
velocities under fixed $E$ and ${\mathcal{M}}$ (see Equations (1) and (2)) as
$\,\Omega_{1}=\Omega_{3}=0.1(2\pi/P),\,\Omega_{2}=\sqrt{1-2\times
0.1^{2}}(2\pi/P)$. For these parameters, $K$ equals $9.9$, and the period
equals $T\sim 6.3\times 10^{6}P$. It can be seen that the $x_{2}$ axis flips
$\pi$ during a relatively short interval of the total period $T$ (see the
bottom of Figure 1 and Figure 2). This is the “Dzhanibekov effect”, which
occurs in the case that a triaxial rigid body rotates and the rotation around
the $x_{2}$ axes is dominant initially.

For the flip to occur, the period $T$ should be shorter than the spin-down
timescale. The spin-down timescale of the neutron star is (Shapiro &
Teukolsky, 1983)

$\displaystyle t_{\rm sd}\sim\frac{3I_{2}c^{3}}{2B_{\rm NS}{}^{2}R_{\rm
NS}{}^{6}\Omega^{2}}$ (9)

The condition $t_{\rm sd}\gtrsim T$ gives the lower limit on the spin period
as

$\displaystyle P\gtrsim 3\times 10^{-4}{\rm s}\,K_{1}\delta_{-6}{}^{-1}B_{\rm
NS,15}{}^{2}R_{\rm NS,6}{}^{4}M_{\rm NS,33.3}{}^{-1},$ (10)

which is much shorter than the spin period of the known magnetars, $\sim
1\,{\rm s}$.

##  3 Burst event associated with the flip

###  3.1 Euler Force during The Flip

---
Figure 3:  Magnitude of the Euler force on the crust, $F_{\rm Eul}$,
normalized by $\delta F_{\rm Ela}$ (see Equation (12)). The left panel shows
the force at $t=0$ (before the flip), the middle panel shows it at
$t=1.6\times 10^{6}P$ (during the flip), and the right panel shows it at
$t=3\times 10^{6}P$ (after the flip) (see also Figures 1 and 2). The black
allow shows the $x_{2}$ axis. During the flip of the $x_{2}$ axis (middle
panel), the Euler force suddenly becomes stronger than before and after the
flip.  Figure 4:  Direction and strength of the Euler force seen from a
positive $Z$. The color contour shows the strength of the Euler force
normalized by $\delta F_{\rm ela}$, and the arrows show the direction of the
Euler forces on the crust. On the lines from $(X,Y)\sim(-0.6,-0.6)$ to
$\sim(0,1)$ and from $(X,Y)\sim(0,-1)$ to $\sim(0.6,0)$, the direction of the
Euler forces suddenly changes, and the crack may occur.

During the rotation, the magnetar surface is affected by the inertia force due
to the rotation. We define the surface of the magnetar at which the density is
lower than $10^{10}\rm g\,cm^{-3}$. This density range is the so-called crust
(e.g., Chamel & Haensel, 2008). The inertia force is composed of three
components, the Euler force, the Coriolis force, and the centrifugal force
(Landau & Lifshitz, 1969). The Euler force per unit mass is
$\vec{r}\times\dot{\vec{\Omega}}$, where $\vec{r}$ is the radius vector of the
particle in the rotating frame. The Coriolis force per unit mass is
$2\dot{\vec{r}}\times\vec{\Omega}$, and the centrifugal force per unit mass is
$\vec{\Omega}\times(\vec{r}\times\vec{\Omega})$. The Coriolis force does not
appear because the crust would not move in the rotating frame (i.e.,
$\dot{\vec{r}}=\vec{0}$). The force balance among the centrifugal force, the
pressure gradient, the elastic force, and the Lorentz force on the crust
determines the shape of the magnetar. The Euler force appears only during the
flip because it is proportional to $\dot{\vec{\Omega}}$. Hereafter, we
evaluate the Euler force on the crust during the flip and indicate that the
crust can clack during the flip.

Figure 3 shows the Euler force on the crust for $t=0$ (before the flip, left
panel), $t=1.6\times 10^{6}P$ (during the flip, middle panel), and $t=3\times
10^{6}P$ (after the flip, right panel) (see also Figures 1 and 2). The black
allow shows the $x_{2}$ axis and the color contour shows the magnitude of the
Euler force on the crust. During the flip of the $x_{2}$ axis, the Euler force
suddenly becomes stronger than before. The Euler force after the flip is as
weak as before the flip. The magnitude of the Euler force per unit volume
element of the crust during the flip of the $x_{2}$ axis is

$\displaystyle F_{\rm Eul}\sim\rho R_{\rm NS}\dot{\Omega}\sim\rho R_{\rm
NS}\Omega^{2}\delta,$ (11)

where we use Equations (3)–(5).

The sudden appearance of the Euler force may break the force balance on the
crust. Before the flip of the $x_{2}$ axis ($0<t\lesssim 10^{6}P$ in Figure 2
for the flip at $t\sim 1.6\times 10^{6}P$), the Lorentz force, $F_{\rm Lor}$,
and the elastic force, $F_{\rm ela}$, would be in force balance (e.g., Chamel
& Haensel, 2008; Kojima et al., 2021). If the force balance between the
Lorentz force and the elastic force determines the triaxial shape
characterized by $\delta$, we may obtain

$\displaystyle|F_{\rm Lor}-F_{\rm ela}|$ $\displaystyle\sim$
$\displaystyle\delta F_{\rm ela}\sim\delta\frac{\mu\sigma^{2}}{\ell},$ (12)

where $\mu$ and $\sigma,$ are shear modulus and shear-strain of the crust
(e.g., Kojima et al., 2021). $\ell$ is the minimal length scale of the cracks
that we will estimate in the following. When the Euler force becomes as strong
as $|F_{\rm Lor}-F_{\rm ela}|$, the magnetar crust will crack. From Equations
(11) and (12), the condition of the cracking is estimated as

$\displaystyle\frac{F_{\rm Eul}}{|F_{\rm Lor}-F_{\rm ela}|}$
$\displaystyle\sim$ $\displaystyle 4\,\ell_{3}R_{\rm
NS,6}P_{,0}{}^{-2}\sigma_{c,-2}{}^{-2}\left(\frac{\mu}{\rho}\right)^{-1}_{14}>1,$

where $\sigma_{c}$ is the critical shear-strain when the crust cracks and we
adopt $\sigma_{c}\sim 10^{-2}$ and $\mu/\rho\sim 10^{14}\,{\rm
cm^{2}s^{-2}}(A/56)^{-4/3}(Z/26)^{2}\rho_{,4}{}^{1/3}$ (Ogata & Ichimaru,
1990; Chamel & Haensel, 2008). The ratio of the shear modulus to the mass
density, $\mu/\rho$, is calculated assuming that the magnetar crust is an
isotropic body-centered cubic polycrystal and that the crystal is cold. The
maximum shear-strain, $\sigma_{c}$, is not well known and $\sigma_{c}$ varies
from $\sim 10^{-5}$ to $\sim 10^{-1}$ depending on the approach (summarized in
Kojima et al., 2021). In this paper, we adopt $\sigma_{c}\sim 10^{-2}$.
Equation (LABEL:eq:balance) shows that the Euler force can break the force
balance between the Lorentz force and the elastic force if $\ell$ is larger
than $\sim 10^{3}\,{\rm cm}$ for $\sigma_{c}=10^{-2}$. Figure 4 shows the
directions of the Euler force at the surface of the magnetar. We can find many
regions where the directions of the Euler force suddenly change. We expect
that the force balance is disturbed there and the crust cracks. We note that
this $\ell$ is the lower limit of the length scale at which the Euler force
can disturb the force balance on the crust, and the length scale of the crack
can be larger than $\ell$. Thus, the flip results in the crack of the crust
with the length scale of $\ell\gtrsim 10^{3}\,{\rm cm}$.

###  3.2 X-ray Short Bursts Associated with The Flip

We will discuss the consequence of the crack of the magnetar crust for the
X-ray short bursts. The crack may also correspond to the starquake. The
crack/starquake releases some of the magnetic energy, and the released energy
can create electron-positron pairs, forming an optically thick trapped
fireball (Thompson & Duncan, 1995). The X-rays emitted from this trapped
fireball can be observed as the X-ray short burst from a magnetar.

An expanding fireball can also be created if the fireball is formed near the
magnetic pole (Thompson & Duncan, 2001; Ioka, 2020; Yang & Zhang, 2021; Wada &
Ioka, 2023). Assuming that this expanding fireball is formed at a part of the
magnetar surface whose area is $x^{2}\sim 10^{8}\,{\rm cm^{2}}$,111 $x\sim
10^{4}\,{\rm cm}$ are chosen so that the isotropic X-ray luminosity and the
cut-off energy equal the values observed in the X-ray short burst associated
with Galactic FRB 20200428A. These values satisfy $L_{X}\simeq\pi x^{2}a_{\rm
rad}T^{4}c$ where $a_{\rm rad}$ is the radiation constant. that the
temperature of the fireball is about $T\sim 80\,{\rm keV}$, and that the
duration is $\tau\sim 1\,{\rm s}$, the X-ray isotropic luminosity is
$L_{X}\sim 10^{41}\,{\rm erg\,s^{-1}}$ (Ioka, 2020). The specific enthalpy
(the ratio of the photon energy density to the rest mass energy density) is

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{L_{X}}{\dot{M}c^{2}}\sim\frac{L_{X}\tau}{\rho
c^{2}x^{2}H}\sim
100L_{X,41}\tau_{,0}\rho_{0,4}{}^{-1}H_{,2.4}{}^{-3.5}x_{,4}{}^{-2}$

where $H$ is the surface depth of the loaded baryon and we used
$\dot{M}\sim\rho x^{2}h/\tau$. We also used the relation between the mass
density, $\rho$, and the density scale height, $H_{\rho}$,
$\rho\sim\rho_{0}(H_{\rho}/10\,{\rm cm})^{2.5}$. This equation is derived by
the equation of hydrostatic equilibrium and the polytope equation of state
with an adiabatic index $1.4$, and we set $\rho_{0}\sim 10^{4}\,{\rm
g\,cm^{-3}}$. These adiabatic index and $\rho_{0}$ are chosen so that the
density profile coincides with the ones derived from effective field theories
in nuclear physics (Cehula et al., 2023). Hereafter we also assume that $H$
equals the density scale height of the crust, i.e., $H\sim H_{\rho}$.

