# LARGE-EDDY SIMULATIONS OF TURBULENT COMPRESSIBLE SUPERSONIC JET FLOWS USING
DISCONTINUOUS GALERKIN METHODS

Diego F. Abreu1,†,∗ Carlos Junqueira-Junior2 Eron T. V. Dauricio1,‡ and João
Luiz F. Azevedo3 1 Instituto Tecnológico de Aeronáutica, 12228–900, São José
dos Campos, SP, Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS>and 2
Arts et Métiers Institute of Technology, DynFluid, CNAM, HESAM University, 151
Boulevard de l’Hôpital, 75013, Paris, France<EMAIL_ADDRESS>and 3
Instituto de Aeronáutica e Espaço, 12228–904, São José dos Campos, SP, Brazil,
<EMAIL_ADDRESS>
###### Abstract

In this work, a discontinuous Galerkin scheme is employed to perform LES
simulations of supersonic jet flows. A total of four simulations are performed
with different meshes and order of accuracy. The number of degrees of freedom
from the simulations varies from $50\times 10^{6}$ to $400\times 10^{6}$. The
results indicate that by increasing the resolution of simulation, in general,
the results got closer to experimental data. The jet lipline is the only
region in which this behavior is not observed. It investigated a procedure of
using lower-order simulations to initialize high-order simulations and
consequently reduce the total time of the simulation using high-order schemes.
This strategy is successful and allows to perform high-order simulations with
only $5\%$ more computational effort than a complete second-order simulation.

###### keywords:

Large-Eddy Simulation, Turbulent Flow, Jet Flow, Discontinuous Galerkin
Methods

Diego F. Abreu, Carlos Junqueira-Junior, Eron T. V. Dauricio and João Luiz F.
Azevedo

##  1 INTRODUCTION

The Reynolds-Averaged Navier-Stokes (RANS) formulation has difficulty
representing some types of fluid motions predominantly governed by free shear
flows or wall-bounded flows with separated boundary layers. This difficulty is
related to constructive assumptions of the formulation, characterized by the
modeling of all turbulent quantities. The recent progress of computational
power is enabling the employment of large-eddy simulations (LES) to simulate
the problems that RANS formulation fails to model important aspects of the
flow. Recent studies show the capability of LES simulations for reproducing
free shear layer [4, 21] and detached flows [13, 22]. Another advantage of
using LES is its capability to produce high-frequency unsteady information,
which is necessary for aerodynamics, acoustics, loads, and heat transfer
analyses.

The authors are interested in the simulation of supersonic jet flows for
performing aerodynamic analyses of the shear layer regarding velocity and
pressure fluctuations to improve the design of nozzles and adjacent
structures. Different numerical options are employed to obtain the solution of
LES formulation for jet flows. For example, low-order accuracy [18] and high-
order accuracy [3, 9] finite difference schemes on structured meshes were
employed to perform LES simulations of subsonic and supersonic jet flows. Low-
order finite volume approach on unstructured meshes [23, 6] is another option
employed.

Due to the employment of structured meshes, the finite difference schemes have
difficulty handling complex geometries. The finite volume schemes are applied
to unstructured meshes, which make it easier to represent complex geometries,
however, it is difficult to implement high-order discretizations with these
schemes [15]. In this context, the discontinuous Galerkin schemes are gaining
relevance, because they are easily implemented with high-order accuracy
discretizations and can be employed with unstructured meshes. Some work is
already simulating jet flows with discontinuous Galerkin schemes [1, 8] or
using similar strategies, for example, the Flux Reconstruction schemes [25].

The discontinuous Galerkin schemes have multiple options for implementation.
For example, one may choose to represent the solution by nodal or modal
polynomials. It is possible to choose between different options of test
functions that could better suits different types of elements, which are
utilized to discretize the computational domain. One set of choices for the
discontinuous Galerkin formulation is named discontinuous Galerkin spectral
element method (DGSEM) [19, 16]. The DGSEM, implemented in a numerical
framework called FLEXI [20], was investigated for performing LES of a
supersonic round jet flows with Mach number equal to $1.4$ and Reynolds number
based on jet inlet diameter of $1.58\times 10^{6}$ [1].

The simulations were performed with two numerical meshes with $6.2\times
10^{6}$ and $1.8\times 10^{6}$ elements with second-order and third-order
accurate discretizations, respectively. The two simulations were performed
with nearly $50\times 10^{6}$ degrees of freedom (DOF). They presented similar
results, with the simulation performed with third-order accuracy requiring
twice the time to perform the same simulation time as the second-order
accurate simulation. When comparing the results to experimental data,
excessive dissipation is observed, which led to shorter potential cores. The
potential core of the jet is the length in the centerline of the jet where the
velocity reaches $0.95$ of jet velocity. Other aspects of the flow, for
example, the root mean square (RMS) of velocity fluctuations in the centerline
and lipline of the jet, also presented some differences with experimental
data.

In this work, the results obtained using a new mesh are presented. The new
mesh has a larger refinement and improved topology than the meshes utilized in
previous work. The new mesh is simulated with second-order and third-order
accurate discretizations. Discussions regarding the quality and improvement of
the simulations are presented. A discussion of computational efficiency
utilizing discontinuous Galerkin methods is also performed to develop
guidelines for future works.

##  2 NUMERICAL FORMULATION

###  2.1 GOVERNING EQUATIONS

The work has an interest in the solution of the filtered Navier-Stokes
equations. The filtering strategy is based on a spatial filtering process that
separates the flow into a resolved part $\bar{(\cdot)}$ and a non-resolved
part $(\cdot)^{\prime}$. Implicit filter size is obtained from the mesh size.
The filtered Navier-Stokes equations in conservative form can be written by

$\frac{\partial\mathbf{\bar{Q}}}{\partial
t}+\nabla\cdot\mathbf{F}(\mathbf{\bar{Q}},\nabla\mathbf{\bar{Q}})=0,$ (1)

where
$\mathbf{\bar{Q}}=[\bar{\rho},\bar{\rho}\tilde{u},\bar{\rho}\tilde{v},\bar{\rho}\tilde{w},\bar{\rho}\check{E}]^{T}$
is the vector of filtered conserved variables and $\mathbf{F}$ is the flux
vector. The flux vector can be divided into the Euler fluxes and the viscous
flux, $\mathbf{F}=\mathbf{F}^{e}-\mathbf{F}^{v}$. The fluxes with the filtered
variables may be written as

$\mathbf{F}_{i}^{e}=\left[\begin{array}[]{c}\bar{\rho}\tilde{u}_{i}\\\
\bar{\rho}\tilde{u}\tilde{u}_{i}+\delta_{1i}\bar{p}\\\
\bar{\rho}\tilde{v}\tilde{u}_{i}+\delta_{2i}\bar{p}\\\
\bar{\rho}\tilde{w}\tilde{u}_{i}+\delta_{3i}\bar{p}\\\
(\bar{\rho}\check{E}+\bar{p})\tilde{u}_{i}\end{array}\right]\hskip
42.67912pt\mathbf{F}_{i}^{v}=\left[\begin{array}[]{c}0\\\ \tau_{1i}^{mod}\\\
\tau_{2i}^{mod}\\\ \tau_{3i}^{mod}\\\
\tilde{u}_{j}\tau_{ij}^{mod}-q_{i}^{mod}\end{array}\right]\hskip
42.67912pt\mbox{ , for }i=1,2,3,$ (2)

where $\tilde{u}_{i}$ or $(\tilde{u},\tilde{v},\tilde{w})$ are the Favre
averaged velocity components, $\bar{\rho}$ is the filtered density, $\bar{p}$
is the filtered pressure and $\bar{\rho}\check{E}$ is the filtered total
energy per unit volume. The terms $\tau_{ij}^{mod}$ and $q_{i}^{mod}$ are the
modified viscous stress tensor and heat flux vector, respectively, and
$\delta_{ij}$ is the Kronecker delta. The filtered total energy per unit
volume, according to the definition proposed by Vreman [28] in its ”system I”
approach, is given by

$\bar{\rho}\check{E}=\frac{\bar{p}}{\gamma-1}+\frac{1}{2}\bar{\rho}\tilde{u}_{i}\tilde{u}_{i}.$
(3)

The filtered pressure, Favre averaged temperature and filtered density are
correlated using the ideal gas equation of state
$\bar{p}=\bar{\rho}R\tilde{T}$, and $R$ is the gas constant, written as
$R=c_{p}-c_{v}$. The properties $c_{p}$ and $c_{v}$ are the specific heat at
constant pressure and volume, respectively. The modified viscous stress tensor
may be written as

$\tau_{ij}^{mod}=(\mu+\mu_{SGS})\left(\frac{\partial\tilde{u}_{i}}{\partial
x_{j}}+\frac{\partial\tilde{u}_{j}}{\partial
x_{i}}\right)-\frac{2}{3}(\mu+\mu_{SGS})\left(\frac{\partial\tilde{u}_{k}}{\partial
x_{k}}\right)\delta_{ij}$ (4)

where $\mu$ is the dynamic viscosity coefficient, calculated by Sutherland’s
Law, and $\mu_{SGS}$ is the SGS dynamic viscosity coefficient, which is
provided by the subgrid-scale model. The strategy of modeling the subgrid-
scale contribution as an additional dynamic viscosity coefficient is based on
the Boussinesq hyphotesis. The modified heat flux vector, using the same
modeling strategy, is given by

$q_{i}^{mod}=-(k+k_{SGS})\frac{\partial\tilde{T}}{\partial x_{i}}$ (5)

where $k$ is the thermal conductivity coefficient of the fluid and $k_{SGS}$
is the SGS thermal conductivity coefficient given by

$k_{SGS}=\frac{\mu_{SGS}c_{p}}{Pr_{SGS}}$ (6)

and $Pr_{SGS}$ is the SGS Prandtl number. The present work employs the static
Smagorinsky model [26] in order to calculate the subgrid-scale contribution.

###  2.2 NODAL DISCONTINUOUS GALERKIN METHOD

The nodal discontinuous Galerkin method used in this work is based on the
modeling called discontinuous Galerkin spectral element method [19, 16]. In
this modeling strategy, the domain is divided into multiple hexahedral
elements. This choice of elements permits the interpolating polynomial to be
defined as a tensor product basis with degree $N$ in each space direction.
This set of options leads to an algorithm that can be easily implemented and
also produce a high level of computational efficiency due to reduced
calculations.

