1. Introduction and Preliminaries
The Hermite and Legendre polynomials are pivotal in classical boundary value problems, particularly within parabolic regions and coordinate systems. These problems arise in diverse fields ranging from heat conduction to fluid dynamics, where understanding the behavior of functions within specific geometrical boundaries is essential and also offer elegant solutions to such problems, aiding in formulating and analyzing mathematical models in various physical and engineering disciplines [
1,
2,
3,
4,
5,
6,
7].
A lot of research has been done on Frobenius-Euler polynomials, also known as Eulerian polynomials, and their generalizations, called as the Apostol type Frobenius-Euler polynomials, see [
8,
9,
10,
11,
12,
13,
14,
15,
16,
17]. These inquiries have covered a broad range of subjects, including identities, recurrence relations, differential equations, integral representations, explicit formulas, extensions in single and double variables, and fractional operational approach, among others. A modern and useful area of research is the study of the fusion of various polynomial types to produce creative multivariate generalized polynomials. The noteworthy characteristics of these polynomials which include recurrence and explicit relationships, functional and differential equations, summation formulas, symmetric and convolution properties, and determinant representations make this field of study especially pertinent. The significance of these characteristics extends across various academic domains, making multivariate convoluted special polynomials a compelling subject of exploration.
The generalized Apostol type Frobenius-Euler polynomials
are generated by the function [
13] is given by
When
,
are called the generalized Apostol type Frobenius-Euler numbers.
The explicit representation for the generalized Apostol type Frobenius-Euler polynomials
[
13] is given by
where
is the binomial coefficient.
The generalized Apostol-Euler polynomials
[
18] are given by
In the special case when
, the generalized Apostol type Frobenius-Euler polynomials reduced to the generalized Frobenius-Euler polynomials [
19,
20] given by
and generalized Apostol-Euler polynomials reduce to generalized Euler polynomials given by
For
, the polynomials
and
reduce to the Eulerian polynomials
[
19,
20] and classical Euler polynomials
[
21].
In addition to introducing new families of special polynomials, several helpful identities can be derived by utilizing the majority of special functions of mathematical physics and their generalizations as proposed by physical problems. Several other fields, including quantum mechanics and statistics, have also made use of these special polynomials for the description and analysis of complex systems. Polynomial sequences are essential to many areas of mathematics, including combinatorics, entropy, and algebraic combinatorics. The Legendre, Hermite, Chebyshev, Laguerre, and Jacobi polynomials are a few instances of polynomial sequences that are solutions to particular ordinary differential equations. The Legendre polynomials defined on the interval
with regard to the weight function
are the significant class of orthogonal polynomials with numerous applications in mathematics and physics [
22]. One key feature of the Legendre polynomials is their ability to satisfy a recurrence relation. This property makes it possible to calculate higher-degree polynomials from lower-degree ones, which facilitates effective polynomial generation and numerical computations. In addition, the generating function of Legendre polynomials aids in solving differential equations and closed-form formulae [
23]. Among the applications of the Legendre polynomials are the solutions of the Schrodinger equation for the hydrogen atom and other quantum systems with spherical symmetry in mathematics, physics, and engineering. These polynomials also arise from problems related to diffusion equations, wave propagation, and heat conduction, see also [
6,
7].
The 2-variable Legendre polynomials (2VLeP)
are generated by the function (see [
24])
where
is the Bessel function of first kind of order
defined by (see [
22])
The 2VLeP
are represented by the following series expansion:
where
is the greatest integer function.
Additionally, we observe that
is the inverse derivative operator.
In view of relation
, we get the classical Legendre polynomials
(see [
24]).
With roots in classical mathematics, the Hermite polynomials are regarded as one of the most important and ancient special functions in mathematics. Because of their many uses and versatility, these polynomials have proven to be highly useful in a wide range of fields. Hermite polynomials are used extensively in quantum mechanics, primarily as solutions to differential equations controlling harmonic oscillators. These equations, like the Schrödinger equation, are essential to comprehending how particles behave in quantum systems. Hermite polynomials are essential tools in theoretical physics because they solve these kinds of equations, which makes it easier to analyze and forecast quantum mechanical phenomena, see [
1,
2,
4,
5,
25].