For $H\lesssim 300\,{\rm cm}$, $\eta$ is larger than $\sim 100$. In the
expanding fireball, a plasma outflow is accelerated by the radiative force
from the X-ray photons (Meszaros et al., 1993; Grimsrud & Wasserman, 1998;
Nakar et al., 2005). For an expanding fireball with $\eta\gtrsim 100$, $T\sim
80\,{\rm keV}$, and $x\sim 10^{4}\,{\rm cm}$, the final Lorentz factor of the
plasma outflow is $\sim 100$ (Wada & Ioka, 2023). This relativistic outflow
can be an energy budget of FRBs associated with the X-ray short burst (Wada &
Ioka, 2023).

The baryons loaded on the fireball can create circular polarization in the
X-ray short burst. This is because the cross-section of the X-ray photon
depends on its polarization in electron-proton plasma. The circularly
polarized photons whose rotation is in the same rotational direction of the
electron gyro-motion in the magnetic field are strongly scattered while the
photons with different circular polarization escape from the fireball more
easily (e.g., Meszaros, 1992). As a result, a circularly polarized X-ray short
burst can be created. Whether the observed rotational direction of the
circular polarization is left- or right-handed depends on the direction of the
magnetic field with respect to our line of sight. Thus, the flip can be
identified by the burst-to-burst reverse of the X-ray circular polarization.

###  3.3 Fast Radio Bursts Associated with the Flip

FRBs are coherent radio bursts with a duration of about milliseconds (Lorimer
et al., 2007; Thornton et al., 2013). The magnetars are strong candidates for
the source of the FRBs (e.g., Cordes & Chatterjee, 2019; Platts et al., 2019;
Zhang, 2020; Lyutikov, 2021; Petroff et al., 2021, for reviews) If some of
FRBs are triggered by the crack, they would be frequently observed during the
flip. Then, the source can be classified as a periodically modulated FRBs as
discussed in Levin et al. (2020); Zanazzi & Lai (2020); Yang & Zou (2020). In
addition to the periodicity, the frequency-dependent active window, which is
observed in FRB 20180916B (CHIME/FRB Collaboration et al., 2020b; Pleunis et
al., 2021; Pastor-Marazuela et al., 2020) can be realized as discussed in Li &
Zanazzi (2021). The period of FRB equals $T$ in Equation (8). We note that the
periodicity and frequency-dependent active window can be explained by the
orbital period of a binary (Wada et al., 2021).

The emission mechanism of the radio wave in FRB is unclear. It has been
observed that electromagnetic waves are emitted when crystals break on the
earth. If a similar process occurs during the crack of the magnetar crust,
electromagnetic radiation might be expected (Perkins et al., 1970; Fukui et
al., 2001). In this case, the observation of the FRBs may become a new clue to
the physics of the relativistic condensed matter under the strong magnetic
field.

The precession in this paper might be observed like a glitch of the magnetar
if it is only observed around the flip timing. The magnetr’s burst associated
with glitches (e.g., Younes et al., 2023; Ge et al., 2022; Sathyaprakash et
al., 2024) might be associated with the flip. The emission signature of the
FRBs (and also the X-ray bursts) around the flip will be studied in separate
papers.

###  3.4 Other Multi-Messenger Astrophysical Object

We consider the possibility that the crack of the crust is the source of the
high-energy cosmic rays. When the crust cracks, the electric fields may occur
on timescales of about the inverse of the plasma oscillation frequency. The
electric field parallel to the magnetic field accelerates the ions which are
ejected from the cracking crust. The possible maximum electric field is
estimated as

$eE_{\rm max}\sim 4\pi\frac{e^{2}\rho}{m_{p}}H\sim 1\times 10^{16}\,{\rm
dyn}\,\rho_{0,4}H_{,2.4}{}^{3.5},$ (15)

where we assume that complete charge separation occurs. If a proton is
accelerated by this electric field for a length scale $d$, which depends on
the details of the microphysics, the possible maximum energy of the ion is

$\varepsilon_{\rm max}\sim eE_{\rm max}d\sim 10^{20}\,{\rm
eV}\rho_{0,4}H_{,2.4}{}^{3.5}d_{,-8}.$ (16)

If that is the case, the crack is a source of the ultrahigh-energy cosmic rays
such as the Amaterasu particle (Telescope Array Collaboration et al., 2023).
In this scenario, the crack should occur near the pole of the magnetar dipole
field in order to avoid cooling via synchrotron radiation and curvature
radiation.

The precession of triaxial magnetar can be the source of the continuous
gravitational wave (Zimmermann, 1980). The waveform of the continuous
gravitational waves and the timing of the bursting activity of the magnetar
may provide a link between the flips and the magnetar busts.

##  4 Conclusion

We investigate the free precession of a triaxial magnetar and the
astrophysical transients associated with the precession. We show that the
triaxial magnetar can flip under the initial condition that the spin around
the second principal axis is dominant (Section 2), and derive the interval of
the flip. Using the time evolution of the angular velocities, we evaluate the
inertial force on the magnetar crust and show that the Euler force is
strongest during the flip (Section 3.1). The Euler force can be strong enough
to break the balance of the forces on the crust. If the Euler force cracks the
crust of the magnetar, the X-ray short bursts possibly occur (Section 3.2).
Then, the baryons at the surface of the magnetar are loaded on the fireball
which emits X-rays. Because the cross-section of the X-ray photon depends on
its circular polarization in electron-proton plasma, the X-ray short burst can
be circularly polarized. The rotational direction of the circular polarization
depends on the direction of the magnetic field, and thus, the flip can be
identified by the burst-to-burst reverse of the X-ray circular polarization.
The periodicity of some FRBs may be explained by the precession as already
discussed in previous studies (Section 3.3). If a strong electric field is
induced during the crack, the emitted nucleons can be accelerated resulting in
a source of the high-energy cosmic rays. The triaxial magnetars also can be a
source of the gravitational waves (Section 3.4). The sudden arising of the
Euler force in the flip can trigger astrophysical transients, such as X-ray
short bursts, flares, FRBs, ultrahigh-energy cosmic-ray events, and
gravitational-wave events.

## Acknowledgement

We thank K. Asano, R. Goto, Y. Ichinose, T. Kawashima, Y. Kusafuka, K.
Nishiwaki for daily discussions. This work is supported by Grants-in-Aid for
Scientific Research No. 22K20366 (TW) from the Ministry of Education, Culture,
Sports, Science and Technology (MEXT) of Japan.