In this method, the elements from the physical domain are mapped onto a
reference unit cube elements $E=[-1,1]^{3}$. The equations, presented in (1)
need also to be mapped to this new reference domain, leading to

$J\frac{\partial\mathbf{\bar{Q}}}{\partial
t}+\nabla_{\xi}\cdot\bar{\mathcal{F}}=0,$ (7)

where $\nabla_{\xi}$ is the divergence operator with respect to the reference
element coordinates, $\mathbf{\xi}=(\xi^{1},\xi^{2},\xi^{3})^{T}$,
$J=\arrowvert\partial\mathbf{x}/\partial\mathbf{\xi}\arrowvert$ is the
Jacobian of the coordinate transformation and $\bar{\mathcal{F}}$ is the
contravariant flux vector.

The discontinuous Galerkin formulation is obtained multiplying (7) by the test
function $\psi=\psi(\xi)$ and integrating over the reference element $E$

$\int_{E}J\frac{\partial\mathbf{\bar{Q}}}{\partial t}\psi
d\xi+\int_{E}\nabla_{\xi}\cdot\bar{\mathcal{F}}\psi d\xi=0.$ (8)

It is possible to obtain the weak form of the scheme by integrating by parts
the second term in (8)

$\frac{\partial}{\partial t}\int_{E}J\mathbf{\bar{Q}}\psi d\xi+\int_{\partial
E}(\bar{\mathcal{F}}\cdot\vec{N})^{*}\psi
dS-\int_{E}\bar{\mathcal{F}}\cdot(\nabla_{\xi}\psi)d\xi=0,$ (9)

where $\vec{N}$ is the unit normal vector of the reference element faces.
Because the discontinuous Galerkin scheme allows discontinuities in the
interfaces, the surface integral above is ill-defined. In this case, a
numerical flux, $\bar{\mathcal{F}}^{*}$, is defined, and a Riemann solver is
used to compute the value of this flux based on the discontinuous solutions
given by the elements sharing the interface.

For the nodal form of the discontinuous Galerkin formulation, the solution in
each element is approximated by a polynomial interpolation of the form

$\mathbf{\bar{Q}}(\xi)\approx\sum_{p,q,r=0}^{N}\mathbf{\bar{Q}}_{h}(\xi_{p}^{1},\xi_{q}^{2},\xi_{r}^{3},t)\phi_{pqr}(\xi),$
(10)

where $\mathbf{\bar{Q}}_{h}(\xi_{p}^{1},\xi_{q}^{2},\xi_{r}^{3},t)$ is the
value of the vector of conserved variables at each interpolation node in the
reference element and $\phi_{pqr}(\xi)$ is the interpolating polynomial. For
hexahedral elements, the interpolating polynomial is a tensor product basis
with degree N in each space direction

$\phi_{pqr}(\xi)=l_{p}(\xi^{1})l_{q}(\xi^{2})l_{r}(\xi^{3}),\hskip
10.0ptl_{p}(\xi^{1})=\prod_{\begin{subarray}{c}i=0\\\ i\neq
p\end{subarray}}^{N_{p}}\frac{\xi^{1}-\xi_{i}^{1}}{\xi_{p}^{1}-\xi_{i}^{1}}.$
(11)

The definitions presented are applicable to other two directions.

The numerical scheme used in the simulation additionally presents the split
formulation [24], with the discrete form [11], to enhance the stability of the
simulation. The split formulation is employed for Euler fluxes only. The
solution and the fluxes are interpolated and integrated at the nodes of a
Gauss-Lobatto Legende quadrature, which presents the summation-by-parts
property, that is necessary to employ the split formulation.

The Riemann solver used in the simulations is a Roe scheme with entropy fix
[14] to ensure that the second law of thermodynamics is respected, even with
the split formulation. For the viscous flux, since the discontinuous Galerkin
scheme is not suitable for discretizing the high order derivative operator,
the lifting scheme of Bassi and Rebay [2] is used, which is also known for
BR2. The time marching method chosen is a five-stage, fourth-order explicit
Runge-Kutta scheme [7]. The shock waves that appear in the simulation are
stabilized using the finite-volume sub-cell shock-capturing method [27]. The
shock indicator of Jameson, Schmidt, and Turkel [17] is utilized.

##  3 EXPERIMENTAL CONFIGURATION

The experimental work [5] provides a good characterization of the flow
properties for jet flows. Many configurations were analyzed. In this work, the
interest is to simulate the fully expanded free jet flow configuration with a
Mach number of $1.4$. In this configuration the jet flow has a static pressure
in the nozzle exit section that equals the ambient static pressure with a
supersonic velocity, for this reason, it is possible to avoid the use of
nozzle wall geometries and also the shock waves are weaker when compared to
other operating conditions.

The experimental apparatus for analyzed configuration is composed of a
convergent-divergent nozzle designed with the method of characteristics [5].
The nozzle exit diameter is $50.8$ mm. The Reynolds number based on nozzle
exit diameter is approximately $1.58\times 10^{6}$, which is large when
compared to other jet experiments available in the literature.

The data acquisition in the tests applies Time-Resolved Particle Image
Velocimetry (TRPIV) operated primarily with a $10$ kHz sample rate. The
experiment uses two sets of cameras, one positioned to capture the flow along
the nozzle centerline and the other positioned to capture the flow of the
mixing layer along the nozzle lipline.

##  4 NUMERICAL SETUP

###  4.1 GEOMETRY AND MESH CONFIGURATION

The geometry used for the calculations in the work presents a divergent shape
and axis length of $40D$, where $D$ is the jet inlet diameter and has external
diameters of $16D$ and $25D$. Figure 1 illustrates a 2-D representation of the
computational domain indicating the inlet surface in red, the far-field region
in blue, the lipline in gray, and the centerline in black.

The computational grids used in the work are named M-1, M-2, and M-3. The M-1
and M-2 meshes are adaptations of the mesh utilized in previous work [18] due
to the different restrictions of each computational code. The M-3 mesh is
generated with topological differences from M-1 and M-2 meshes. The M-3 mesh
topology presents a high refinement level around the jet inlet boundary
external diameter that transitions to a uniform distribution when moving
forward in the longitudinal direction. In addition to the new topology, the
M-3 mesh also presents a larger number of elements. The mesh generation uses a
multiblock strategy since the FLEXI solver only handles hexahedral elements.

Fig. 2 exhibits a cut plane of the M-2 and M-3 meshes. M-2 mesh is presented
to illustrate the topological differences between the two strategies. M-1 mesh
is not presented because it only differs in the number of elements from M-2
mesh. The M-1 and M-2 meshes have a total of $6.2\times 10^{6}$ and $1.8\times
10^{6}$ elements that are simulated with second and third-order accuracy,
respectively, resulting in simulations with $50\times 10^{6}$ DOF. The M-2
mesh has $15.4\times 10^{6}$ elements and is simulated with second and third-
order accuracy, resulting in approximately $120\times 10^{6}$ and $410\times
10^{6}$ DOF. All the meshes utilized in the work are generated with the GMSH
[12] generator.

Figure 1: 2-D schematic representation of the computational domain used on the
jet flow simulations.

(a) M-2 mesh.

(b) M-3 mesh.

Figure 2: Visualization of the half-plane longitudinal cut planes for the
meshes used in the present work.

###  4.2 BOUNDARY CONDITIONS

different reference states to characterize the jet inflow, $(\cdot)_{jet}$,
and the far-field, $(\cdot)_{ff}$. The inflow and the far-field surfaces are
indicated in Fig. 1 in red and blue, respectively. A weakly enforced solution
of a Riemann problem with a Dirichlet condition is enforced at the boundaries.
The flow is characterized as perfectly expanded and unheated, i.e.
$p_{jet}/p_{ff}=T_{jet}/T_{ff}=1$, where $p$ stands for pressure and $T$ for
temperature. The Mach number of the jet at the inlet is $M_{jet}=1.4$ and the
Reynolds number based on the diameter of the nozzle is $Re_{jet}=1.58\times
10^{6}$. A small velocity component with $M_{ff}=0.01$ in the streamwise
direction is imposed at the far-field to avoid numerical issues. A sponge zone
[10] is employed close to all far-field boundaries to dump any oscillation
that could reach the boundaries.

###  4.3 SIMULATION SETTINGS

A total of 4 simulations are compared in this work. The development of the
simulations utilized 3 different meshes with two orders of accuracy obtained
by changing the degree of the polynomial representing the solution. The S-1
simulation utilizes the M-1 mesh with second-order accuracy. The S-2
simulation utilizes the M-2 mesh with third-order accuracy. The S-3 and S-4
simulations utilize the M-3 mesh with second and third-order accuracy,
respectively. Table 1 summarizes the simulations performed and the total
number of degrees of freedom in each of them.

Table 1: Summary of simulations settings. Simulation | Meshes | Order of | DOF/cell | Cells | Total # of DOF
---|---|---|---|---|---
|  | Accuracy |  | ($10^{6}$) | ($10^{6}$)
S-1 | M-1 | 2nd order | 8 | $6.2$ | $\approx 50$
S-2 | M-2 | 3rd order | 27 | $1.8$ | $\approx 50$
S-3 | M-3 | 2nd order | 8 | $15.4$ | $\approx 120$
S-4 | M-3 | 3rd order | 27 | $15.4$ | $\approx 410$

###  4.4 CALCULATION OF STATISTICAL PROPERTIES

Two different approaches are taken to perform the 4 simulations. In the first
approach, utilized for S-1, S-2, and S-3 simulations, the procedure involves
three steps. The first one is to clean off the domain since the computation
starts with a quiescent flow initial condition. The simulations run three
flow-through times (FTT) to develop the jet flow. One FTT is the time required
for one particle with the jet velocity to cross the computational domain. In
the sequence, the simulations run an additional three FTT to produce a
statistically steady condition. Then, in the last step, data are collected
with a sample of approximately $250$ kHz for another FTT to obtain the
statistical properties of the flow. In the second approach, utilized for the
S-4 simulation, the solution obtained from the S-3 simulation is utilized as
the initial condition. The simulation is performed for 0.5 FTT to clean the
second-order accuracy solution and allow it to provide a third-order accuracy
solution. Then 2 additional FTT are simulated to extract data for the
analysis. The cost of S4 simulation is higher than other simulations and the
authors had some difficulties to stabilize the simulation, which consumed some
available computational resources, for this reason it was not possible to run
3 FTT to obtain the statistics.

The mean and the root mean square (RMS) fluctuations of properties of the flow
are calculated along the centerline, lipline, and different domain surfaces in
the streamwise direction. The centerline is defined as the line in the center
of the geometry $y/D=0$, whereas the lipline is a surface parallel to the
centerline and located at the nozzle diameter, $y/D=0.5$. The results from the
lipline are an azimuthal mean from six equally spaced positions. The four
surfaces in the streamwise positions are $x/D=2.5$, $x/D=5.0$, $x/D=10.0$, and
$x/D=15.0$. Fig. 3 illustrates a snapshot of the jet flow with the lines and
surfaces of data extraction. Mach number contours are presented in the figure.