Apart from their importance in quantum mechanics, Hermite polynomials are essential in classical boundary-value problems, especially in coordinate systems and parabolic regions. These issues come up in a variety of disciplines, such as fluid dynamics and heat conduction, where it’s crucial to comprehend how functions behave within particular geometrical bounds. Hermite polynomials provide sophisticated solutions to these kinds of issues, supporting the development and examination of mathematical models in a range of scientific and technical fields. Furthermore, Hermite polynomials are used in wavelet transform analysis as basis functions in signal processing. Wavelets are mathematical functions that are applied to various scales of signal analysis and information extraction, see [
3,
26,
27,
28]. Researchers can effectively analyze and process complex data by representing signals in terms of Hermite wavelets, which makes Hermite polynomials indispensable in domains like data compression, communication systems, and image processing. Hermite polynomials also appear in contexts like the Edgeworth series and Brownian motion, which have broad implications in probabilistic studies. Understanding random phenomena and modeling uncertainty in a variety of scientific and financial applications require an understanding of these stochastic processes. Researchers can create intricate probabilistic models and precisely analyze complex systems by utilizing Hermite polynomials. Hermite polynomials are also very important in combinatorics, where they are used to examine discrete structures and solve counting problems. Because of their combinatorial qualities, researchers can study combinatorial structures and algorithms more easily by counting arrangements, permutations, and combinations in various contexts, see also [
28,
29,
30].
The 2-variable Hermite Kampé de Feriet polynomials (2VHKdFP)
are generated by the function [
31]
The 2VHKdFP
are also defined by the following series expansion:
In view of relation
, we get the classical Hermite polynomials
(see [
22]).
Also, in view of the Equations (
7), (
10) and (
11), we find that
where
are the 2-variable Hermite-Kampé de Fériet polynomials.
The ability of generalized convoluted polynomials to establish explicit relationships and recurrence is one of their key characteristics. This indicates that specific behaviors or patterns repeat, giving researchers a valuable tool for understanding and predicting mathematical phenomena. Furthermore, these polynomials’ capacity to be used to create differential and functional equations increases their usefulness in the solution of challenging mathematical issues. This feature is particularly helpful in applications that require the modeling and analysis of dynamic relationships or rates of change. Another important feature of summation formulas is their ability to concisely represent series or sequences, which makes complicated mathematical expressions easier to understand. The symmetric and convolutional characteristics of these polynomials increase their adaptability and allow researchers to investigate a wide range of mathematical operations and manipulations.
Motivated by the above facts, the article is organized as follows. In
Section 2, the convolution of multivariate Legendre-Hermite polynomials is taken with generalized Apostol type Frobenius-Euler polynomials to construct a new class, namely the multivariate generalized Apostol type Legendre-Hermite-Frobenius-Euler polynomials (MVGATLeHFEP) and its operational view is presented.
Section 3 and
Section 4 are devoted for providing several recurrence relations, connection and summation formulae and symmetric identities for these polynomials. In
Section 5, some algebraic notions are imposed to provide a nice standard to these polynomials.
2. Convoluted Multivariate Legendre-Hermite Polynomials
The Hermite polynomials have far-reaching implications in probabilistic studies, appearing in contexts such as the Edgeworth series and Brownian motion. These stochastic processes are essential in understanding random phenomena and modeling uncertainty in various scientific and financial applications. By leveraging Hermite polynomials, researchers can develop sophisticated probabilistic models and analyze complex systems with precision. Additionally, Hermite polynomials play a crucial role in combinatorics, where they are employed to solve counting problems and analyze discrete structures. Their combinatorial properties enable researchers to enumerate arrangements, permutations, and combinations in diverse settings, facilitating the study of combinatorial structures and algorithms.
To construct the convoluted form, we expand the exponential function and replace the powers
in Equation (
11) by the polynomials
and after using Equation (
7), we find the multivariate Legendre-Hermite polynomials (MVLeHP)
.
The generating function for the MVLeHP
is given by
Using Equations (
11) and (
8) with Equation (
7) and expansion of
in Equation (
14), we find that the explicit series expansions for the MVLeHP
are given by
which in view of Equations (
9) and (
12) becomes:
Now, fix the parameter
and choose a concrete value
and present the shapes by drawing the surface plots of the MVLeHP
for even and odd index
n of polynomials (
Figure 1 and
Figure 2).