## References

  * Abbott et al. (2022) Abbott, R., Abe, H., Acernese, F., et al. 2022, Physical Review D, 106, 042003
  * Ashbaugh et al. (1991) Ashbaugh, M. S., Chicone, C. C., & Cushman, R. H. 1991, Journal of Dynamics and Differential Equations, 3, 67, doi: 10.1007/BF01049489
  * Bochenek et al. (2020) Bochenek, C. D., Ravi, V., Belov, K. V., et al. 2020, Nature, 587, 59, doi: 10.1038/s41586-020-2872-x
  * Cehula et al. (2023) Cehula, J., Thompson, T. A., & Metzger, B. D. 2023, arXiv e-prints, arXiv:2311.05681, doi: 10.48550/arXiv.2311.05681
  * Chamel & Haensel (2008) Chamel, N., & Haensel, P. 2008, Living Reviews in Relativity, 11, 10, doi: 10.12942/lrr-2008-10
  * CHIME/FRB Collaboration et al. (2020a) CHIME/FRB Collaboration, Andersen, B. C., Bandura, K. M., et al. 2020a, Nature, 587, 54, doi: 10.1038/s41586-020-2863-y
  * CHIME/FRB Collaboration et al. (2020b) CHIME/FRB Collaboration, Amiri, M., Andersen, B. C., et al. 2020b, Nature, 582, 351, doi: 10.1038/s41586-020-2398-2
  * Cordes & Chatterjee (2019) Cordes, J. M., & Chatterjee, S. 2019, ARA&A, 57, 417, doi: 10.1146/annurev-astro-091918-104501
  * Duncan & Thompson (1992) Duncan, R. C., & Thompson, C. 1992, ApJ, 392, L9, doi: 10.1086/186413
  * Enoto et al. (2019) Enoto, T., Kisaka, S., & Shibata, S. 2019, Reports on Progress in Physics, 82, 106901, doi: 10.1088/1361-6633/ab3def
  * Fujisawa et al. (2022) Fujisawa, K., Kisaka, S., & Kojima, Y. 2022, MNRAS, 516, 5196, doi: 10.1093/mnras/stac2585
  * Fukui et al. (2001) Fukui, K., Okubo, S., & Terashima, T. 2001, SHIGEN-TO-SOZAI, 117
  * Ge et al. (2022) Ge, M., Yang, Y.-P., Lu, F., et al. 2022, arXiv e-prints, arXiv:2211.03246, doi: 10.48550/arXiv.2211.03246
  * Grimsrud & Wasserman (1998) Grimsrud, O. M., & Wasserman, I. 1998, MNRAS, 300, 1158, doi: 10.1046/j.1365-8711.1998.02004.x
  * Heyl et al. (2023) Heyl, J., Doroshenko, V., González-Caniulef, D., et al. 2023, arXiv e-prints, arXiv:2311.03667, doi: 10.48550/arXiv.2311.03667
  * Ioka (2020) Ioka, K. 2020, ApJ, 904, L15, doi: 10.3847/2041-8213/abc6a3
  * Kaspi & Beloborodov (2017) Kaspi, V. M., & Beloborodov, A. M. 2017, ARA&A, 55, 261, doi: 10.1146/annurev-astro-081915-023329
  * Kojima et al. (2021) Kojima, Y., Kisaka, S., & Fujisawa, K. 2021, MNRAS, 506, 3936, doi: 10.1093/mnras/stab1848
  * Kolesnikov et al. (2022) Kolesnikov, D., Shakura, N., & Postnov, K. 2022, MNRAS, 513, 3359, doi: 10.1093/mnras/stac1107
  * Landau & Lifshitz (1969) Landau, L. D., & Lifshitz, E. M. 1969, Mechanics
  * Levin et al. (2020) Levin, Y., Beloborodov, A. M., & Bransgrove, A. 2020, ApJ, 895, L30, doi: 10.3847/2041-8213/ab8c4c
  * Li et al. (2021) Li, C. K., Lin, L., Xiong, S. L., et al. 2021, Nature Astronomy, doi: 10.1038/s41550-021-01302-6
  * Li & Zanazzi (2021) Li, D., & Zanazzi, J. J. 2021, ApJ, 909, L25, doi: 10.3847/2041-8213/abeaa4
  * Lorimer et al. (2007) Lorimer, D. R., Bailes, M., McLaughlin, M. A., Narkevic, D. J., & Crawford, F. 2007, Science, 318, 777, doi: 10.1126/science.1147532
  * Lyutikov (2021) Lyutikov, M. 2021, arXiv e-prints, arXiv:2110.08435.  https://arxiv.org/abs/2110.08435
  * Mardešić et al. (2020) Mardešić, P., Guillen, G. J. G., Van Damme, L., & Sugny, D. 2020, Phys. Rev. Lett., 125, 064301, doi: 10.1103/PhysRevLett.125.064301
  * Mereghetti et al. (2020) Mereghetti, S., Savchenko, V., Ferrigno, C., et al. 2020, ApJ, 898, L29, doi: 10.3847/2041-8213/aba2cf
  * Meszaros (1992) Meszaros, P. 1992, High-energy radiation from magnetized neutron stars (University of Chicago press)
  * Meszaros et al. (1993) Meszaros, P., Laguna, P., & Rees, M. J. 1993, ApJ, 415, 181, doi: 10.1086/173154
  * Nakar et al. (2005) Nakar, E., Piran, T., & Sari, R. 2005, ApJ, 635, 516, doi: 10.1086/497296
  * Ogata & Ichimaru (1990) Ogata, S., & Ichimaru, S. 1990, Phys. Rev. A, 42, 4867, doi: 10.1103/PhysRevA.42.4867
  * Pastor-Marazuela et al. (2020) Pastor-Marazuela, I., Connor, L., van Leeuwen, J., et al. 2020, arXiv e-prints, arXiv:2012.08348.  https://arxiv.org/abs/2012.08348
  * Perkins et al. (1970) Perkins, R., Green, S., & Friedman, M. 1970, Uniaxial stress behavior of porphyritic tonalite at strain rates to 103/second
  * Petroff et al. (2021) Petroff, E., Hessels, J. W. T., & Lorimer, D. R. 2021, arXiv e-prints, arXiv:2107.10113.  https://arxiv.org/abs/2107.10113
  * Platts et al. (2019) Platts, E., Weltman, A., Walters, A., et al. 2019, Phys. Rep., 821, 1, doi: 10.1016/j.physrep.2019.06.003
  * Pleunis et al. (2021) Pleunis, Z., Michilli, D., Bassa, C. G., et al. 2021, ApJ, 911, L3, doi: 10.3847/2041-8213/abec72
  * Ridnaia et al. (2021) Ridnaia, A., Svinkin, D., Frederiks, D., et al. 2021, Nature Astronomy, 5, 372, doi: 10.1038/s41550-020-01265-0
  * Sathyaprakash et al. (2024) Sathyaprakash, R., Parent, E., Rea, N., et al. 2024, arXiv e-prints, arXiv:2401.08010, doi: 10.48550/arXiv.2401.08010
  * Shakura et al. (1998) Shakura, N. I., Postnov, K. A., & Prokhorov, M. E. 1998, A&A, 331, L37, doi: 10.48550/arXiv.astro-ph/9801033
  * Shapiro & Teukolsky (1983) Shapiro, S. L., & Teukolsky, S. A. 1983, Black holes, white dwarfs, and neutron stars : the physics of compact objects (John Wiley & Sons)
  * Tananbaum et al. (1972) Tananbaum, H., Gursky, H., Kellogg, E. M., et al. 1972, ApJ, 174, L143, doi: 10.1086/180968
  * Tavani et al. (2021) Tavani, M., Casentini, C., Ursi, A., et al. 2021, Nature Astronomy, 5, 401, doi: 10.1038/s41550-020-01276-x
  * Telescope Array Collaboration et al. (2023) Telescope Array Collaboration, Abbasi, R. U., Allen, M. G., et al. 2023, Science, 382, 903, doi: 10.1126/science.abo5095
  * Thompson & Duncan (1995) Thompson, C., & Duncan, R. C. 1995, MNRAS, 275, 255, doi: 10.1093/mnras/275.2.255
  * Thompson & Duncan (1996) —. 1996, ApJ, 473, 322, doi: 10.1086/178147
  * Thompson & Duncan (2001) —. 2001, ApJ, 561, 980, doi: 10.1086/323256
  * Thornton et al. (2013) Thornton, D., Stappers, B., Bailes, M., et al. 2013, Science, 341, 53, doi: 10.1126/science.1236789
  * Wada & Ioka (2023) Wada, T., & Ioka, K. 2023, MNRAS, 519, 4094, doi: 10.1093/mnras/stac3681
  * Wada et al. (2021) Wada, T., Ioka, K., & Zhang, B. 2021, ApJ, 920, 54, doi: 10.3847/1538-4357/ac127a
  * Yang & Zou (2020) Yang, H., & Zou, Y.-C. 2020, ApJ, 893, L31, doi: 10.3847/2041-8213/ab800f
  * Yang & Zhang (2021) Yang, Y.-P., & Zhang, B. 2021, ApJ, 919, 89, doi: 10.3847/1538-4357/ac14b5
  * Younes et al. (2023) Younes, G., Baring, M. G., Harding, A. K., et al. 2023, Nature Astronomy, 7, 339, doi: 10.1038/s41550-022-01865-y
  * Zanazzi & Lai (2020) Zanazzi, J. J., & Lai, D. 2020, ApJ, 892, L15, doi: 10.3847/2041-8213/ab7cdd
  * Zhang (2020) Zhang, B. 2020, ApJ, 890, L24, doi: 10.3847/2041-8213/ab7244
  * Zimmermann (1980) Zimmermann, M. 1980, Phys. Rev. D, 21, 891, doi: 10.1103/PhysRevD.21.891

##  Appendix A Analytic solution

In this section, we review the analytic solution of the Euler equations
(Equations (3)–(5)). The analytic solutions of Equations (3)–(5) for
$M^{2}>2EI_{2}$ are (Landau & Lifshitz, 1969)

$\displaystyle\Omega_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2EI_{3}-M^{2}}{I_{1}(I_{3}-I_{1})}}{\rm
cn}(\tau;\,k),$ (A1) $\displaystyle\Omega_{2}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2EI_{3}-M^{2}}{I_{2}(I_{3}-I_{2})}}{\rm
sn}(\tau;\,k),$ (A2) $\displaystyle\Omega_{3}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{M^{2}-2EI_{1}}{I_{3}(I_{3}-I_{1})}}{\rm
dn}(\tau;\,k)$ (A3)

where

$\displaystyle\tau$ $\displaystyle=$ $\displaystyle
t\sqrt{\frac{(I_{3}-I_{2})(M^{2}-2EI_{1})}{I_{1}I_{2}I_{3}}},$ (A4)
$\displaystyle k^{2}$ $\displaystyle=$
$\displaystyle\frac{(I_{2}-I_{1})(2EI_{3}-M^{2})}{(I_{3}-I_{2})(M^{2}-2EI_{1})},$
(A5)

and ${\rm sn}(x)$, ${\rm cn}(x)$, and ${\rm dn}(x)$ are the Jacobi elliptic
functions. The Jacobi elliptic functions are periodic with respect to the real
argument $t$ and its period is

$\displaystyle T$ $\displaystyle=$ $\displaystyle
4K\sqrt{\frac{I_{1}I_{2}I_{3}}{(I_{3}-I_{2})(M^{2}-2EI_{1})}}$ (A6)

where

$\displaystyle K$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{ds}{\sqrt{(1-s^{2})(1-k^{2}s^{2})}}$ (A7)

If we set $I_{1}=I_{2}(1-\delta),\,I_{3}=I_{2}(1+\delta)$, and
$\Omega_{2,0}\gg\Omega_{1,0},\Omega_{3,0}$, $k^{2}$ is expressed as

$\displaystyle k^{2}$ $\displaystyle=$
$\displaystyle\frac{1+2(\Omega_{1,0}/\Omega_{2,0})^{2}}{1+2(\Omega_{3,0}/\Omega_{2,0})^{2}}$
$\displaystyle\times$
$\displaystyle\left[1-\left(\frac{2(\Omega_{1,0}/\Omega_{2,0})^{2}}{1+2(\Omega_{1,0}/\Omega_{2,0})^{2}}+\frac{2(\Omega_{3,0}/\Omega_{2,0})^{2}}{1+2(\Omega_{3,0}/\Omega_{2,0})^{2}}\right)\delta\right],$

up to the first order of $\delta$, and the period is

$\displaystyle T$ $\displaystyle\sim$ $\displaystyle
4K\delta^{-1}\Omega_{2,0}^{-1}\sim 0.2\,{\rm yr}\,K_{1}\delta_{-6}^{-1}P_{0}.$
(A9)

$K$ is a monotonously increasing function for $k$, $K$ equals $\pi/2$ for
$k=0$, $K\simeq 8.3$ for $k^{2}=1-10^{-6}$, and $K\simeq 13$ for
$k^{2}=1-10^{-10}$. We adopt $K=10$ in this paper.


# generalized degenerate Stirling numbers arising from degenerate Boson normal
ordering

Taekyun Kim 1 Department of Mathematics, Kwangwoon University, Seoul 139-701,
Republic of Korea<EMAIL_ADDRESS>,  Dae San Kim 2 Department of Mathematics,
Sogang University, Seoul 121-742, Republic of Korea<EMAIL_ADDRESS>and
Hye Kyung Kim 3,∗ Department of Mathematics Education, Daegu Catholic
University, Gyeongsan 38430, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

It is remarkable that, in recent years, intensive studies have been done for
degenerate versions of many special polynomials and numbers and have yielded
many interesting results. The aim of this paper is to study the generalized
degenerate $(r,s)$-Stirling numbers of the second and their natural extensions
to polynomials, namely the generalized degenerate $(r,s)$-Bell polynomials,
arising from certain ‘degenerate boson normal ordering.’ We derive some
properties, explicit expressions and generating functions for those numbers
and polynomials. The generalized degenerate $(r,s)$-Stirling numbers of the
second and the degenerate boson normal ordering are respectively degenerate
versions of the generalized $(r,s)$-Stirling numbers of the second and the
boson normal ordering studied earlier by Blasiak-Person-Solomon.