Figure 3: Snapshot of the jet simulation with the two longitudinal lines and
three crossflow lines along which data is extracted. Mach number contours are
shown.

##  5 RESULTS

###  5.1 ANALYSIS OF NUMERICAL RESULTS

The results from S1, S2, S3, and S4 simulations are presented in this section,
which is divided into two parts to group different types of comparisons. In
the first part, contours of mean longitudinal velocity, RMS longitudinal
velocity fluctuation, and mean density are presented for each simulation. In
the second part, the distribution of mean longitudinal velocity and RMS of
longitudinal velocity fluctuation are presented along the jet centerline and
lipline for the four simulations and compared to experimental data. In the
final results, the mean longitudinal velocity, RMS of longitudinal velocity
fluctuation, RMS of radial velocity fluctuation, and shear-stress tensor are
presented in four spanwise lines for all the simulations and compared to
experimental data.

In the first part, three main aspects can be analyzed from the different
contours investigated and each contour is better to discuss one of the three
aspects. The length of the potential core cannot be directly assessed from
visual inspection, so the authors prefer to refer to the region of high
velocity that can be easily inspected from the results of mean longitudinal
velocity. The development of the shear layer can be visualized in all results,
however, the one in which its intensity can be better visualized is in the
results of RMS of longitudinal velocity fluctuation. The last aspect that can
be assessed from the contours is the development of the series of shock and
expansion waves in the early stages of the jet.

Figure 4 presents the contours for the mean longitudinal velocity for all
simulations. In Figs. 4a and 4b the contours of velocity are very similar,
with the high velocity region in Fig. 4b being slightly longer. Analyzing the
results in Fig. 4c, one can observe that the high-velocity region has
increased significantly when compared to previous results. The improvement in
the results obtained shows the importance of distributing elements where they
are necessary. Finally, in Fig. 4d, the results from S4 simulation are
presented. It is possible to observe that the high-velocity region is the
longest among all the simulations, which is indicative that it was lacking
resolution in previous simulations to adequately capture the development of
the jet flow by adding too much dissipation.

(a) S1 simulation.

(b) S2 simulation.

(c) S3 simulation.

(d) S4 simulation.

Figure 4: Contours of mean longitudinal velocity component on cutplane in
$z/D=0$.

In Fig. 5 the contours of RMS of longitudinal velocity fluctuation is
presented. Once more, the results presented in Figs. 5a and 5b from S1 and S2
simulations are similar, with the shear-layer development starting
approximately $1D$ far from the jet inlet section. Just after the initial of
the shear layer development, one can observe that the peak of RMS fluctuation
occurs, which can be related to the large difference between the velocities
and possibly the transition of the shear layer from laminar to turbulent. The
results presented in Fig. 5c from the S3 simulation have significant
differences from the other two previously discussed. The development of the
shear layer is starting closer to the jet inlet section with smaller peaks of
RMS of velocity fluctuation and with a smaller spreading. One can visualize
that the two mixing layers are crossing in the center of the jet farther in
Fig. 5c than in Figs. 5a and 5b even presenting a sooner development.
Analyzing the results in Fig. 5d, it can be observed that tendencies from the
previous investigation continued to increase, which means that the beginning
of the development of the shear layer got even closer to the jet inlet section
and the crossing of the two mixing layers is happening farther from jet inlet
section than the results from S3 simulation, Fig. 5c. S4 simulation is the one
with smaller spreading and early development of the shear layer among all the
simulations.

(a) S1 simulation.

(b) S2 simulation.

(c) S3 simulation.

(d) S4 simulation.

Figure 5: Contours of RMS of longitudinal velocity fluctuation on cutplane in
$z/D=0$.

Finally, in Fig. 6, the contours of mean density are presented for all
simulations. In these results, it is possible to better visualize the
development of the series of shocks and expansion waves. Different from what
has been observed in Figs. 4 and 5, where the results from S1 and S2
simulation are very similar, in Figs. 6a and 6b the results of mean pressure
for S1 and S2 simulations, one can observe clear differences regarding the
series of shock and expansion waves. In Fig. 6a only three sets of shocks and
expansion waves are clearly visible, while in Fig. 6b it is possible to
observe more than 6 sets. It is possible to observe also that the sets of
shocks and expansion waves from the S2 simulation are stronger than those from
the S1 simulation. Analyzing Fig. 6c, it is possible to observe that the S3
simulation produced even more sets of shocks and expansion waves than the S2
simulation, Fig. 6b, with larger intensity, that is evaluated by the level of
variation of density produced by the shocks and expansion waves. Another
aspect that can be observed is that the first set of shock and expansion waves
in the S3 simulation is occurring closer to the jet inlet section than in S2
and S1 simulations, and appears to be a relation to the first set of shock and
expansion waves with the beginning of the development of the shear layer. In
Fig. 6d the results from S4 simulation is presented. One can observe the
largest set of shock and expansion waves among all the simulations and also a
thinner representation of the shocks and expansion waves, which can be closely
related to the increased resolution of the simulation. It is also possible to
observe a reduction in the intensity of the sets of shock and expansion waves
when compared to the S3 simulation, Fig. 6c. The first part of this section is
concluded with the comparison of the contours of mean longitudinal velocity,
RMS of longitudinal velocity fluctuation, and mean density among the
simulations. In the second part, the numerical results are compared with
experimental data.

(a) S1 simulation.

(b) S2 simulation.

(c) S3 simulation.

(d) S4 simulation.

Figure 6: Contours of mean density on cutplane in $z/D=0$.

In the second part, in Fig. 7, the numerical results of all simulations are
compared to the experimental data. In Fig. 7a the distribution of mean
longitudinal velocity $<U>/U_{j}$ is presented in the centerline of the jet.
One can observe in the figure that results from S1 and S2 simulations are
almost equal. Results from the S3 simulation present a significant improvement
when compared to previously performed simulations and the S4 simulation could
almost capture the behavior observed in the experiments. In Fig. 7b the RMS of
longitudinal velocity fluctuation $u_{rms}/U_{j}$ is presented in the
centerline of the jet. It is possible to observe once more in these results
how well the resolution influenced the numerical results regarding the
proximity to experimental data. The differences between S1 and S2 simulations
are small, the S3 simulation got closer to experimental data and the S4
simulation once more presented the best match with experimental data. It can
be observed in the results from the S4 simulation a double peak that does not
appear in any other simulation or the experimental data. The authors believe
that this result is a consequence of the fewer FFTs of the simulation in which
data was gathered. Once more data could be used, authors believe that this
characteristic would disappear and only one peak would be formed.

While the results for the jet centerline present always improvements in the
simulations with increased resolution, in the lipline that behavior is not
always observed. In Fig. 7c, where the mean longitudinal velocity $<U>/U_{j}$
is presented in the lipline of the jet one may observe that far from the jet
inlet section, the increased resolution produced a monotonic improvement in
the numerical results, close to the jet inlet section, S3 simulation was the
one that could better capture experimental data. However, it is not from the
mean results where the greatest differences are observed. When analyzing the
distribution of the RMS of longitudinal velocity distribution along the
lipline of the jet, Fig. 7d, one can observe that monotonically the increased
resolution pushed the results away from experimental data. While in the
experimental data it is possible to observe a smooth growth of RMS of
longitudinal velocity fluctuation and almost a plateau from $x/D=5$ to
$x/D=15$, in all the simulations there is a sudden increase in the RMS of
longitudinal velocity fluctuation and after the peak, it is possible only to
observe a reduction on the values. The authors believe that the differences
observed in these results are related to the choice of the boundary condition
imposed for the jet inlet that represents neither the boundary layer profile
from the nozzle nor the turbulent intensity in the nozzle exit section.

(a) Centerline

(b) Centerline

(c) Lipline

(d) Lipline

Figure 7: Results of mean streamwise velocity component distribution (left)
and RMS of streamwise velocity fluctuation (right) in the jet centerline
$y/D=0$ (top) and lipline $y/D=0.5$ (bottom).

The results in Fig. 8 present different statistical properties of the flow in
different longitudinal positions. The first set of results, in Figs. 8a to 8d,
concerns the mean of longitudinal velocity. The S1 simulation is in agreement
with the experimental data at $x/D=2.5$, Fig. 8a. In the position $x/D=5$,
Fig. 8b all the simulations produce very similar results. Moving forward and
analyzing the results in position $x/D=10$, Fig. 8c it is possible to observe
a similar behavior between S1 and S2 simulations, S3 simulation presenting
improvements to the other two and S4 simulation presenting the best match with
experimental data. In the last position $x/D=15$, Fig. 8a, a monotonically
improvement is observed with increased resolution the the simulations.

(a) $x/D=2.5$

(b) $x/D=5$

(c) $x/D=10$

(d) $x/D=15$

(e) $x/D=2.5$

(f) $x/D=5$

(g) $x/D=10$

(h) $x/D=15$

(i) $x/D=2.5$

(j) $x/D=5$

(k) $x/D=10$

(l) $x/D=15$

(m) $x/D=2.5$

(n) $x/D=5$

(o) $x/D=10$

(p) $x/D=15$

Figure 8: Profiles of mean streamwise velocity component, RMS of streamwise
velocity fluctuation, RMS of radial velocity fluctuation, and mean shear-
stress tensor component (from top to bottom) at four streamwise positions
$x/D=2.5$, $x/D=5$, $x/D=10$ and $x/D=15$ (from left to right).

The profiles of RMS values of longitudinal velocity fluctuation are presented
in Figs. 8e to 8h. The simulation results at $x/D=2.5$, Fig. 8e, present a
similar profile with differences only in the peak of RMS of longitudinal
velocity fluctuation with a monotonically decrease in the peak with increased
resolution and proximity to experimental data. Similar behavior is observed in
the next position at $x/D=5$, Fig. 8f. In the position $x/D=10$, Fig. 8g the
main aspects of the flow are captured except for the smaller values of RMS of
longitudinal velocity fluctuation in the center of the jet that is only badly
captured by S4 simulation. The results for all simulations and experimental
data are very similar in the last position $x/D=15$, Fig. 8h

Profiles of RMS values of radial velocity component fluctuation are presented
in Figs. 8i to 8l. They exhibit similar behavior as the longitudinal velocity
fluctuation. One can also observe the positive effects of the increased
resolution on the profiles of the mean shear-stress tensor component, Figs. 8m
to 8p. The profiles from the S4 simulation are in good agreement with the
experimental data and they indicate considerable improvement when compared to
the simulations with smaller resolution.