Remark 1. Considering (10) and (14) with relation (13), we find thatwhere denotes the multivariate double Hermite polynomials. The generating function and the series expansion for the multivariate double Hermite polynomials are given by For some concrete value
, the shapes of the multivariate double Hermite polynomials
for even and odd index
n of polynomials are as follows (
Figure 3 and
Figure 4):
Further, by convoluting the MVLeHP with the ATFEP using replacement method, we generate the multivariate generalized Apostol type Legendre-Hermite Frobenius-Euler polynomials (MVGATLeHFEP) denoted by .
The MVGATLeHFEP
are generated by the function
Through utilizing Equation (
1) in conjunction with (
2) and (
17) within Equation (
22), we find that the series expansion for the MVGATLeHFEP is given by
The multivariate generalized Legendre-Hermite-Apostol-Euler polynomials (MVGLeHAEP) are given by
In the special case when
, the MVGLeHATFEP reduced to the multivariate generalized Legendre-Hermite-Frobenius-Euler polynomials (3VGLeHFEP) given by
and MVGLeHAEP reduce to the multivariate generalized Legendre-Hermite-Euler polynomials (MVGLeHEP) given by
For
, the MVGLeHFEP reduce to the multivariate Legendre-Hermite-Frobenius-Euler polynomials
and MVGLeHEP multivariate Legendre-Hermite-Euler polynomials
. We reflect the shapes of the multivariate Legendre-Hermite-Euler polynomials
by utilizing its series expansion given as follows:
where
are the Euler numbers. For a concrete value
, the shapes of the multivariate Legendre-Hermite-Euler polynomials
for even and odd index
n of polynomials are as follows (
Figure 5 and
Figure 6):
Remark 2. In view of Equations (10) and (22) with relation (13), we find thatwhere denotes the multivariate generalized Apostol type double Hermite Frobenius-Euler polynomials (MVGATDHFEP). The generating function and the series expansion for these polynomials are given by The multivariate generalized double Hermite-Apostol-Euler polynomials (MVGDHAEP) are given by
In the special case when
, the MVGATDHFEP reduced to the multivariate generalized double Hermite-Frobenius-Euler polynomials (MVGMHFEP) given by
and MVGDHAEP reduce to the multivariate generalized double Hermite-Euler polynomials (MVGDHEP) given by
For
, the MVGDHFEP reduce to the multivariate double Hermite-Frobenius-Euler polynomials
and MVGDHEP multivariate double Hermite-Euler polynomials
. We reflect the shapes of the multivariate double Hermite-Euler polynomials
by utilizing its series expansion given as follows:
For a concrete value
, the shapes of the multivariate double Hermite-Euler polynomials
for even and odd index
n of polynomials are as follows (
Figure 7 and
Figure 8):
Remark 3. By taking in generating Equation (22) such thatyields the 2-variable generalized Apostol type Hermite-Frobenius-Euler polynomials (2VGATHFEP) [10]. Remark 4. By taking in generating Equation (22) such thatyields the 2-variable generalized Apostol type Legendre-Frobenius-Euler polynomials (2VGATLeFEP) . Note. We note that for , the results for the 2VGATHFEP and 2VGATLeFEP reduce to the results for the 2-variable generalized Hermite-Apostol-Euler polynomials (2VGHAEP) and 2-variable generalized Legendre-Apostol-Euler polynomials (2VGLeAEP) .
The properties of both ordinary and generalized special functions can be derived using the operational techniques associated with the study of generalized special functions, including multivariate cases, which can offer a unified study of these polynomials. The majority of special functions and their generalizations have their roots in physical issues.
We recall that the 2VLeP
are demonstrated to be natural solutions of a specific set of partial differential equations, and that they are of considerable interest because of their numerous applications in physics and mathematics:
and are defined by the following operational representation [
24]:
with
.
In many areas of pure and applied mathematics and physics, the 2VHKdFP
are widely employed. They have been used to solve issues with quantum mechanics and optical beam transport. These polynomials are the solutions of the heat equation:
and are defined by means of the operational representation:
We establish the operational representation for the MVLeHP
. Use of Equation (
40) with (
9) in (
15) yields the following operational representation for the MVLeHP:
Again, with use of Equation (
38) with (
12) in (
16) yields the following:
Also, using Equation (
38) (or (
40)) in (
41) (or (
42)) yields the following operational representation for the MVLeHP:
Next, we find the operational representation for the MVGATLeHFEP
. We note that in view of generating Equation (
22), the MVGATLeHFEP are the solutions of the following equations:
under the following initial condition:
Consequently, we determine the operational representation between the MVGATLeHFEP and the MVLeATFEP as follows in light of the aforementioned equations:
Again, from generating Equation (
22), we find that the MVGATLeHFEP are the solutions of the following equations:
under the following initial condition:
Consequently, we determine the operational representation between the MVGATLeHFEP and the MVHATFEP as follows in light of the aforementioned equations:
Note. As we can see, for
, the operational representations (
46) and (
49) for the MVGATLeHFEP gives the following operational representation between the MVGATLeHEP and 2VGLeAEP and MVGATLeHEP and 2VGHAEP such that
We derive some formulas and recurrence relations for the MVGATLeHFEP and their relatives in the following section.