###### Key words and phrases:

generalized degenerate $(r,s)$-Stirling numbers of the second kind;
generalized degenerate $(r,s)$-Bell polynomials; generalized (r,s)-Stirling
numbers of the second kind

###### MSC2020 Mathematics Subject Classification:

11B73; 11B83

* is corresponding author

##  1\. Introduction

The Stirling number of the second kind $S_{2}(n,k)$ is the number of ways to
partition a set of $n$ objects into $k$ nonempty subsets (see (3)). The
Stirling numbers of the second kind arise in various different contexts and
have numerous applications, for example to enumerative combinatorics and
quantum mechanics. In recent years, intensive explorations have been done for
degenerate versions of many special numbers and polynomials, which was
initiated by Carlitz in his work on degenerate Bernoulli and degenerate Euler
polynomials. They have been studied by using such tools as combinatorial
methods, generating functions, $p$-adic analysis, umbral calculus techniques,
probability theory, mathematical physics, operator theory, special functions,
analytic number theory and differential equations. The degenerate Stirling
numbers of the second kind $S_{2,\lambda}(n,k)$ appear naturally when we
replace the power $x^{n}$ by the generalized falling factorial polynomial
$(x)_{n,\lambda}$ in the defining equation of $S_{2}(n,k)$ (see (3), (6)). It
turns out that they appear very frequently when we study degenerate versions
of many special polynomials and numbers.

The normal ordering of $(a^{{\dagger}}a)^{n}$ in terms of boson operators $a$
and $a^{{\dagger}}$ can be written in the form

(1) $(a^{{\dagger}}a)^{n}=\sum_{l=0}^{n}S_{2}(n,l)(a^{{\dagger}})^{l}a^{l}.$

The normal ordering of $(a^{{\dagger}}a)_{n,\lambda}$ in terms of boson
operators $a$ and $a^{{\dagger}}$, which is a degenerate version of (1), can
be written in the form

$(a^{{\dagger}}a)_{n,\lambda}=\sum_{l=0}^{n}S_{2,\lambda}(k,l)(a^{{\dagger}})^{l}a^{l},$

where the generalized falling factorials $(x)_{n,\lambda}$ are given by (5).

Let $r,s$ be positive integers with $r\geq s$, and let $n$ be any positive
integer. Blasiak-Person-Solomon introduced the generalized $(r,s)$-Stirling
numbers of the second kind $S_{r,s}(n,k)$, which boil down to
$S_{1,1}(n,k)=S_{2}(n,k)$, for $r=s=1$, by considering the boson normal
ordering of $((a^{{\dagger}})^{r}a^{s})^{n}$:

(2)
$\begin{split}((a^{{\dagger}})^{r}a^{s})^{n}=(a^{{\dagger}})^{n(r-s)}\sum_{k=s}^{ns}S_{r,s}(n,k)(a^{{\dagger}})^{k}a^{k}.\end{split}$

In this paper, we introduce the generalized degenerate $(r,s)$-Stirling
numbers of the second kind, which are degenerate versions of the generalized
$(r,s)$-Stirling numbers of the second kind, by considering a degenerate
version of (2), namely the boson normal ordering of
$\prod_{k=0}^{n-1}\big{(}(a^{{\dagger}})^{r}a^{s}-k\lambda(a^{{\dagger}})^{r-s}\big{)}$:

$\begin{split}\prod_{k=0}^{n-1}\Big{[}(a^{{\dagger}})^{r-s}\Big{(}(a^{{\dagger}})^{s}a^{s}-k\lambda\Big{)}\Big{]}&=\prod_{k=0}^{n-1}\Big{(}(a^{{\dagger}})^{r}a^{s}-k\lambda(a^{{\dagger}})^{r-s}\Big{)}\\\
&=(a^{{\dagger}})^{n(r-s)}\sum_{k=0}^{ns}S_{\lambda}^{(r,s)}(n,k)(a^{{\dagger}})^{k}a^{k}.\end{split}$

The aim of this paper is to derive some properties, explicit expressions and
generating functions for the generalized degenerate $(r,s)$-Stirling numbers
of the second kind and their natural extensions to polynomials, namely the
generalized degenerate $(r,s)$-Bell polynomials. The novelty of this paper is
that the generalized degenerate $(r,s)$-Stirling numbers of the second kind
are introduced in a natural manner by considering the ‘degenerate boson normal
ordering.’ We think that these new numbers will play an important role in the
study of various degenerate versions of many special polynomials and numbers.

In more detail, the outline of this paper is as follows. We derive several
expressions for the generalized degenerate $(r,s)$-Bell polynomials
$\phi_{n,\lambda}^{(r,s)}(x)$ (see (19)) in Theorem 2 and the generalized
degenerate $(r,s)$-Bell numbers
$\phi_{n,\lambda}^{(r,s)}=\phi_{n,\lambda}^{(r,s)}(1)$ in Theorems 2 and 3. We
obtain several expressions for the generalized degenerate $(r,s)$-Stirling
numbers of the second kind in Theorems 4-6.
$\phi_{n,\lambda}^{(r,r)}(|z|^{2})$ and its generating function
$\sum_{n=0}^{\infty}\phi_{n,\lambda}^{(r,r)}(|z|^{2})\frac{t^{n}}{n!}$ are
expressed in terms of bra-ket notation respectively in Theorem 7 and Theorem
8. We deduce the generating function
$\sum_{n=0}^{\infty}\phi_{n,\lambda}^{(r)}(|z|^{2})\frac{t^{n}}{n!}$ of the
degenerate $r$-Bell polynomials $\phi_{n,\lambda}^{(r)}(x)$, which are
different from $\phi_{n,\lambda}^{(r,r)}(x)$ and natural extension to
polynomials of the degenerate $r$-Stirling numbers of the second kind (see
(43), (44)). Some recurrence relations for $\phi_{n,\lambda}^{(r)}(|z|^{2})$
are obtained in Theorem 10. Another expression for
$\phi_{n,\lambda}^{(r,r)}(|z|^{2})$ is obtained in Theorem 11 by using the
representation of the coherent state in terms of the number states. Finally,
we define by introducing two new notations the unsigned degenerate Lah numbers
and the signed degenerate Lah numbers, which are respectively degenerate
versions of the Lah numbers and the signed Lah numbers. For the rest of this
section, we recall the facts that are needed throughout this paper.

For $n\geq 0$, the Stirling numbers of the second kind are defined by

(3) $\begin{split}x^{n}=\sum_{k=0}^{n}S_{2}(n,k)(x)_{k},\quad(n\geq
0),\quad{(\text{see [1,4,5,10,14,16]})}.\end{split}$

For any $\lambda\in\mathbb{R}$, the degenerate exponentials are given by

(4)
$\begin{split}e_{\lambda}^{x}(t)=\sum_{k=0}^{\infty}(x)_{k,\lambda}\frac{t^{k}}{k!},\quad
e_{\lambda}(t)=e_{\lambda}^{1}(t),\quad{(\text{see
\cite[cite]{[\@@bibref{}{10,12}{}{}]}})},\end{split}$

where the generalized falling factorials are given by

(5)
$\begin{split}(x)_{0,\lambda}=1,(x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),\quad(n\geq
1).\end{split}$

Recently, the degenerate Stirling numbers of the second kind are defined by

(6)
$\begin{split}(x)_{n,\lambda}=\sum_{k=0}^{n}S_{2,\lambda}(n,k)(x)_{k},\quad(n\geq
0),\quad{(\text{see [7,8,10,11,12]})},\end{split}$

where $(x)_{0}=1,(x)_{n}=x(x-1)\cdots(x-n+1),\quad(n\geq 1)$.
Note that $\lim_{\lambda\rightarrow 0}S_{2,\lambda}(n,k)=S_{2}(n,k),\quad(n,\
k\geq 0)$. From (6), we note that

(7)
$\begin{split}\frac{1}{k!}(e_{\lambda}(t)-1)^{k}=\sum_{n=k}^{\infty}S_{2,\lambda}(n,k)\frac{t^{n}}{n!},\quad(k\geq
0),\quad{(\text{see \cite[cite]{[\@@bibref{}{10}{}{}]}})}.\end{split}$

It is well known that the ordinary Bell polynomials are defined by

(8)
$\begin{split}e^{x(e^{t}-1)}=\sum_{n=0}^{\infty}\phi_{n}(x)\frac{t^{n}}{n!},\quad{(\text{see
\cite[cite]{[\@@bibref{}{9, 13, 14}{}{}]}})}.\end{split}$

When $x=1,\phi_{n}=\phi_{n}(1),\quad(n\geq 0)$, are called the Bell numbers.
From (8), we note that $\phi_{n}(x)=\sum_{k=0}^{n}S_{2}(n,k)x^{k},\quad(n\geq
0)$.

Recently, the degenerate Bell polynomials are given by

(9)
$\begin{split}e^{x(e_{\lambda}(t)-1)}=\sum_{n=0}^{\infty}\phi_{n,\lambda}(x)\frac{t^{n}}{n!},\quad{(\text{see
\cite[cite]{[\@@bibref{}{8,9,11,12}{}{}]}})}.\end{split}$

By (9), we get

(10)
$\begin{split}\phi_{n,\lambda}(x)=\sum_{k=0}^{n}S_{2,\lambda}(n,k)x^{k},\quad(n\geq
0).\end{split}$

When $x=1,\phi_{n,\lambda}=\phi_{n,\lambda}(1),\quad(n\geq 0)$, are called the
degenerate Bell numbers.