This result concludes our analysis of the numerical results from the
simulations. It was possible to observe, in general, that the improved
resolution of the simulations produced better results compared to experimental
data. The simulation with the highest resolution, the S4 simulation, was the
one that better matched the experimental data. Only in the jet lipline, this
behavior is not observed. The authors have strong confidence that this has
nothing to do with the effect of the resolution, instead, it may be strongly
related to the choice of the boundary condition for the jet inlet condition,
that does not reproduce the boundary layer developed inside the nozzle nor the
turbulent intensity in the region. To improve the quality of the simulations
in the jet lipline, a new boundary condition or a different simulation
strategy should be adopted.

###  5.2 ANALYSIS OF COMPUTATIONAL EFFORT

At this point of the work, it is important to discuss some other aspects of
computational effort to be able to improve the computational efficiency of the
simulations. The main parameter utilized to measure the efficiency of a
simulation is the Performance Index $PID$, which can be calculated by

$PID=\frac{wall\hskip 2.0ptclock\hskip 2.0pttime\hskip
3.0ptn_{cores}}{n_{DOF}\hskip 3.0ptn_{time\hskip 1.0ptsteps}\hskip 3.0ptn_{RK-
stages}},$ (12)

where $wall\hskip 2.0ptclock\hskip 2.0pttime$ is the time the simulation
needed to perform $n_{time\hskip 1.0ptsteps}$ time steps, $n_{cores}$ is the
quantity of cores used in the simulation, $n_{DOF}$ is the number of DOF of
the simulation and $n_{RK-stages}$ is the number of stages from the Runge-
Kutta scheme. The PID was calculated for all four simulations and the results
are presented in Tab. 2.

Table 2: Summary of Performance Index $PID$ from all simulations. Simulation | PID ($\mu s$)
---|---
S1 | $8$
S2 | $15$
S3 | $5$
S4 | $2$

It is important to clarify to the author that the numerical solver presented
some improvements during the execution of the simulations and they can be
related to the improvements in the $PID$ reduction from S3 and S4 simulations
compared to S1 and S2 simulations. It is also possible to argue that with the
increased number of degrees of freedom it is expected that more computation is
performed with a similar number of cores, which also should increase the
efficiency of the simulations. If we compare only S1 and S2 simulations, for
the same number of DOF, the third-order accurate simulations cost almost twice
the effort of a second-order accurate simulation. If these values are employed
in the two simulations procedure performed in this analysis, the total cost of
the simulation can be compared.

The first simulation procedure involves the whole calculation of the 9 FTT
with a mesh of $50\times 10^{6}$ elements that produces $\approx 400\times
10^{6}$ DOF when simulated with second-order accurate discretization, which
could produce a very similar result to those of S4 simulation. If the
procedure for S4 simulation could be completely performed, it could initially
start its 5 FTT with second-order accurate discretization with a total of
$120\times 10^{6}$ DOF. At this point, we do not consider the effect of the
number of degrees of freedom in the PID, only the order of discretization. For
this 5 FTT, the total time would be $3.34\times$ smaller than those to perform
the first simulation procedure due to the reduced number of degrees of
freedom. Then, it is possible to consider the next 4 FTT calculated with
third-order accurate discretization and a total of $\approx 400\times 10^{6}$
DOF. In this 4FTT the cost of the third-order accurate discretization is twice
of the second-order accurate simulation. If the total time of the second
procedure is calculated, it costs $\approx 5\%$ more than the procedure with
second-order accuracy.

What is discussed here is that, once it is possible to start the high-order
simulation with a previous result from another order of accuracy with the same
mesh, it is possible to reduce the time required to obtain the desired data
with high-order simulation and consequently reduce the cost of the total
simulation. In the proposed procedure, the cost of the total third-order
accurate simulation was only $5\%$ larger than those of a second-order
accurate simulation. This result is expressive and very interesting for high-
order simulations.

Another important point to present is that the computational code utilized
presented very good scalability with the number of cores. The tests for the S4
simulation varied the number of cores from a few hundred cores to a few
thousand cores and the $PID$ was always close to $2\mu s$.

##  6 CONCLUDING REMARKS

In this work, the employment of a discontinuous Galerkin framework called
FLEXI was investigated for the LES simulation of supersonic free round jets. A
total of four simulations are performed with 3 different meshes and second and
third-order accuracy. The range of degrees of freedom from the simulations
varies from $50\times 10^{6}$ to $400\times 10^{6}$. All the simulations are
performed for the same geometric model and with the same boundary conditions.

The results of the simulations are firstly compared visually between
themselves to compare how they are capturing the main features of the flow:
extension of the high-velocity region, development of the shear layer, and
development of the sets of shocks and expansion waves. The results showed that
with increased resolution the high-velocity regions got longer. The
development of the shear layer starts closer to the jet inlet section and
presents a smaller spreading. The number of sets of shocks and expansion waves
increased and the visual of the shocks and expansion waves are thinner.

The numerical results showed that, in general, the increase in the resolution
of the simulation, especially the number of degrees of freedom, produced
better results when compared to experimental data. This behavior is observed
in the results of mean longitudinal velocity distribution and RMS of
longitudinal velocity fluctuation distribution in the centerline. It is also
observed in the results from the four spanwise planes. The only results that
do not follow this behavior are the mean longitudinal and RMS of longitudinal
velocity fluctuation distributions along the jet lipline. In these regions,
with increased resolutions, the results are pushed away from experimental
data. The authors believe this behavior is related to the lack of resolution
from the jet inlet boundary condition that could not reproduce the boundary
layer and turbulent intensity from the experiments.

The analysis of the computational effort of the simulation showed that even
utilizing a high-order method that costs more than a second-order method for
the same number of degrees of freedom it was possible to reproduce a third-
order simulation with only $5\%$ more computational cost of total simulation
by initializing the simulation with a smaller order of accuracy.

The work reached is objective of identifying the guidelines for performing LES
simulations of supersonic jet flows using a discontinuous Galerkin scheme with
adequate results with a reasonable computational cost. The open point on the
jet inlet condition is the next step in the development of the work.

## ACKNOWLEDGMENTS

The authors acknowledge the support for the present research provided by
Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, under the
Research Grant No. 309985/2013-7. The work is also supported by the
computational resources from the Center for Mathematical Sciences Applied to
Industry, CeMEAI, funded by Fundação de Amparo à Pesquisa do Estado de São
Paulo, FAPESP, under the Research Grant No. 2013/07375-0. The authors further
acknowledge the National Laboratory for Scientific Computing (LNCC/MCTI,
Brazil) for providing HPC resources of the SDumont supercomputer. This work
was also granted access to the HPC resources of IDRIS under the allocation
2020-A0092A12067 / 2021-A0112A12067 made by GENCI. The first author
acknowledges authorization by his employer, Embraer S.A., which has allowed
his participation in the present research effort. The doctoral scholarship
provide by FAPESP to the third author, under the Grant No. 2018/05524-1, is
thankfully acknowledged. Additional support to the fourth author under the
FAPESP Research Grant No. 2013/07375-0 is also gratefully acknowledged. This
study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior - Brasil (CAPES) - Finance Code 001.

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# Analysis of Bell Based Euler polynomial and their Application

Nabiullah Khan and Saddam Husain Nabiullah Khan: Department of Applied
Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University,
Aligarh 202002, India<EMAIL_ADDRESS>Saddam Husain: Department of
Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim
University, Aligarh 202002, India<EMAIL_ADDRESS>
###### Abstract.

In the present article, we study Bell based Euler polynomial of order $\alpha$
and investigate some useful correlation formula, summation formula and
derivative formula . Also, we introduce some relation of string number of the
second kind. Moreover, we drive several important formulae of bell based Euler
polynomial by using umbral calculus.

###### Key words and phrases:

Bell polynomial, Euler polynomial, String polynomial, String number of second
kind, Sheffer sequence

###### 2010 Mathematics Subject Classification:

11B68, 33B15, 33C05, 33C10, 33C15, 33C45, 33E20

##  1\. Introduction

The polynomials and number play an important role in the multifarious area of
science such as mathematics, applied science, physics and engineering sciences
and some related research area involving number theory, quantum mechanics,
differential equation and mathematical physics (see [1, 3, 4]). Some of the
most important polynomials are Bell polynomial, Genocchi polynomial, Euler
polynomial, Bernaulii polynomial and Hermite polynomial.

We know that, properties of polynomials with the help of umbral calculus
studied by many authors. Kim et al. [11] study some properties of Euler
polynomials by using umbral Calculus and give some useful identities of Euler
polynomials. Kim et al. [12] study Bernoulli, Euler and Abel polynomials by
using umbral calculus and gives interesting identities of it. Kim et al. [14]
investigate useful identities for partially degenerate Bell Number and
polynomials associated with umbral calculus and Kim et al. [15] drived some
important identities and properties of umbral calculus associated with
degenerate orderd Bell number and polynomials.

Recently, S. Araci et al. [7] defined a Bell based Bernaoulli polynomials.
Motivated by above mention work in the present article we introduce Bell based
Euler polynomials and investigate some useful correlation formula, summation
formula and derivative formula of Bell based Euler polynomial of order
$\alpha$. Also, we acquire some implicit summation formula and some special
cases of Bell based Euler polynomials. Further, we drived some interesting
result of Bell based Euler polynomials associate with umbral calculus.

##  2\. Preliminaries

In the present paper, we take the symbols
$\mathbb{Z},\mathbb{N},\mathbb{N}_{0},\mathbb{R},$ and $\mathbb{C}$ to be set
of integer, set of natural number, set of non negative integers, set of real
number and set of complex number respectively.