5. Algebraic Matrix Approach
Determinants have wide range of applications in linear algebra, geometry, calculus, and other fields of mathematics. They provide valuable tools for solving problems, characterizing matrix properties, and understanding the behavior of mathematical systems. Numerical computations and the solution of linear interpolation problems to ascertain the coefficients of polynomial interpolation benefit from the use of the determinant representations of the special polynomials. Determinants offer a methodical and effective approach to calculating the coefficients of the Lagrange polynomial, thereby transforming interpolation into a feasible and dependable method for approximating functions with restricted data. In order to establish the orthogonality connections between the functions which are essential for numerous applications in physics and mathematics determinants are also used. They guarantee that the squared magnitudes of the functions integrate into unity and aid in normalizing the functions. They are employed in the development of generating functions for particular functions. Recursion relations, difference equations, and other characteristics of special functions can be obtained effectively with their help. For computational purposes, the algebraic method involving determinants for the complex special polynomials can be useful. Several important research to explore algebraic properties of various polynomials can be seen in [
36,
37,
38,
39,
40].
The 2-variable Legendre-Hermite polynomials (2VLeHP)
are first represented by their determinant. From the generating equation of multivariate Legendre-Hermite polynomials
, we deduce the following generating function for the 2-variable Legendre-Hermite polynomials provided
, which is given by
Now, to establish the determinants for the 2-variable polynomials, we are in search of classical polynomials determinants forms.
Lets recall the determinant form for the classical Hermite polynomials
[
41,
42]:
In the r.h.s. of Equation (
98), replace the powers
by the polynomials
, and
by the polynomial
in the l.h.s. Using the following relation:
in the l.h.s. of resultant equation, we find the following determinant form of the 2VLeHP
:
In view of relation
, we find that
where
denotes the 2-variable double Hermite polynomials (2VMHP). The generating function for the 2VMHP is given by
Now, using Equations (
100) and (
13) in determinant form (
99) yields the following determinant form for the 2-variable double Hermite polynomials:
In order to determine the determinant form of the multivariate generalized Legendre-Hermite-Euler polynomials (MVGLeHEP)
, we substitute the polynomials
in the r.h.s. and
by the polynomial
in the l.h.s. of the determinant form of generalized Euler polynomials (see [
42]) and then using relation
in the l.h.s. of the resulting equation, we find the following determinant form of the MVGLeHEP:
where
are the Stirling numbers of the second kind defined by
By finding the values of
from Equation (
104) and substituting it in Equation (
103) together with
, we find the following determinant form for the multivariate Legendre-Hermite-Euler polynomials (MVLeHEP)
:
where
.
In view of the following relation:
we find the following determinant form for the multivariate generalized double Hermite-Euler polynomials (MVGDHEP)
:
By finding the values of
from Equation (
104) and substituting it in Equation (
106) together with
, we find the following determinant form for the multivariate double Hermite-Euler polynomials (MVDHEP)
:
where
.
Note. We also that the analytical approach developed in previous
Section 2 can also be utilized to provide matrix representations for the two variable and three variable polynomials. If we expand the determinant (
98) of Hermite polynomials along first row and then applying operational Equations (
38) and (
42) in resultant equation and after summing up the terms, we can obtain the matrix representation (
99) of the 2VLeHP. In the same way, by appropriately applying the operational rules of the polynomials, we can construct the matrix representations of the polynomials, which we have derived by replacement technique in
Section 5. Hence, both the analytical and matrix approaches are connected with each other in the sense that one can derive the matrix representations of the convoluted polynomials by utilizing operational rules of polynomials on respective matrix representations. Such techniques of obtaining matrix representations are carried in various researches, see [
43,
44].