Recall that $a$ and $a^{\dagger}$ are the boson annihilation and creation
operators such that

(11)
$\begin{split}[a,a^{\dagger}]=aa^{\dagger}-a^{\dagger}a=1,\quad{(\text{see
[2,3,6-8,11-13,15]})}.\end{split}$

The number states $|m\rangle,m=0,1,\cdots,$ are given by

(12) $\begin{split}a|m\rangle=\sqrt{m}|m-1\rangle,\
a^{\dagger}|m\rangle=\sqrt{m+1}|m+1\rangle.\end{split}$

The coherent state $|z\rangle$, where $z$ is complex number, satisfies
$a|z\rangle=z|z\rangle,\langle z|z\rangle=1$. To show a connection to coherent
states, we recall that the harmonic oscillator has Hamiltonian
$H=a^{\dagger}a$ (neglecting the zero point energy) and the usual eigenstates
$|n\rangle(n\in\mathbb{N})$ satisfying

(13) $\begin{split}H|n\rangle=n|n\rangle\ \ {\rm{and}}\ \ \langle
m|n\rangle=\delta_{m,n},\quad\ \ {(\text{see
\cite[cite]{[\@@bibref{}{8,11,13}{}{}]}})},\end{split}$

where $\delta_{m,n}$ is Kronecker’s symbol.

The normal ordering of a degenerate integral power of the number operator
$a^{\dagger}a$ in terms of the boson operators $a$ and $a^{\dagger}$ can be
written in the form

(14)
$\begin{split}(a^{\dagger}a)_{n,\lambda}=\sum_{k=0}^{n}S_{2,\lambda}(n,k)(a^{\dagger})^{k}a^{k},\quad(n\geq
0),\quad\ \ {(\text{see \cite[cite]{[\@@bibref{}{7,8,11}{}{}]}})}.\end{split}$

We note that the standard bosonic commutation relation
$[a,a^{\dagger}]=aa^{\dagger}-a^{\dagger}a=1$ can be considered formally, in a
suitable space of functions $f$, by letting $a=\frac{d}{dx}$ and
$a^{\dagger}=x$ (the operator of multiplication by $x$).

By (14), we get

$\begin{split}\bigg{(}x\frac{d}{dx}\bigg{)}_{n,\lambda}f(x)=\sum_{k=0}^{n}S_{2,\lambda}(n,k)x^{k}(\frac{d}{dx})^{k}f(x),\quad{(\text{see
\cite[cite]{[\@@bibref{}{7,8,11}{}{}]}})}.\end{split}$

From the definition of coherent states, we note that $a|z\rangle=z|z\rangle$,
equivalently $\langle z|a^{\dagger}=\langle z|\overline{z}$, where
$z\in\mathbb{C}$ and $\overline{z}$ is the complex conjugate of $z$.

In [2, 3], Blasiak-Person-Solomon considered the generalized Stirling numbers
of the second kind $S_{r,s}(n,k)$ which are given by

(15)
$\begin{split}((a^{{\dagger}})^{r}a^{s})^{n}=(a^{{\dagger}})^{n(r-s)}\sum_{k=s}^{ns}S_{r,s}(n,k)(a^{{\dagger}})^{k}a^{k},\end{split}$

where $r,s$ are positive integers with $r\geq s$ and $n$ is any positive
integer. They also considered the polynomials, which we call the $(r,s)$-Bell
polynomials, given by

(16)
$\begin{split}\phi_{n}^{(r,s)}(x)=\sum_{k=s}^{ns}S_{r,s}(n,k)x^{k}.\end{split}$

For $x=1$, $\phi_{n}^{(r,s)}=\phi_{n}^{(r,s)}(1)$ are called the $(r,s)$-Bell
numbers.

##  2\. Generalized degenerate Stirling numbers arising from degenerate boson
normal ordering

In this section, unless otherwise stated, $r,s$ are positive integers with
$r\geq s$ and $n$ is any positive integer. In light of (15), we introduce the
generalized degenerate $(r,s)$-Stirling numbers of the second kind arising
from the degenerate boson normal ordering

(17)
$\begin{split}\prod_{k=0}^{n-1}\bigg{[}x^{r-s}\bigg{(}x^{s}\bigg{(}\frac{d}{dx}\bigg{)}^{s}-k\lambda\bigg{)}\bigg{]}&=\prod_{k=0}^{n-1}\bigg{(}x^{r}\bigg{(}\frac{d}{dx}\bigg{)}^{s}-k\lambda
x^{r-s}\bigg{)}\\\
&=x^{n(r-s)}\sum_{k=0}^{ns}S_{\lambda}^{(r,s)}(n,k)x^{k}\bigg{(}\frac{d}{dx}\bigg{)}^{k}.\end{split}$

From (15) and (17), we see that $\lim_{\lambda\rightarrow
0}S_{\lambda}^{(r,s)}(n,k)=S_{r,s}(n,k)$.

######  Remark 1.

(a) In (17) and below, the product of operators is understood to be written in
the order of the factors corresponding to 1, 2, $\dots$, to $n-1$, from the
left to the right.
(b) If $r=s$, then we see from (15) and (17) that
$S_{\lambda}^{(r,r)}(n,k)=0$, for $0\leq k<r$.

Example: Let $n=2,\,r=4,\,s=2$ in (17). Then, with $x=X,\,\frac{d}{dx}=D$, we
have:

$\begin{split}X^{4}D^{2}(X^{4}D^{2}-\lambda X^{2})=(x^{4}D^{2})^{2}-\lambda
X^{4}D^{2}X^{2},\end{split}$

where

$\begin{split}D^{2}X^{2}&=D(XD+1)X=(DX)^{2}+DX=(XD+1)^{2}+XD+1\\\
&=(XD)^{2}+3XD+2=X(XD+1)D+3XD+2=X^{2}D^{2}+4XD+2.\end{split}$

Thus, from (15), we have

$\begin{split}X^{4}D^{2}(X^{4}D^{2}-\lambda X^{2})&=(x^{4}D^{2})^{2}-\lambda
X^{6}D^{2}-4\lambda X^{5}D-2\lambda X^{4}\\\
&=X^{4}\sum_{k=2}^{4}S_{4,2}(2,k)X^{k}D^{k}-\lambda X^{6}D^{2}-4\lambda
X^{5}D-2\lambda X^{4}\\\
&=X^{4}\Big{(}S_{4,2}(2,4)X^{4}D^{4}+S_{4,2}(2,3)X^{3}D^{3}+(S_{4,2}(2,2)-\lambda)X^{2}D^{2}-4\lambda
XD-2\lambda\Big{)}\\\
&=X^{4}\Big{(}X^{4}D^{4}+8X^{3}D^{3}+(12-\lambda)X^{2}D^{2}-4\lambda
XD-2\lambda\Big{)}.\end{split}$

Thus
$S_{\lambda}^{(4,2)}(2,4)=1,\,S_{\lambda}^{(4,2)}(2,3)=8,\,S_{\lambda}^{(4,2)}(2,2)=12-\lambda,\,S_{\lambda}^{(4,2)}(2,1)=-4\lambda,\,S_{\lambda}^{(4,2)}(2,0)=-2\lambda$.

By applying the operators in (17) to $x^{p}$ and letting $x=1$, for any
positive integer $p$, we obtain (18) with $x$ replaced by $p$ and hence it
holds as polynomials.

(18)
$\begin{split}\prod_{k=1}^{n}[(x+(k-1)(r-s))_{s}-(n-k)\lambda]=\sum_{k=0}^{ns}S_{\lambda}^{(r,s)}(n,k)(x)_{k}.\end{split}$

From (6) and (18), we note that $S_{\lambda}^{(1,1)}(n,k)=S_{2,\lambda}(n,k),\
\ (n,\ k\geq 0)$.

In view of (10), we define the generalized degenerate $(r,s)$-Bell polynomials
given by

(19)
$\begin{split}\phi_{n,\lambda}^{(r,s)}(x)=\sum_{k=0}^{ns}S_{\lambda}^{(r,s)}(n,k)x^{k}.\end{split}$

When $x=1,\phi_{n,\lambda}^{(r,s)}=\phi_{n,\lambda}^{(r,s)}(1)$ are called the
generalized degenerate $(r,s)$-Bell numbers. Here we see that
$\lim_{\lambda\rightarrow 0}\phi_{n,\lambda}^{(r,s)}(x)=\phi_{n}^{(r,s)}(x)$.

We observe that

(20)
$\begin{split}e^{-x}\sum_{k=0}^{\infty}\frac{1}{k!}&\bigg{(}\prod_{j=1}^{n}[(k+(j-1)(r-s))_{s}-(n-j)\lambda]\bigg{)}x^{k}\\\
&=e^{-x}\sum_{k=0}^{\infty}\frac{1}{k!}\sum_{l=0}^{ns}S_{\lambda}^{(r,s)}(n,l)(k)_{l}x^{k}\\\
&=\sum_{l=0}^{ns}S_{\lambda}^{(r,s)}(n,l)e^{-x}\sum_{k=0}^{\infty}\frac{(k)_{l}}{k!}x^{k}\\\
&=\sum_{l=0}^{ns}S_{\lambda}^{(r,s)}(n,l)e^{-x}x^{l}\bigg{(}\frac{d}{dx}\bigg{)}^{l}e^{x}\\\
&=\sum_{l=0}^{ns}S_{\lambda}^{(r,s)}(n,l)x^{l}e^{-x}e^{x}=\phi_{n,\lambda}^{(r,s)}(x).\end{split}$

Therefore, by (20), we obtain the following theorem.