The bivariate Bell polynomials are describe by the following generating
function, which is defined as follows:

$\sum_{n\geq
0}{\mathcal{B}_{n}}(x;y)\frac{{t}^{n}}{n!}=e^{x{t}}\,e^{y(e^{{t}}-1)}.$ (2.1)

When we take $x=0$, ${\mathcal{B}_{n}}(0;y)={\mathcal{B}_{n}}(y)$ are called
classical Bell polynomials (or exponential polynomials) which is describe by
following generating function (see [2, 3, 4, 13, 17]) which is defined as
follows:

$\sum_{n\geq 0}{\mathcal{B}_{n}}(y)\frac{{t}^{n}}{n!}=e^{y(e^{{t}}-1)}.$ (2.2)

If we take $y=1$ in (2.2) i.e.
${\mathcal{B}_{n}}(0;1)={\mathcal{B}_{n}}(1)={\mathcal{B}_{n}}$ are called
Bell number which is defined as follows (see [2, 3, 4, 13, 17]):

$\sum_{n\geq 0}{\mathcal{B}_{n}}\frac{{t}^{n}}{n!}=e^{(e^{{t}}-1)}.$ (2.3)

The generating function of an Euler polynomials of order $\alpha$ (see[8, 9,
10, 11, 18, 19]) is as folows:

$\sum_{n\geq
0}\mathcal{E}_{n}^{(\alpha)}(x)\frac{{t}^{n}}{n!}=e^{x{t}}\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}\,\,\,\,\,\,\,(|{t}|<2\pi).$
(2.4)

If we take $x=0$ in (2.4) i.e.
$\mathcal{E}_{n}^{(\alpha)}(0)=\mathcal{E}_{n}^{(\alpha)}$ are called Euler
number (see[8, 9, 10, 11, 18, 19]) which is shown as follows:

$\sum_{n\geq
0}\mathcal{E}_{n}^{(\alpha)}\frac{{t}^{n}}{n!}=\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}.$
(2.5)

The generating function of second kind string polynomials
$\mathcal{S}_{2}(n,k;x)$ and string number $\mathcal{S}_{2}(n,k)$ are defined
as (see[2, 3]):

$\sum_{n\geq
0}\mathcal{S}_{2}(n,k;x)\frac{{t}^{n}}{n!}=\left(\frac{e^{{t}}-1}{k!}\right)^{k}e^{{t}x}.$
(2.6)

When $x=0$ in (2.6) i.e. $\mathcal{S}_{2}(n,k;0)=\mathcal{S}_{2}(n,k)$ are
called string number and defined by following exponential generating function
(see[2, 3]):

$\sum_{n\geq
0}\mathcal{S}_{2}(n,k)\frac{{t}^{n}}{n!}=\left(\frac{e^{{t}}-1}{k!}\right)^{k}.$
(2.7)

##  3\. Bell based Euler polynomials and Number

In this segment, we introduced Bell based Euler polynomials of order $\alpha$
and investigate numerous correlation formulae like implicit summation
formulae, derivative formulae.

For any $n\in\mathbb{N}$ and $\alpha\in\mathbb{C}$, we define Bell based Euler
polynomial of order $\alpha$ as:

$\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)\frac{{t}^{n}}{n!}=\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{{x{t}}+y(e^{t}-1)}\,\,\,\,\,\,\,(|{t}|<2\pi).$
(3.1)

If x=0 and y=1 in (3.1) then we get a Bell Based Euler number of order
$\alpha$, which is defined as follows:

$\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}\frac{{t}^{n}}{n!}=\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{(e^{t}-1)}\,\,\,\,\,\,\,(|{t}|<2\pi).$
(3.2)

###  3.1. Special Cases:

In this section, we introduce some special types of Bell based Euler
polynomials of order $\alpha$, which is obtain by putting particular value in
(3.1) and defined as follows:

  1. (1)

If we choose x=0 in (3.1), we get Bell based Euler polynomials of order
$\alpha$, which are an extension of Euler polynomials of order $\alpha$
defined in (2.4) as follows:

$\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(y)\frac{{t}^{n}}{n!}=\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{y(e^{t}-1)}.$

  2. (2)

In case y=0 in (3.1) the Bell based Euler polynomials of order $\alpha$
reduced to the famililar Euler polynomials $\mathcal{E}_{n}^{(\alpha)}(x)$ of
order $\alpha$ defined in (2.4)

$\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x)\frac{{t}^{n}}{n!}=\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{x{t}}.$

  3. (3)

In case y=0 and $\alpha$=1 in (3.1) the Bell Based Euler polynomials
${{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)$ reduced to usual Euler
polynomials $\mathcal{E}_{n}(x)$ defined as:

$\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}}(x)\frac{{t}^{n}}{n!}=\left(\frac{2}{e^{{t}}+1}\right)e^{x{t}}.$

######  Theorem 3.1.

The following relation hold true for $\alpha\in\mathbb{C}$ and
$n\in\mathbb{N}$;

${{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)=\sum\limits_{k=0}^{n}\binom{n}{k}{{\mathcal{E}}_{k}^{(\alpha)}}(x){\mathcal{B}}_{n-k}(y).$
(3.3)

###### Proof.

By using relation (3.1), we have

$\displaystyle\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)\frac{{t}^{n}}{n!}=$
$\displaystyle\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{{x{t}}+y(e^{t}-1)}$
$\displaystyle=$
$\displaystyle\left\\{\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{x{t}}\right\\}\left\\{e^{y(e^{t}-1)}\right\\}$
$\displaystyle=$ $\displaystyle\left\\{\sum_{k\geq
0}{\mathcal{E}_{k}^{(\alpha)}}(x)\frac{{t}^{k}}{k!}\right\\}\left\\{\sum_{n\geq
0}{\mathcal{B}_{n}}(y)\frac{{t}^{n}}{n!}\right\\}.$

Now, using series rearrangement method, we get

$\displaystyle\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)\frac{{t}^{n}}{n!}=$
$\displaystyle\sum_{n\geq
0}\left\\{\sum\limits_{k=0}^{n}\binom{n}{k}{{\mathcal{E}}_{k}^{(\alpha)}}(x){\mathcal{B}}_{n-k}(y)\right\\}\frac{{t}^{n}}{n!}.$

By equating same power of t both side, we get desired result (3.3). ∎

######  Theorem 3.2.

Bell based Euler polynomials of order $\alpha$ satisfy following relation for
any $\alpha\in\mathbb{C}$ and $n\in\mathbb{N}$:

${{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)=\sum\limits_{k=0}^{n}\binom{n}{k}{{\mathcal{E}}_{k}^{(\alpha)}}{\mathcal{B}}_{n-k}(x;y).$
(3.4)

###### Proof.

By using generating function (3.1), we have

$\displaystyle\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)\frac{{t}^{n}}{n!}=$
$\displaystyle\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{{x{t}}+y(e^{t}-1)}$
$\displaystyle=$
$\displaystyle\left\\{\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}\right\\}\left\\{e^{x{t}+y(e^{t}-1)}\right\\}$
$\displaystyle=$ $\displaystyle\left\\{\sum_{k\geq
0}{\mathcal{E}_{k}^{(\alpha)}}\frac{{t}^{k}}{k!}\right\\}\left\\{\sum_{n\geq
0}{\mathcal{B}_{n}}(x;y)\frac{{t}^{n}}{n!}\right\\}$

After applying series rearrangement technique and compare same power of t, we
get desired result (3.4). ∎

######  Theorem 3.3.

If $\alpha\in\mathbb{C}$ and $n\in\mathbb{N}$, following relation hold true;

${{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)=\sum\limits_{k=0}^{n}\binom{n}{k}{{\mathcal{E}}_{k}^{(\alpha)}}(y)\,\,x^{n-k}.$
(3.5)

###### Proof.

Using relation (3.1), we have

$\displaystyle\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)\frac{{t}^{n}}{n!}=$
$\displaystyle\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{{x{t}}+y(e^{t}-1)}$
$\displaystyle=$
$\displaystyle\left\\{\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{y(e^{{t}}-1)}\right\\}\left\\{e^{x{t}}\right\\}$
$\displaystyle=$ $\displaystyle\left\\{\sum_{k\geq
0}{{}_{\mathcal{B}}}{\mathcal{E}_{k}^{(\alpha)}}(y)\frac{{t}^{k}}{k!}\right\\}\left\\{\sum_{n\geq
0}\frac{(x{t})^{n}}{n!}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\sum_{n\geq 0}\sum_{k\geq
0}{{}_{\mathcal{B}}}{\mathcal{E}_{k}^{(\alpha)}}(y)\,\,\frac{x^{n}}{n!}\frac{{t}^{n+k}}{k!}\right\\}.$

By using series rearrangement, we get

$\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)\frac{{t}^{n}}{n!}=\sum_{n\geq
0}\left\\{\sum_{k=0}^{n}\binom{n}{k}{{}_{\mathcal{B}}}{\mathcal{E}_{k}^{(\alpha)}}(y)\,\,x^{n-k}\right\\}\frac{{t}^{n}}{n!}.$

By equating same power of t both side, we get desired result (3.5). ∎

##  4\. Implicit summation formulae

In this section, we discus some useful implicit summation formulae for Bell
Based Euler polynomials of order $\alpha$, which is defined in following
theorem as follows:

######  Theorem 4.1.

If $\alpha_{1},\alpha_{2}\in\mathbb{C}$ and $n\in\mathbb{N}$, following
relation hold true;

${{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha_{1}+\alpha_{2})}}(x_{1}+x_{2};y_{1}+y_{2})=\sum\limits_{k=0}^{n}\binom{n}{k}{{}_{\mathcal{B}}{\mathcal{E}}_{k}^{(\alpha_{1})}}(x_{1};y_{1})\,\,{{}_{\mathcal{B}}}{\mathcal{E}}^{(\alpha_{2})}_{n-k}(x_{2};y_{2}).$
(4.1)

###### Proof.

Using the following identity

$\left(\frac{2}{e^{{t}}+1}\right)^{\alpha_{1}+\alpha_{2}}e^{{(x_{1}+x_{2})t}+(y_{1}+y_{2})(e^{t}-1)}$

$=\left\\{\left(\frac{2}{e^{{t}}+1}\right)^{\alpha_{1}}e^{x_{1}{t}}+y_{1}(e^{t}-1)\right\\}\left\\{\left(\frac{2}{e^{{t}}+1}\right)^{\alpha_{2}}e^{x_{2}{t}}+y_{2}(e^{t}-1)\right\\}.$

By using generating function (3.1), we have

$\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha_{1}+\alpha_{2})}}(x_{1}+x_{2};y_{1}+y_{y})\frac{{t}^{n}}{n!}$
$\displaystyle=$
$\displaystyle\left(\frac{2}{e^{{t}}+1}\right)^{\alpha_{1}+\alpha_{2}}e^{{(x_{1}+x_{2}){t}}+(y_{1}+y_{2})(e^{t}-1)}$
$\displaystyle=$
$\displaystyle\left\\{\left(\frac{2}{e^{{t}}+1}\right)^{\alpha_{1}}e^{x_{1}{t}}+y_{1}(e^{t}-1)\right\\}\left\\{\left(\frac{2}{e^{{t}}+1}\right)^{\alpha_{2}}e^{x_{2}{t}}+y_{2}(e^{t}-1)\right\\}$
$\displaystyle=$ $\displaystyle\left\\{\sum_{k\geq
0}{{}_{\mathcal{B}}}{\mathcal{E}_{k}^{(\alpha_{1})}}(x_{1};y_{1})\frac{{t}^{k}}{k!}\right\\}\left\\{\sum_{n\geq
0}{{}_{\mathcal{B}}}{\mathcal{E}_{n}^{(\alpha_{2})}}(x_{2};y_{2})\frac{{t}^{n}}{n!}\right\\}$
$\displaystyle=$ $\displaystyle\left\\{\sum_{n\geq 0}\sum_{k\geq
0}{{}_{\mathcal{B}}}{\mathcal{E}_{k}^{(\alpha_{1})}}(x_{1};y_{1})\,\,{{}_{\mathcal{B}}}{\mathcal{E}_{n}^{(\alpha_{2})}}(x_{2};y_{2})\,\,\frac{{t}^{n+k}}{n!k!}\right\\}$