######  Theorem 2.

For $n\geq 1$ and $r\geq s\geq 1$, we have

(21)
$\begin{split}\phi_{n,\lambda}^{(r,s)}(x)=\frac{1}{e^{x}}\sum_{k=0}^{\infty}\frac{1}{k!}\bigg{(}\prod_{j=1}^{n}[(k+(j-1)(r-s))_{s}-(n-j)\lambda]\bigg{)}x^{k}.\end{split}$

In particular, for $x=1$, we get

(22)
$\begin{split}\phi_{n,\lambda}^{(r,s)}=\frac{1}{e}\sum_{k=0}^{\infty}\frac{1}{k!}\bigg{(}\prod_{j=1}^{n}[(k+(j-1)(r-s))_{s}-(n-j)\lambda]\bigg{)}.\end{split}$

From (21), we observe that

(23)
$\begin{split}\phi_{n,\lambda}^{(r,r)}(x)=\frac{1}{e^{x}}\sum_{k=0}^{\infty}\frac{((k)_{r})_{n,\lambda}}{k!}x^{k}.\end{split}$

In particular, from (10), (19) and (23), we note that

(24)
$\begin{split}\phi_{n,\lambda}^{(1,1)}(x)=\frac{1}{e^{x}}\sum_{k=0}^{\infty}\frac{(k)_{n,\lambda}}{k!}x^{k}=\phi_{n,\lambda}(x).\end{split}$

From (22), we observe that

(25) $\begin{split}\phi_{n}^{(r,s)}&=\lim_{\lambda\rightarrow
0}\phi_{n,\lambda}^{(r,s)}=\frac{1}{e}\sum_{k=0}^{\infty}\frac{1}{k!}\prod_{j=0}^{n-1}(k+j(r-s))_{s}\\\
&=\frac{1}{e}\sum_{k=0}^{\infty}\frac{1}{k!}\prod_{j=0}^{n-1}\big{(}k+j(r-s)\big{)}\big{(}k-1+j(r-s)\big{)}\cdots\big{(}k-s+1+j(r-s)\big{)}\\\
&=\frac{1}{e}\sum_{k=0}^{\infty}\frac{1}{k!}\prod_{j=0}^{n-1}(r-s)^{s}\bigg{(}j+\frac{k}{r-s}\bigg{)}\bigg{(}j+\frac{k-1}{r-s}\bigg{)}\cdots\bigg{(}j+\frac{k-s+1}{r-s}\bigg{)}\\\
&=\frac{1}{e}\sum_{k=0}^{\infty}\frac{(r-s)^{sn}}{k!}\prod_{j=0}^{n-1}\prod_{l=1}^{s}\bigg{(}j+\frac{k-l+1}{r-s}\bigg{)}=\frac{1}{e}\sum_{k=0}^{\infty}\frac{(r-s)^{sn}}{k!}\prod_{l=1}^{s}\bigg{(}n-1+\frac{k-l+1}{r-s}\bigg{)}_{n}\\\
&=\frac{1}{e}\sum_{k=0}^{\infty}\frac{(r-s)^{sn}}{k!}\prod_{l=1}^{s}\frac{\Gamma(n+\frac{k-l+1}{r-s})}{\Gamma(\frac{k-l+1}{r-s})}=\frac{(r-s)^{sn}}{e}\sum_{k=0}^{\infty}\frac{1}{k!}\prod_{l=1}^{s}\frac{\Gamma(n+\frac{k-l+1}{r-s})}{\Gamma(\frac{k-l+1}{r-s})}.\end{split}$

Thus we obtain the following alternative expression for $\phi_{n}^{(r,s)}$.

######  Theorem 3.

For $r>s\geq 1$ and $n\geq 1$, we have the following expression:

(26)
$\begin{split}\phi_{n}^{(r,s)}=\frac{(r-s)^{sn}}{e}\sum_{k=0}^{\infty}\frac{1}{k!}\prod_{l=1}^{s}\frac{\Gamma(n+\frac{k-l+1}{r-s})}{\Gamma(\frac{k-l+1}{r-s})}.\end{split}$

From (18) and (21), we have

(27)
$\begin{split}\sum_{k=0}^{ns}&S_{\lambda}^{(r,s)}(n,k)x^{k}=\phi_{n,\lambda}^{(r,s)}(x)\\\
&=e^{-x}\sum_{p=0}^{\infty}\frac{1}{p!}\bigg{(}\prod_{j=1}^{n}[(p+(j-1)(r-s))_{s}-(n-j)\lambda]\bigg{)}x^{p}\\\
&=\sum_{m=0}^{\infty}(-1)^{m}\frac{x^{m}}{m!}\sum_{p=0}^{\infty}\bigg{(}\prod_{j=1}^{n}[(p+(j-1)(r-s))_{s}-(n-j)\lambda]\bigg{)}\frac{x^{p}}{p!}\\\
&=\sum_{k=0}^{\infty}\sum_{p=0}^{k}\frac{(-1)^{k-p}k!}{(k-p)!p!}\bigg{(}\prod_{j=1}^{n}[(p+(j-1)(r-s))_{s}-(n-j)\lambda]\bigg{)}\frac{x^{k}}{k!}\\\
&=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\sum_{p=0}^{k}(-1)^{p}\binom{k}{p}\bigg{(}\prod_{j=1}^{n}[(p+(j-1)(r-s))_{s}-(n-j)\lambda]\bigg{)}x^{k}.\end{split}$

Thus, by (27), we obtain the following theorem.

######  Theorem 4.

For $r\geq s\geq 1$ and $n\geq 1$, we have

(28)
$\begin{split}\frac{(-1)^{k}}{k!}\sum_{p=0}^{k}(-1)^{p}\binom{k}{p}\bigg{(}\prod_{j=1}^{n}[(p+(j-1)(r-s))_{s}&-(n-j)\lambda]\bigg{)}\\\
&=\left\\{\begin{split}&S_{\lambda}^{(r,s)}(n,k),\ \ {\rm{if}}\ 0\leq k\leq
ns,\\\ &0,\quad{\rm{if}}\ k>ns.\end{split}\right.\end{split}$

From (17) and (28), we note that

(29)
$\begin{split}S_{\lambda}^{(r,s)}(n,k)&=\frac{(-1)^{k}}{k!}\sum_{p=0}^{k}(-1)^{p}\binom{k}{p}\prod_{j=1}^{n}\big{[}(p+(j-1)(r-s))_{s}-(n-j)\lambda\big{]}\\\
&=\frac{(-1)^{k}}{k!}\sum_{p=0}^{k}(-1)^{p}\binom{k}{p}\prod_{j=0}^{n-1}\bigg{[}x^{r}\bigg{(}\frac{d}{dx}\bigg{)}^{s}-j\lambda
x^{r-s}\bigg{]}x^{p}\Big{|}_{x=1}\\\
&=\frac{(-1)^{k}}{k!}\prod_{j=0}^{n-1}\bigg{[}x^{r}\bigg{(}\frac{d}{dx}\bigg{)}^{s}-j\lambda
x^{r-s}\bigg{]}\sum_{p=0}^{k}(-1)^{p}\binom{k}{p}x^{p}\Big{|}_{x=1}\\\
&=\frac{(-1)^{k}}{k!}\prod_{j=0}^{n-1}\bigg{[}\bigg{(}x^{r}\bigg{(}\frac{d}{dx}\bigg{)}^{s}-j\lambda
x^{r-s}\bigg{)}\bigg{]}(1-x)^{k}\Big{|}_{x=1}.\end{split}$

Therefore, by (29), we obtain the following theorem.

######  Theorem 5.

For $r\geq s\geq 1$ and $n\geq 1$, we have

(30)
$\begin{split}S_{\lambda}^{(r,s)}(n,k)=\frac{(-1)^{k}}{k!}\prod_{j=0}^{n-1}\bigg{[}\bigg{(}x^{r}\bigg{(}\frac{d}{dx}\bigg{)}^{s}-j\lambda
x^{r-s}\bigg{)}\bigg{]}(1-x)^{k}\Big{|}_{x=1}.\end{split}$

From (30), we note that

$\begin{split}S_{2,\lambda}(n,k)=S_{\lambda}^{(1,1)}(n,k)&=\frac{(-1)^{k}}{k!}\prod_{j=0}^{n-1}\bigg{[}\bigg{(}x\frac{d}{dx}-j\lambda\bigg{)}\bigg{]}(1-x)^{k}\Big{|}_{x=1}\\\
&=\frac{(-1)^{k}}{k!}\bigg{(}x\frac{d}{dx}\bigg{)}_{n,\lambda}\sum_{p=0}^{k}\binom{k}{p}(-1)^{p}x^{p}\Big{|}_{x=1}\\\
&=\frac{(-1)^{k}}{k!}\sum_{p=0}^{k}\binom{k}{p}(-1)^{p}(p)_{n,\lambda},\quad(n\geq
1).\end{split}$

From (18), we note that

(31)
$\begin{split}\sum_{k=0}^{nr}S_{\lambda}^{(r,r)}(n,k)(x)_{k}=\prod_{k=1}^{n}[(x)_{r}-(n-k)\lambda]=((x)_{r})_{n,\lambda}.\end{split}$

Thus, by (31), we get

(32)
$\begin{split}((x)_{r})_{n,\lambda}=\sum_{k=0}^{nr}S_{\lambda}^{(r,r)}(n,k)(x)_{k}.\end{split}$

In particular, for $r=1$, we have

(33)
$\begin{split}\sum_{k=0}^{n}S_{\lambda}^{(1,1)}(n,k)(x)_{k}=(x)_{n,\lambda}=\sum_{k=0}^{n}S_{2,\lambda}(n,k)(x)_{k}.\end{split}$

Thus, by (33), we get

(34) $\begin{split}S_{\lambda}^{(1,1)}(n,k)=S_{2,\lambda}(n,k),\ \ (n,\ k\geq
0).\end{split}$

From (30), we note that

(35)
$\begin{split}S_{\lambda}^{(r,r)}(n,k)&=\frac{(-1)^{k}}{k!}\prod_{j=0}^{n-1}\bigg{(}x^{r}\bigg{(}\frac{d}{dx}\bigg{)}^{r}-j\lambda\bigg{)}\sum_{p=0}^{k}(-1)^{p}\binom{k}{p}x^{p}\Big{|}_{x=1}\\\
&=\frac{(-1)^{k}}{k!}\sum_{p=0}^{k}(-1)^{p}\binom{k}{p}((p)_{r})_{n,\lambda}.\end{split}$

Therefore, by (35), we obtain the following theorem.

######  Theorem 6.

For $r,\,n\geq 1$, we have

(36)
$\begin{split}S_{\lambda}^{(r,r)}(n,k)=\frac{(-1)^{k}}{k!}\sum_{p=0}^{k}(-1)^{p}\binom{k}{p}((p)_{r})_{n,\lambda}.\end{split}$

From (36), we note that

(37)
$\begin{split}S_{\lambda}^{(r,r)}(1,r)=\frac{(-1)^{r}}{r!}\sum_{p=r}^{r}\binom{r}{p}(p)_{r}(-1)^{p}=\frac{(-1)^{r}}{r!}\binom{r}{r}(-1)^{r}r!=1.\\\
\end{split}$

We recall that $\phi_{n,\lambda}$ and $S_{2,\lambda}(n,k)$ are related to
special quantum states, called coherent states, defined as linear combinations
of the eigenstates of the harmonic oscillator, $H=a^{\dagger}a$,
$H|n\rangle=n|n\rangle$, $\langle n|m\rangle=\delta_{m,n}$ and defined as
$|z\rangle=e^{-\frac{|z|^{2}}{2}}\sum_{n=0}^{\infty}\frac{z^{n}}{\sqrt{n}}|n\rangle$,
with $\langle z|z\rangle=1$, for complex $z$,
(see [3,7,11-14]).

From (17), we have

(38)
$\begin{split}\prod_{k=0}^{n-1}[(a^{\dagger})^{r-s}((a^{\dagger})^{s}a^{s}-k\lambda)]=(a^{\dagger})^{n(r-s)}\sum_{k=0}^{rs}S_{\lambda}^{(r,s)}(n,k)(a^{\dagger})^{k}a^{k}.\end{split}$

By (24) and (38), we get

(39) $\begin{split}\langle
z|e_{\lambda}^{a^{\dagger}a}(t)|z\rangle&=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\langle
z|(a^{\dagger}a)_{n,\lambda}|z\rangle\\\
&=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\langle
z|\prod_{k=1}^{n}(a^{\dagger}a-(n-k)\lambda)|z\rangle\\\
&=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\sum_{k=0}^{n}S_{2,\lambda}^{(1,1)}(n,k)(\overline{z})^{k}z^{k}\langle
z|z\rangle\\\
&=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\phi_{n,\lambda}^{(1,1)}(|z|^{2})=e^{|z|^{2}(e_{\lambda}(t)-1)}.\end{split}$

From (35) and (38), we note that

(40) $\begin{split}\langle
z|\prod_{k=0}^{n-1}[(a^{\dagger})^{r}a^{r}-k\lambda]|z\rangle&=\sum_{k=0}^{nr}S_{\lambda}^{(r,r)}(n,k)\langle
z|(a^{\dagger})^{k}a^{k}|z\rangle=\sum_{k=0}^{nr}S_{\lambda}^{(r,r)}(n,k)(\overline{z})^{k}z^{k}\langle
z|z\rangle\\\
&=\sum_{k=0}^{nr}S_{\lambda}^{(r,r)}(n,k)(|z|^{2})^{k}=\phi_{n,\lambda}^{(r,r)}(|z|^{2})\\\
&=\sum_{k=0}^{nr}\frac{(-1)^{k}}{k!}\sum_{p=0}^{k}(-1)^{p}\binom{k}{p}((p)_{r})_{n,\lambda}(|z|^{2})^{k}\\\
&=\sum_{p=0}^{nr}\sum_{k=p}^{nr}\frac{(-1)^{k-p}}{k!}\binom{k}{p}((p)_{r})_{n,\lambda}(|z|^{2})^{k}.\end{split}$

In particular, when $|z|=1$, we have

(41) $\begin{split}\langle
z|\prod_{k=0}^{n-1}[(a^{\dagger})^{r}a^{r}-k\lambda]|z\rangle&=\phi_{n,\lambda}^{(r,r)}=\sum_{k=0}^{nr}S_{\lambda}^{(r,r)}(n,k)\\\
&=\sum_{p=0}^{nr}\sum_{k=p}^{nr}\frac{(-1)^{k-p}}{k!}\binom{k}{p}((p)_{r})_{n,\lambda}.\end{split}$

Therefore, by (40) and (41), we obtain the following theorem.

######  Theorem 7.

For $r,\,n\geq 1$, we have

$\begin{split}\phi_{n,\lambda}^{(r,r)}(|z|^{2})=\langle
z|\prod_{k=0}^{n-1}[(a^{\dagger})^{r}a^{r}-k\lambda]|z\rangle=\sum_{p=0}^{nr}\sum_{k=p}^{nr}\frac{(-1)^{k-p}}{k!}\binom{k}{p}((p)_{r})_{n,\lambda}(|z|^{2})^{k}.\end{split}$

In particular, when $|z|=1$, we get

$\begin{split}\phi_{n,\lambda}^{(r,r)}=\langle
z|\prod_{k=0}^{n-1}[(a^{\dagger})^{r}a^{r}-k\lambda]|z\rangle=\sum_{p=0}^{nr}\sum_{k=p}^{nr}\frac{(-1)^{k-p}}{k!}\binom{k}{p}((p)_{r})_{n,\lambda}.\end{split}$

Now, we observe (40) that

(42) $\begin{split}\langle
z|e_{\lambda}^{(a^{\dagger})^{r}a^{r}}(t)|z\rangle&=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\langle
z|((a^{\dagger})^{r}a^{r})_{n,\lambda}|z\rangle\\\
&=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\langle
z|\prod_{k=0}^{n-1}[(a^{\dagger})^{r}a^{r}-k\lambda]|z\rangle\\\
&=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\phi_{n,\lambda}^{(r,r)}(|z|^{2}).\end{split}$

Therefore, by (42), we obtain the following theorem.

######  Theorem 8.

Let $r$ be a positive integer. Then the generating function of
$\phi_{n,\lambda}^{(r,r)}(|z|^{2})$ is given by

$\begin{split}\langle
z|e_{\lambda}^{(a^{\dagger})^{r}a^{r}}(t)|z\rangle=\sum_{n=0}^{\infty}\phi_{n,\lambda}^{(r,r)}(|z|^{2})\frac{t^{n}}{n!}.\end{split}$

We recall that the degenerate $r$-Stirling numbers of the second kind are
define by

(43) $\begin{split}(x+r)_{n,\lambda}=\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}(x)_{k},\ \ (n\geq 0),\ \ {(\text{see
\cite[cite]{[\@@bibref{}{8,12}{}{}]}})}.\end{split}$

The degenerate $r$-Bell polynomials are given by

(44) $\begin{split}\phi_{n,\lambda}^{(r)}(x)=\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}x^{k},\ \ (n\geq 0),\ \ {(\text{see
\cite[cite]{[\@@bibref{}{8,12}{}{}]}})}.\end{split}$

For $x=1$, $\phi_{n,\lambda}^{(r)}=\phi_{n,\lambda}^{(r)}(1)$ are called the
degenerated $r$-Bell numbers.

Now, we recall from [8] that

(45) $\begin{split}(a^{{\dagger}}a+r)_{n,\lambda}=\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}(a^{\dagger})^{k}a^{k},\ \ (n\geq 0).\end{split}$

From (45), we have

(46) $\begin{split}\langle
z|e_{\lambda}^{a^{{\dagger}}a+r}(t)|z\rangle&=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\langle
z|(a^{{\dagger}}a+r)_{n,\lambda}|z\rangle\\\
&=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}\langle z|(a^{\dagger})^{k}a^{k}|z\rangle\\\
&=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}(\overline{z})^{k}z^{k}\langle z|z\rangle\\\
&=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\sum_{k=0}^{n}{n+r\brace
k+r}_{r,\lambda}(|z|^{2})^{k}=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\phi_{n,\lambda}^{(r)}(|z|^{2})\\\
&=e_{\lambda}^{r}(t)e^{|z|^{2}(e_{\lambda}(t)-1)}.\end{split}$

Therefore, by (46), we obtain the following theorem.

######  Theorem 9.

Let $r$ be nonnegative integer. Then we have

$\begin{split}\sum_{n=0}^{\infty}\phi_{n,\lambda}^{(r)}(|z|^{2})\frac{t^{n}}{n!}=\langle
z|e_{\lambda}^{a^{{\dagger}}a+r}(t)|z\rangle=e_{\lambda}^{r}(t)e^{|z|^{2}(e_{\lambda}(t)-1)}.\end{split}$

Let
$g(t)=\sum_{n=0}^{\infty}\phi_{n,\lambda}^{(r)}(|z|^{2})\frac{t^{n}}{n!}=\langle
z|e_{\lambda}^{a^{{\dagger}}a+r}(t)|z\rangle=e_{\lambda}^{r}(t)e^{|z|^{2}(e_{\lambda}(t)-1)}$.
Then from the series expression we have

(47)
$\begin{split}\frac{dg(t)}{dt}=\sum_{n=0}^{\infty}\phi_{n+1,\lambda}^{(r)}(|z|^{2})\frac{t^{n}}{n!}.\end{split}$

From the expression of $g(t)$ in the bracket notation, we get

(48) $\begin{split}\frac{dg(t)}{dt}&=\langle
z|(a^{{\dagger}}a+r)e_{\lambda}^{a^{{\dagger}}a+r-\lambda}(t)|z\rangle\\\
&=e_{\lambda}^{-\lambda}(t)\Big{(}\langle
z|a^{{\dagger}}e_{\lambda}^{aa^{{\dagger}}+r}(t)a|z\rangle+r\langle
z|e_{\lambda}^{a^{{\dagger}}a+r}(t)|z\rangle\Big{)}\\\
&=e_{\lambda}^{-\lambda}(t)\Big{(}|z|^{2}\langle
z|e_{\lambda}^{a^{{\dagger}}a+r+1}(t)|z\rangle+r\langle
z|e_{\lambda}^{a^{{\dagger}}a+r}(t)|z\rangle\Big{)}\\\
&=e_{\lambda}^{-\lambda}(t)\Big{(}|z|^{2}\sum_{k=0}^{\infty}\phi_{k,\lambda}^{(r+1)}(|z|^{2})\frac{t^{k}}{k!}+r\sum_{k=0}^{\infty}\phi_{k,\lambda}^{(r)}(|z|^{2})\frac{t^{k}}{k!}\Big{)}\\\
&=\sum_{l=0}^{\infty}(-\lambda)^{l}l!\frac{t^{l}}{l!}\sum_{k=0}^{\infty}\Big{(}|z|^{2}\phi_{k,\lambda}^{(r+1)}(|z|^{2})+r\phi_{k,\lambda}^{(r)}(|z|^{2})\Big{)}\frac{t^{k}}{k!}\\\
&=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}\binom{n}{k}(-\lambda)^{n-k}(n-k)!\Big{(}|z|^{2}\phi_{k,\lambda}^{(r+1)}(|z|^{2})+r\phi_{k,\lambda}^{(r)}(|z|^{2})\Big{)}\bigg{)}\frac{t^{n}}{n!}.\end{split}$

From the last expression of $g(t)$, we obtain

(49)
$\begin{split}\frac{dg(t)}{dt}&=\Big{(}re_{\lambda}^{-\lambda}(t)+|z|^{2}e_{\lambda}^{1-\lambda}(t)\Big{)}e_{\lambda}^{r}(t)e^{|z|^{2}(e_{\lambda}(t)-1)}\\\
&=\sum_{k=0}^{\infty}\big{(}r(-\lambda)_{k,\lambda}+|z|^{2}(1-\lambda)_{k,\lambda}\big{)}\frac{t^{k}}{k!}\sum_{m=0}^{\infty}\phi_{m,\lambda}^{(r)}(|z|^{2})\frac{t^{m}}{m!}\\\
&=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}\big{(}r(-\lambda)_{k,\lambda}+|z|^{2}(1-\lambda)_{k,\lambda}\big{)}\phi_{n-k,\lambda}^{(r)}(|z|^{2})\frac{t^{n}}{n!}.\end{split}$

Now, from (47), (48) and (49), we obtain the next result.