Using series rearrangement technique, we obtain

$\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha_{1}+\alpha_{2})}}(x_{1}+x_{2};y_{1}+y_{y})\frac{{t}^{n}}{n!}=\sum_{n\geq
0}\left\\{\sum_{k=0}^{n}\binom{n}{k}{{}_{\mathcal{B}}}{\mathcal{E}_{k}^{(\alpha_{1})}}(x_{1};y_{1})\,\,{{}_{\mathcal{B}}}{\mathcal{E}_{n-k}^{(\alpha_{2})}}(x_{2};y_{2})\,\,\right\\}\frac{{t}^{n}}{n!}$

Now, equating same power of ${t}$ both side, we get desired result (4.1). ∎

######  Remark 4.1.

In case, if we choose $\alpha_{1}=\alpha$, $\alpha_{2}=0$, $x_{1}=x$,
$x_{2}=1$, $y_{1}=y$ and $y_{2}=0$ in (4.1), we have

${{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x+1;y)=\sum\limits_{k=0}^{n}\binom{n}{k}{{}_{\mathcal{B}}{\mathcal{E}}_{k}^{(\alpha)}}(x;y),$
(4.2)

which is an extension of Euler polynomials defined by

${\mathcal{E}_{n}}(x+1)=\sum\limits_{k=0}^{n}\binom{n}{k}{\mathcal{E}_{k}}(x;y).$
(4.3)

######  Theorem 4.2.

If $\alpha\in\mathbb{C}$ and $n\in\mathbb{N}$, Bell bassed Euler polynomials
of order $\alpha$ satisfy following summation formula:

${{}_{\mathcal{B}}\mathcal{E}_{n+1}^{(\alpha)}}(x+1;y)-{{}_{\mathcal{B}}\mathcal{E}_{n+1}^{(\alpha)}}(x;y)=\sum\limits_{k=0}^{n}\binom{n+1}{k}{{}_{\mathcal{B}}{\mathcal{E}}_{k}^{(\alpha)}}(x;y)$
(4.4)

###### Proof.

Using relation defined in (3.1), we get

$\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x+1;y)\frac{{t}^{n}}{n!}-\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)\frac{{t}^{n}}{n!}$

$\displaystyle=$
$\displaystyle\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{{(x+1){t}}+y(e^{t}-1)}-\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{{x{t}}+y(e^{t}-1)}$
$\displaystyle=$
$\displaystyle\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{{x{t}}+y(e^{t}-1)}\,\,(e^{{t}}-1)$
$\displaystyle=$ $\displaystyle\left\\{\sum_{k\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{k}^{(\alpha)}}(x;y)\frac{{t}^{k}}{k!}\right\\}\,\,\left\\{\sum_{n\geq
0}\frac{{t}^{n+1}}{(n+1)!}\right\\}.$

After, series rearrangement to compare both side the same power of ${t}$, we
obtain a desired result (4.4) ∎

######  Theorem 4.3.

If $\alpha=1$ and $n\in\mathbb{N}$ then Bell based Euler polynomials satisfy
following relation:

${\mathcal{B}_{n}}(x;y)=\frac{{{}_{\mathcal{B}}\mathcal{E}_{n}}(x+1;y)+{{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)}{2}.$
(4.5)

###### Proof.

Using generating function (3.1) for $\alpha=1$ and definition of bivariate
Bell polynomials, we get

$\displaystyle\sum_{n\geq 0}{\mathcal{B}_{n}}(x;y)\frac{{t}^{n}}{n!}=$
$\displaystyle e^{x{t}+{y(e^{{t}}-1)}}$ $\displaystyle=$
$\displaystyle\frac{e^{{t}}+1}{2}\,\,\left\\{\sum_{k=0}^{\infty}{{}_{\mathcal{B}}\mathcal{E}_{k}}(x;y)\right\\}$
$\displaystyle=$
$\displaystyle\frac{e^{{t}}+1}{2}\left\\{\left(\frac{2}{e^{{t}}+1}\right)e^{{x{t}}+y(e^{t}-1)}\right\\}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{\left(\frac{2}{e^{{t}}+1}\right)e^{{(x+1){t}}+y(e^{t}-1)}+\left(\frac{2}{e^{{t}}+1}\right)e^{{x{t}}+y(e^{t}-1)}\right\\}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left\\{\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}}(x+1;y)\frac{{t}^{n}}{n!}+\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)\frac{{t}^{n}}{n!}\right\\}.$

By equating same power of ${t}$ both side, we get desired result (4.5). ∎

######  Remark 4.2.

The formula (4.5) is generalized form of Euler polynomials defined as

$x^{n}=\frac{{\mathcal{E}_{n}}(x+1)+{\mathcal{E}_{n}}(x)}{2}.$ (4.6)

######  Theorem 4.4.

If $n\geq 0$, then

${{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)=\sum_{j=0}^{n}\sum_{k\geq
0}\binom{n}{j}(x)_{k}\,\mathcal{S}_{2}(j,k)\,{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(y).$
(4.7)

###### Proof.

By using (3.1), we have

$\displaystyle\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)\frac{{t}^{n}}{n!}=$
$\displaystyle\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{{x{t}}+y(e^{t}-1)}$
$\displaystyle=$
$\displaystyle\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{y(e^{t}-1)}\,e^{x{t}}$
$\displaystyle=$
$\displaystyle\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{y(e^{t}-1)}\,(1+e^{{t}}-1)^{x}$
$\displaystyle=$ $\displaystyle\left\\{\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)\frac{{t}^{n}}{n!}\right\\}\left\\{\sum_{k\geq
0}(x)_{k}\,\frac{(e^{{t}}-1)^{k}}{k!}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)\frac{{t}^{n}}{n!}\right\\}\left\\{\sum_{k\geq
0}(x)_{k}\,\sum_{j\geq 0}\mathcal{S}_{2}(j,k)\frac{{t}^{j}}{j!}\right\\}$

By using series rearrangement technique and equating same power of
$\mathfrak{t}$ both side, we get desired result (4.7). ∎

##  5\. Deriavtive formula

######  Theorem 5.1.

The differential operator formula for the bell based Euler polynomials of
order $\alpha$ w.r.t. x defined as follows:

$\frac{\partial}{\partial
x}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)=n\,\,{{}_{\mathcal{B}}\mathcal{E}_{n-1}^{(\alpha)}}(x;y),$
(5.1)

hold for all $n\in\mathbb{N}$.

###### Proof.

we know that

$\frac{\partial}{\partial
x}\,\,{e^{x{t}+y(e^{{t}}-1)}}=t\,\,e^{x{t}+y(e^{{t}}-1)}$ (5.2)

By using definition (3.1) in (5.2), we obtain desired result (5.1). ∎

######  Theorem 5.2.

The difference operator formula for Bell based Euler polynomials of order
$\alpha$ w.r.t. y defined as:

$\frac{\partial}{\partial
y}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)=(-2)\,\left\\{{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)-{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha-1)}}(x;y)\right\\},$
(5.3)

hold for all $n\in\mathbb{N}$.

###### Proof.

By using well known derivative properties

$\frac{\partial}{\partial
y}\,\,{e^{x{t}+y(e^{{t}}-1)}}=(e^{{t}}-1)\,\,e^{x{t}+y(e^{{t}}-1)}$ (5.4)

Now, using definition (3.1), we get

$\displaystyle\frac{\partial}{\partial y}\left\\{\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)\frac{{t}^{n}}{n!}\right\\}=$
$\displaystyle\frac{\partial}{\partial
y}\left\\{\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{{x{t}}+y(e^{t}-1)}\right\\}$
$\displaystyle=$
$\displaystyle\left\\{\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{x{t}+y(e^{{t}}-1)}\right\\}(e^{{t}}-1)$
$\displaystyle=$
$\displaystyle(-2+e^{{t}}+1)\,\left\\{\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{x{t}+y(e^{{t}}-1)}\right\\}$
$\displaystyle=$
$\displaystyle(-2)\left(1-\frac{e^{{t}}+1}{2}\right)\,\left\\{\left(\frac{2}{e^{{t}}+1}\right)^{\alpha}e^{x{t}+y(e^{{t}}-1)}\right\\}$

Now, using (3.1) l.h.s of above equation and equating same power of
$\mathfrak{t}$ both side, we get desired result (5.3). ∎

##  6\. Application of Bell based Euler polynomials from umbral calculus

In this section, we discus the concept of umbral calculus (see[5, 6, 7, 11,
12, 14, 16]) and by using the concept of umbral calculus we drive some useful
relation of Bell Based Euler polynomilas.

Let $\mathcal{G}$ be set of all formal power series in the variable t over
complex field $\mathbb{C}$ with

$\mathcal{G}=\left\\{\mathfrak{g}|\mathfrak{g}(t)=\sum_{k\geq
0}\frac{c_{k}}{k!}{t}^{k}\,\,s.t.\,\,c_{k}\in\mathbb{C}\right\\}.$ (6.1)

Let $\mathcal{P}$ be set of polynomials in the single vriable $t$ and
$\mathcal{P}^{*}$ be set of vector space of all linear functional on
$\mathcal{P}$. In the umbral calculus, we denote
$\left<\mathcal{T}|~{}\mathfrak{q}(x)\right>$ be linear functional
$\mathcal{T}$ on the polynomials $\mathfrak{q}(x)$. Now, we defined the vector
space operations on $\mathcal{P}^{*}$ as follows:

$\left<\mathcal{T}_{1}+\mathcal{T}_{2}|\,\mathfrak{q}(x)\right>=\left<\mathcal{T}_{1}|\,\mathfrak{q}(x)\right>+\left<\mathcal{T}_{2}|\,\mathfrak{q}(x)\right>$

and

$\left<\beta\mathcal{T}|~{}\mathfrak{q}(x)\right>=\beta\left<\mathcal{T}|~{}\mathfrak{q}(x)\right>$

for any constant $\beta$ in $\mathbb{C}$.

The formal power series

$\mathfrak{g}(t)=\sum_{k\geq 0}\frac{c_{k}}{k!}{t}^{k}\,\,\in\mathcal{G}$
(6.2)

defined a linear functional on $\mathcal{P}$ as

$\left<\mathfrak{g}(t)|~{}x^{n}\right>=c_{n}$ (6.3)

for all $n\in\mathbb{N}\cup\\{0\\}$.

If we choose $\mathfrak{g}(t)={t}^{k}$ in (6.2) and (6.3), we get

$\left<{t}^{k}|~{}x^{n}\right>=n!\delta_{n,k},$ (6.4)

for all $n,k\in\mathbb{N}\cup\\{0\\}$ and

$\delta_{n,k}=\left\\{\begin{array}[]{ll}0&if\,\,n\neq k\\\
1,&if\,\,n=k\end{array}.\right.$

Since any linear functional $\mathcal{T}$ in $\mathcal{P}^{*}$ has in the form
of (6.2) i.e.,

$\mathfrak{g}_{\mathcal{T}}({t})=\sum_{k\geq
0}\left<\mathcal{T}|~{}x^{k}\right>\frac{{t}^{k}}{k!}$

and

$\left<\mathfrak{g}_{\mathcal{T}}({t})|~{}x^{n}\right>=\left<\mathcal{T}|~{}x^{n}\right>.$

So, the linear functional $\mathcal{T}=\mathfrak{g}_{\mathcal{T}}({t})$. We
know that, the map $\mathcal{T}\rightarrow\mathfrak{g}_{\mathcal{T}}({t})$ is
a vector space isomorphism from set of vector space of all linear functional
on $\mathcal{P}$ onto set of formal series $\mathcal{G}$. Therefore, set of
formal series $\mathcal{G}$ have vector space of all linear functionals on
$\mathcal{P}$, also $\mathcal{G}$ have algebra of formal power series and so
for $\mathfrak{g}(t)\in\mathcal{G}$ will be treatrd as both a formal power
series and a linear functional. From (6.3), we get

$\left<e^{y{t}}|~{}x^{n}\right>=y^{n}$ (6.5)

and so, we have

$\left<e^{y{t}}|~{}\mathfrak{q}(x)\right>=\mathfrak{q}(y)$ (6.6)

for all $\mathfrak{q}(x)\in\mathcal{P}$.

We know that the order of power series $\mathfrak{g}(t)$
$(i.e.~{}~{}o(\mathfrak{g}(t))$ will be the smallest positive integer $k$ such
that the coefficient of ${t}^{k}$ does not vanish. We know that
$\mathfrak{g}(t)$ is invertible series if the order of formal power series
$\mathfrak{g}(t)$ is zero, also the formal power series $\mathfrak{g(t)}$ is a
delta series if the order of $\mathfrak{g}(t)$ is one
$(i.e.\,o(\mathfrak{g}(t)=1)$. (see[5, 6, 7, 11, 12, 14, 16]).

For $\mathfrak{g}_{1}({t}),...,\mathfrak{g}_{m}({t})\in\mathcal{G}$, then

$\left<\mathfrak{g}_{1}({t}),...,\mathfrak{g}_{m}({t})\arrowvert~{}x^{n}\right>=\sum_{i_{1}+i_{2}+...+i_{m}=n}\binom{n}{i_{1},...,i_{m}}\left<\mathfrak{g}_{1}({t})|~{}x^{i_{1}}\right>...\left<\mathfrak{g}_{m}({t})|~{}x^{i_{m}}\right>,$

where

$\binom{n}{i_{1},...,i_{m}}=\frac{n!}{i_{1}!...i_{m}!}.$

If $\mathfrak{g}(t),\mathfrak{h}(t)\in\mathcal{G}$, then

$\left<\mathfrak{g}(t)\mathfrak{h}(t)|~{}\mathfrak{q}(x)\right>=\left<\mathfrak{g}(t)|~{}\mathfrak{h}(t)\mathfrak{q}(x)\right>=\left<\mathfrak{h}(t)|~{}\mathfrak{g}(t)\mathfrak{q}(x)\right>.$
(6.7)

Therfore, $\forall$ $\mathfrak{g}(t)\in\mathcal{G}$

$\mathfrak{g}(t)=\sum_{k\geq
0}\left<\mathfrak{g}(t)|~{}x^{k}\right>\frac{{t}^{k}}{k!},$ (6.8)

and $\forall$ polynomials $\mathfrak{q}(x)$

$\mathfrak{q}(x)=\sum_{k\geq
0}\frac{\left<{t}^{k}|~{}\mathfrak{q}(x)\right>}{k!}{t}^{k}.$ (6.9)

By using (6.9), we get

$\mathfrak{q}^{k}(x)=D^{k}\mathfrak{q}(x)=\sum_{k\geq
0}\frac{\left<{t}^{l}|~{}\mathfrak{q}(x)\right>}{l!}x^{l-k}\prod_{s=1}^{k}(l-s+1)$
(6.10)

From (6.10), we have

$\mathfrak{q}^{k}(0)=\left<{t}^{k}|~{}\mathfrak{q}(x)\right>\,\,\,\,and\,\,\,\,\,\,\,\mathfrak{q}^{k}(0)=\left<1|~{}\mathfrak{q}^{k}(x)\right>$
(6.11)

By (6.11), we note that

${t}^{k}\mathfrak{q}(x)=\mathfrak{q}^{k}(x).$ (6.12)

Let $\mathfrak{g}(t)$ and $\mathfrak{h}(t)$ be an element of formal powr
series $\mathcal{G}$ such that $\mathfrak{g}(t)$ be a delta and
$\mathfrak{h}(t)$ be an invertible series. Then $\exists$ an unique sequence
$\mathcal{S}_{n}(x)$ of the polynomials with following properties

$\left<\mathfrak{h}(t)\mathfrak{g}(t)^{k}\arrowvert~{}\mathcal{S}_{n}(x)\right>=n!\delta_{n,k}$
(6.13)

For all $n,k\in\mathbb{N}\cup\\{0\\}$, which is orthogonality condition for
the Sheffer sequence (see[5, 6, 7, 11, 12, 14, 16]).

The sequence $\mathcal{S}_{n}(x)$ is said to be Sheffer sequence for
$(\mathfrak{h}(t),\mathfrak{g}(t))$, which is denoted by
$\mathcal{S}_{n}(x)\sim(\mathfrak{h}(t),\mathfrak{g}(t))$.

Let $\mathcal{S}_{n}(x)$ is the Sheffer sequence for
$(\mathfrak{h}(t),\mathfrak{g}(t))$. Then for $\mathfrak{f(t)}\in\mathcal{G}$
and for $\mathfrak{q}(x)$, we have the the following relation;

$\mathfrak{f}(t)=\sum_{k\geq
0}\frac{\left<\mathfrak{f}(t)\arrowvert~{}\mathcal{S}_{k}(x)\right>}{k!}\mathfrak{h}(t)\mathfrak{g}(t)^{k}$
(6.14)

and

$\mathfrak{q}(x)=\sum_{k\geq
0}\frac{\left<\mathfrak{h}(t)\mathfrak{g}(t)^{k}\arrowvert~{}\mathfrak{q}(x)\right>}{k!}\mathcal{S}_{k}(x)$
(6.15)

and the sequence $\mathcal{S}_{n}(x)$ be a Sheffer sequence for
${(\mathfrak{h}(t),\mathfrak{g}(t))}$, iff

$\frac{1}{\mathfrak{h}(\bar{\mathfrak{g}}({t}))}e^{y\bar{\mathfrak{g}}({t})}=\sum_{k\geq
0}\frac{\mathcal{S}_{k}(y)}{k!}{t}^{k},\,\,\,\,\,\,\,\,\forall\,\,\,y\in\mathbb{C}.$
(6.16)

Here, $\bar{\mathfrak{g}}({t})$ is composition inverse of $\mathfrak{g}(t)$
i.e.
$\bar{\mathfrak{g}}(\mathfrak{g}(t))=\mathfrak{g}(\bar{\mathfrak{g}}({t}))={t}$.

Suppose that $\mathcal{S}_{n}(x)$ be an appell sequence for $\mathfrak{h}(t)$.
From (6.16), we get

$\mathcal{S}_{n}(x)=\frac{1}{\mathfrak{h}(t)}x^{n}\Leftrightarrow
t\mathcal{S}_{n}(x)=n\mathcal{S}_{n-1}(x).$ (6.17)

Recently, many author have studied Euler polynomilas, Bernoulli polynomials
and Bell polynomials under theory of umbral calculus (see[5, 6, 7, 11, 12, 14,
15, 16]) .

Recall from (3.1), we have

$\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\alpha)}}(x;y)\frac{{t}^{n}}{n!}=\left(\frac{2}{e^{{t}}+1}\right)e^{{x{t}}+y(e^{t}-1)}\,\,\,\,\,\,\,(|{t}|<2\pi).$
(6.18)

As $t$ goes to zero in (6.18) gives ${{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)$
equal to one $(i.e.\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)=1)$ which means
that $o\left(\left(\frac{2}{e^{{t}}+1}\right)e^{{x{t}}+y(e^{t}-1)}\right)=0$,
which implies that (3.1) is an invertible series and treated as a Sheffer
sequence.

We defined an important and useful properties of Bell based Euler polynomials
by using the definition of umbral calculus. Which is defined as follows:

From (6.16) and (6.18), we have

${{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)\sim\left(\frac{e^{{t}}+1}{2}\,e^{-y(e^{{t}}-1)},{t}\right)$
(6.19)

and

${t}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)=n\,\,{{}_{\mathcal{B}}\mathcal{E}_{n-1}}(x;y)$
(6.20)

Then from (6.19) and (6.20) we say that
${{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)$ is an appell sequence for
$\frac{e^{{t}}+1}{2}\,e^{-y(e^{{t}}-1)}$.