######  Theorem 10.

Let $r,\,n$ be nonnegative integers. Then we have the following recurrence
relations:

$\begin{split}\phi_{n+1,\lambda}^{(r)}(|z|^{2})&=\sum_{k=0}^{n}\binom{n}{k}(-\lambda)^{n-k}(n-k)!\Big{(}|z|^{2}\phi_{k,\lambda}^{(r+1)}(|z|^{2})+r\phi_{k,\lambda}^{(r)}(|z|^{2})\Big{)}\\\
&=\sum_{k=0}^{n}\binom{n}{k}\big{(}r(-\lambda)_{k,\lambda}+|z|^{2}(1-\lambda)_{k,\lambda}\big{)}\phi_{n-k,\lambda}^{(r)}(|z|^{2}).\end{split}$

In particular, for $|z|=1$, we get

$\begin{split}\phi_{n+1,\lambda}^{(r)}&=\sum_{k=0}^{n}\binom{n}{k}(-\lambda)^{n-k}(n-k)!\Big{(}\phi_{k,\lambda}^{(r+1)}+r\phi_{k,\lambda}^{(r)}\Big{)}\\\
&=\sum_{k=0}^{n}\binom{n}{k}\big{(}r(-\lambda)_{k,\lambda}+(1-\lambda)_{k,\lambda}\big{)}\phi_{n-k,\lambda}^{(r)}.\end{split}$

Evaluating the left hand side of (40) by using the representation of the
coherent state in terms of the number states, we have

(50) $\begin{split}&\langle
z|((a^{\dagger})^{r}a^{r})_{k,\lambda}|z\rangle=\langle
z|\Pi_{l=0}^{k-1}[(a^{\dagger})^{r}a^{r}-l\lambda]|z\rangle\\\
&=e^{-\frac{|z|^{2}}{2}}e^{-\frac{|z|^{2}}{2}}\sum_{m,n=0}^{\infty}\frac{(\overline{z})^{m}(z)^{n}}{\sqrt{m!}\sqrt{n!}}((n)_{r})_{k,\lambda}\langle
m|n\rangle\\\
&=e^{-|z|^{2}}\sum_{n=0}^{\infty}\frac{(|z|^{2})^{n}}{n!}((n)_{r})_{k,\lambda}=e^{-|z|^{2}}\sum_{n=1}^{\infty}\frac{(|z|^{2})^{n}}{n!}((n)_{r})_{k,\lambda},\end{split}$

where $r$ and $k$ are positive integer.

Thus, from (40) and (50), we get the next theorem.

######  Theorem 11.

Let $r,\,k$ be positive integers. Then we have

$\begin{split}\phi_{k,\lambda}^{(r,r)}(|z|^{2})=e^{-|z|^{2}}\sum_{n=1}^{\infty}\frac{(|z|^{2})^{n}}{n!}((n)_{r})_{k,\lambda}.\end{split}$

In particular, when $|z|=1$, we get

$\begin{split}\phi_{k,\lambda}^{(r,r)}=\frac{1}{e}\sum_{n=1}^{\infty}\frac{1}{n!}((n)_{r})_{k,\lambda}.\end{split}$

Lastly, we introduce two notations given by

(51) $\begin{split}\langle\langle x\rangle\rangle_{0,\lambda}=1,\langle\langle
x\rangle\rangle_{n,\lambda}=\prod_{k=1}^{n}(x+(k-1)-(n-k)\lambda),\quad(n\geq
1).\end{split}$

and

(52)
$\begin{split}((x))_{0,\lambda}=1,((x))_{n,\lambda}=\prod_{k=1}^{n}(x-(k-1)+(n-k)\lambda),\quad(n\geq
1).\end{split}$

We may consider the unsigned degenerate Lah numbers defined by

(53) $\begin{split}\langle\langle
x\rangle\rangle_{n,\lambda}=\sum_{k=0}^{n}L_{\lambda}(n,k)(x)_{k},\quad(k\geq
0),\quad(n\geq 0).\end{split}$

In addition, the signed degenerate Lah numbers are given by

(54) $\begin{split}((x))_{n,\lambda}=\sum_{k=0}^{n}L_{\lambda}^{1}(n,k)\langle
x\rangle_{k},\quad(n\geq 0).\end{split}$

Note that $\lim_{\lambda\rightarrow 0}L_{\lambda}(n,k)=L(n,k),\quad(n,\ k\geq
0)$ are the ordinary Lah numbers given by $\langle
x\rangle_{n}=\sum_{k=0}^{n}L(n,k)(x)_{k}$, where $\langle
x\rangle_{0}=1,\langle x\rangle_{n}=x(x+1)\cdots(x+n-1),\quad(n\geq 1)$, (see
[3]). Moreover, $\lim_{\lambda\rightarrow 0}L_{\lambda}^{1}(n,k)=L^{1}(n,k)$,
where $L^{1}(n,k)$ are the signed Lah numbers given by
$(x)_{n}=\sum_{k=0}^{n}L^{1}(n,k)\langle x\rangle_{k}$. We also observe from
(18) and (53) that

$\begin{split}S_{\lambda}^{(2,1)}(n,k)=L_{\lambda}(n,k),\quad(n\geq
1).\end{split}$

##  3\. Conclusion

In recent years, studying degenerate versions of some special numbers and
polynomials has drawn the attention of many mathematicians and yielded many
interesting results. These degenerate versions include the degenerate Stirling
numbers of the first and second kinds, degenerate Bernoulli numbers of the
second kind and degenerate Bell numbers and polynomials.

In this paper, we introduced the generalized degenerate $(r,s)$-Stirling
numbers of the second kind and the generalized degenerate $(r,s)$-Bell
polynomials by considering the boson normal ordering of
$\prod_{k=0}^{n-1}\big{(}(a^{{\dagger}})^{r}a^{s}-k\lambda(a^{{\dagger}})^{r-s}\big{)}$.
We studied some properties, explicit expressions and generating functions of
those numbers and polynomials. They are degenerate versions of the
corresponding ones in the earlier works by Blasiak-Person-Solomon (see [4]).

These new numbers are expected to play an important role in the study of
various degenerate versions of many special polynomials and numbers, just as
the degenerate Stirling numbers of the second kind have played an important
role in that study.

We would like to continue to study various degenerate versions of many special
polynomials and numbers and their applications to physics, science and
engineering as well as to mathematics.

Acknowledgments
The authors thank Jangjeon Institute for Mathematical Science for the support
of this research.

Availability of data and material
Not applicable.

Funding
The third author is supported by the Basic Science Research Program, the
National Research Foundation of Korea, (NRF-2021R1F1A1050151).

Ethics approval and consent to participate
The authors declare that there is no ethical problem in the production of this
paper.

## References

  * [1] Aydin, M. S.; Acikgoz, M.; Araci, S. A new construction on the degenerate Hurwitz-zeta function associated with certain applications. Proc. jangjeon Math. Soc. 25 (2022), no. 2, 195-203.
  * [2] Blasiak, P.; Horzela, A.; Person, K. A.; Solomon, A. I.; Duchamp, G. H. E. Combinatorics and Boson normal ordering: a gentle introduction. Amer. J. Phys. 75 (2007), no. 7, 639-646.
  * [3] Blasiak, P.; Penson, K. A.; Solomon, A. I.; The general boson normal ordering problem. Phys. Lett. A 309 (2003), no. 3-4, 198-205.
  * [4] Carlitz, Leonard On Arrays of Numbers. Amer. J. Math. 54 (1932), no. 4, 739-752.
  * [5] Comtet, L. Advanced combinatorics. The art of finite and infinite expansions. Recised and enlarged edition. D. Reidel Publishing Co., Dordrecht, 1974.
  * [6] Katriel, J. Bell numbers and coherent states. Phys Lett A. 2000;273(3):159-161.
  * [7] Kim, T.; Kim, D. S.; Some Identities on Degenerate $r$-Stirling numbers via Boson Operators. Russ. J. Math. Phys. 29 (2022), no. 4, 508-517.
  * [8] Kim, T.; Kim, D. S. Degenerate $r$-Whitney numbers and degenerate $r$-Dowling polynomials via boson operators. Adv. in Appl. Math. 140 (2022), Paper No. 102394, 21 pp.
  * [9] Kim, T.; Kim, D. S. Some identities on truncated polynomials associated with degenerate Bell polynomials. Russ. J. Math. Phys. 28 (2021), no. 3, 342-355
  * [10] Kim, T.; Kim, D. S.; Dolgy, D. V.; Park, J.-W. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials. Open Math. 19 (2021), no. 1, 1588-1597.
  * [11] Kim, T.; Kim, D. S.; Kim, H. K. Normal ordering of degenerate integral oowers of number operator and its applications. Appl. Math. Sci. Eng. 30 (2022), no. 1, 440-447.
  * [12] Kim, T.; Kim, D. S.; Kim, H. K. Degenerate r-Bell polynomials arising from degenerate normal ordering. J. Math. 2022, Art. ID 2626249, 6 pp.
  * [13] Perelomov A. Generalized coherent states and their applications. Texts and Monographs in Physics. Berlin: Springer-Verlag; 1986.
  * [14] Roman, S. The umbral calculus. Pure and Applied Mathematics, 111. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984.
  * [15] Schork, M. Recent developments in combinatorial aspects of normal ordering. Enumer. Comb. Appl. 1 (2021), no. 2, Paper No. S2S2, 40 pp.
  * [16] Simsek, Y. On $q$-deformed Stirling numbers. Int. J. Math. Comput. 15 (2012), no. 2, 70-80.