######  Theorem 6.1.

If $\mathfrak{q}(x)\in\mathcal{P}$, there exist a constant
$b_{0},b_{1},...,b_{n}$ such that

$\mathfrak{q}(x)=\sum_{k=0}^{n}b_{k}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y),$
(6.21)

where

$b_{k}=\frac{1}{k!}\left<\frac{e^{{t}}+1}{2}\,e^{-y(e^{{t}}-1)}{t}^{k}\,\,\arrowvert\,\,\mathfrak{q}(x)\right>$
(6.22)

###### Proof.

From (6.13), (6.16) and (6.19), we noted that

$\left<\frac{e^{{t}}+1}{2}\,e^{-y(e^{{t}}-1)}{t}^{k}\,\,\arrowvert\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)\right>=n!\delta_{n,k}\,\,\,\,\,\,\,n,k\in\mathbb{N}\cup\\{0\\},$

By using (6.21), we obtain

$\displaystyle\left<\frac{e^{{t}}+1}{2}\,e^{-y(e^{{t}}-1)}{t}^{k}\,\,\arrowvert\,\,\mathfrak{q}(x)\right>=$
$\displaystyle\left<\frac{e^{{t}}+1}{2}\,e^{-y(e^{{t}}-1)}{t}^{k}\,\,\arrowvert\,\,\sum_{l=0}^{n}b_{l}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)\right>$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{n}b_{l}\left<\frac{e^{{t}}+1}{2}\,e^{-y(e^{{t}}-1)}{t}^{k}\,\,\arrowvert\,\,\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)\right>$
$\displaystyle=$ $\displaystyle\sum_{l=0}^{n}b_{l}{l!}\delta_{l,k}=k!b_{k}$

Which is desired result (6.22). ∎

######  Theorem 6.2.

If $n\in\mathbb{N}$, we have

$\int_{x}^{x+z}{{}_{\mathcal{B}}\mathcal{E}_{n}}(v;y)\,dv=\frac{e^{z{t}}-1}{{t}}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)$
(6.23)

###### Proof.

By using (6.20), we have

$\displaystyle\int_{x}^{x+z}{{}_{\mathcal{B}}\mathcal{E}_{n}}(v;y)\,dv=$
$\displaystyle\frac{1}{n+1}\,\,\left\\{{{}_{\mathcal{B}}\mathcal{E}_{n+1}}(x+z;y)-{{}_{\mathcal{B}}\mathcal{E}_{n+1}}(x;y)\right\\}$
$\displaystyle=$ $\displaystyle\frac{1}{n+1}\sum_{k\geq
1}\binom{n+1}{k}{{}_{\mathcal{B}}\mathcal{E}_{n+l-k}}(x;y)\,z^{k}$
$\displaystyle=$ $\displaystyle\frac{1}{n+1}\left(\sum_{k\geq
0}\frac{z^{k}}{k!}\,t^{k}-1\right){{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)$
$\displaystyle=$
$\displaystyle\frac{e^{z{t}}-1}{{t}}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)$

we get desired result. ∎

######  Corollary 6.1.

If $n\in\mathbb{N}\cup\\{0\\}$, we have

$\int_{0}^{z}{{}_{\mathcal{B}}\mathcal{E}_{n}}(v;y)\,dv=\left<\frac{e^{z{t}}-1}{{t}}\arrowvert\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}}(y)\right>$
(6.24)

###### Proof.

From (6.20), we get

${{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)=\frac{{t}}{n+1}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n+1}}(x;y)$
(6.25)

and by using (6.25), we have

$\displaystyle\left<\frac{e^{z{t}}-1}{t}\arrowvert\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}}(y)\right>=$
$\displaystyle\left<\frac{e^{z{t}}-1}{{t}}\,\arrowvert\,\,\frac{{t}}{n+1}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n+1}}(y)\right>$
$\displaystyle=$
$\displaystyle\left<e^{z{t}}-1\,\arrowvert\,\,\frac{1}{n+1}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n+1}}(y)\right>$
$\displaystyle=$
$\displaystyle\frac{1}{n+1}\left\\{{{}_{\mathcal{B}}\mathcal{E}_{n+1}}(z;y)-{{}_{\mathcal{B}}\mathcal{E}_{n+1}}(0;y)\right\\}$
$\displaystyle=$
$\displaystyle\int_{0}^{z}{{}_{\mathcal{B}}\mathcal{E}_{n}}(v;y)\,\,dv$

∎

For any $\mu\in\mathbb{N}\cup\\{0\\}$ from (3.1) the Bell based Euler
polynomials of order $\mu$ are given as:

$\sum_{n\geq
0}{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\mu)}}(x;y)\frac{{t}^{n}}{n!}=\left(\frac{2}{e^{{t}}+1}\right)^{\mu}e^{{x{t}}+y(e^{t}-1)}\,\,\,\,\,\,\,(|{t}|<2\pi).$
(6.26)

When ${t}$ goes to zero then
${{}_{\mathcal{B}}\mathcal{E}_{n}^{(\mu)}}(x;y)=1$, which means that
$o\left(\left(\frac{2}{e^{{t}}+1}\right)^{\mu}e^{{x{t}}+y(e^{t}-1)}\right)=0$.
Hence the generating function (6.26) of order $\mu$ is an invertible and it
will be treated as Sheffer sequence.

Suppose that

$\mathfrak{g}^{\mu}({t};y)=\frac{(e^{{t}}+1)^{\mu}}{2^{\mu}}\,e^{-y(e^{{t}}-1)}$

We know that $\mathfrak{g}^{\mu}(t;y)$ is an invertible series. From (6.26) we
says that ${{}_{\mathcal{B}}\mathcal{E}_{n}^{(\mu)}}(x;y)$ is an appell
sequence for $\mathfrak{g}^{\mu}(t;y)$. Hence from (6.17), we have

${{}_{\mathcal{B}}\mathcal{E}_{n}^{(\mu)}}(x;y)=\frac{1}{\mathfrak{g}^{\mu}({t};y)}\,\,x^{n},$

and

${t}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}^{\mu}}(x;y)=n\,\,{{}_{\mathcal{B}}\mathcal{E}_{n-1}^{\mu}}(x;y)$

Thus, we have

${{}_{\mathcal{B}}\mathcal{E}_{n}^{\mu}}(x;y)\sim\left(\frac{(e^{{t}}+1)^{\mu}}{2^{\mu}}\,e^{-y(e^{{t}}-1)},{t}\right)$

Now, using above result we discuse some interesting theorem.

######  Theorem 6.3.

If $n\geq 0$, then

${{}_{\mathcal{B}}\mathcal{E}_{n}^{(\mu)}}(y)=\sum_{i_{1}+...+i_{\mu}=n}\binom{n}{i_{1},...,i_{\mu}}\,\,{{}_{\mathcal{B}}\mathcal{E}_{i_{\mu}}}(y)\prod_{j=1}^{\mu-1}\mathcal{E}_{i_{j}}$
(6.27)

###### Proof.

By using (6.3) and (6.26), we have

$\left<\frac{2^{\mu}}{(e^{{t}}+1)^{\mu}}\,e^{z{t}+y(e^{{t}}-1)}\arrowvert\,\,x^{n}\right>={{}_{\mathcal{B}}\mathcal{E}_{n}^{(\mu)}}(z;y)=\sum_{l=0}^{n}\binom{n}{l}{{}_{\mathcal{B}}\mathcal{E}_{n-l}^{(\mu)}}(y)\,z^{l}$
(6.28)

and

$\displaystyle\left<\frac{2^{\mu}}{(e^{{t}}+1)^{\mu}}\,e^{z{t}+y(e^{{t}}-1)}\arrowvert\,\,x^{n}\right>=$
$\displaystyle\left<\frac{2}{e^{{t}}+1}\times...\times\frac{2}{e^{{t}}+1}e^{y(e^{{t}}-1)}\arrowvert\,x^{n}\right>$
(6.29) $\displaystyle=$
$\displaystyle\sum_{i_{1}+...+i_{\mu}=n}\binom{n}{i_{1},...,i_{\mu}}{{}_{\mathcal{B}}\mathcal{E}_{i_{\mu}}^{(\mu)}}(y)\times\mathcal{E}_{i_{1}}\times...\times\mathcal{E}_{i_{\mu-1}}.$

From (6.28) and (6.29), we get result (6.27). ∎

######  Theorem 6.4.

If $\,\mathfrak{q}(x)\in\mathcal{P}_{n}$, then

$\mathfrak{q}(x)=\sum_{k=0}^{n}b_{k}^{\mu}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\mu)}}(x;y)\,\in\mathcal{P}_{n}$

where

$b_{k}^{\mu}=\frac{1}{2^{\mu}k!}\left<(e^{{t}}+1)^{\mu}e^{-y(e^{{t}}-1)}{t}^{k}\arrowvert\,\mathfrak{q}(x)\right>$
(6.30)

###### Proof.

Let as assume that

$\mathfrak{q}(x)=\sum_{k=0}^{n}b_{k}^{\mu}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}^{(\mu)}}(x;y)\,\in\mathcal{P}_{n}$
(6.31)

with the help of (6.31), we can write

$\displaystyle\left<\left(\frac{e^{{t}}+1}{2}\right)^{\mu}\,e^{-y(e^{{t}}-1)}{t}^{k}\,\,\arrowvert\,\,\mathfrak{q}(x)\right>=$
$\displaystyle\left<\left(\frac{e^{{t}}+1}{2}\right)^{\mu}\,e^{-y(e^{{t}}-1)}{t}^{k}\,\,\arrowvert\,\,\sum_{l=0}^{n}b_{l}^{\mu}\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}^{\mu}}(x;y)\right>$
(6.32) $\displaystyle=$
$\displaystyle\sum_{l=0}^{n}b_{l}^{\mu}\left<\left(\frac{e^{{t}}+1}{2}\right)^{\mu}\,e^{-y(e^{{t}}-1)}{t}^{k}\,\,\arrowvert\,\,\,\,{{}_{\mathcal{B}}\mathcal{E}_{n}}(x;y)\right>$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{n}b_{l}^{\mu}{l!}\delta_{l,k}=k!b_{k}^{\mu},$

and we know that

$\left<\left(\frac{e^{{t}}+1}{2}\right)^{\mu}\,e^{-y(e^{{t}}-1)}{t}^{k}\,\,\arrowvert\,\,\mathfrak{q}(x)\right>=\frac{1}{2^{\mu}}\left<(e^{{t}}+1)^{\mu}e^{-y(e^{{t}}-1)}{t}^{k}\arrowvert\,\mathfrak{q}(x)\right>.$
(6.33)

Hence from (6.32) and (6.33), we obtain the result (6.30). ∎

##  7\. Conclusions

In this paper, we introduced the Bell based Euler polynomials of order
$\alpha$ and study their various correlation, implicit sumation and derivative
formula. Also, we investigate various application of Bell based Euler
polynomials of order $\alpha$ by using defintion of umbral calculus. Now, it
is very useful because polynomials is solution of various differential
equation and also play an importnat role in multifarious area like
mathematics, physics and engineering sciences.

Conflicts of Interest: The authors declare no conflict of interest.

Ethical approval: This article does not contain any studies with human
participants or animals performed by any of the authors.

Author Contributions: All authors contributed equally to this manuscript.

Acknowledgments: The authors express their deep gratitude to the anonymous
referees for their critical comments and suggestions to improve this paper to
its current form.

Availability of data and material: Data sharing is not applicable to this
article as no datasets were generated or analysed during the current study.

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