Differential and Integral Equations Involving Multivariate Special Polynomials with Applications in Computer Modeling

Abstract

This work introduces a new family of multivariate hybrid special polynomials, motivated by their growing relevance in mathematical modeling, physics, and engineering. We explore their core properties, including recurrence relations and shift operators, within a unified structural framework. By employing the factorization method, we derive various governing equations such as differential, partial differential, and integrodifferential equations. Additionally, we establish a related fractional Volterra integral equation, which broadens the theoretical foundation and potential applications of these polynomials. To support the theoretical development, we carry out computational simulations to approximate their roots and visualize the distribution of their zeros, offering practical insights into their analytical behavior.

1. Introduction

Special polynomials play a central role in both pure and applied mathematics due to their rich algebraic structures and wide-ranging applications. These include classical families such as Hermite, Laguerre, Legendre, Chebyshev, and Jacobi polynomials, each satisfying specific differential equations and orthogonality conditions. Special polynomials often arise in solutions to physical problems involving wave propagation, quantum mechanics, heat conduction, and signal processing. Their properties, such as recurrence relations, generating functions, and operational representations, provide powerful tools for analytical and numerical computations. In modern mathematical analysis, generalized and multivariate extensions of these polynomials continue to emerge, offering deeper insights into functional equations, approximation theory, and spectral methods.

Significant research has systematically introduced and hybridized special polynomials, spanning both classical and generalized forms. Various analytical techniques have been employed, as seen in [1,2,3,4,5,6]. A notable class of Hermite–Frobenius–Euler polynomials was recently explored by Araci et al. in [7], alongside other hybrid special polynomials in [8,9], using the generating function method as a structured framework. The significance of these convoluted special polynomials lies in their essential characteristics, which include recurrence and explicit relations, summation formulas, symmetric and convolution identities, and fundamental algebraic properties. These attributes enhance their versatility, making them valuable tools for various mathematical applications.

One particularly notable feature of these polynomials is their explicit recurrence relations. These connections enable efficient computations, allowing subsequent terms in a series to be derived from preceding ones. This property simplifies their analysis and facilitates further research and computational methods. Additionally, the existence of summation formulas allows these polynomials to be represented as either finite or infinite series. Such formulations enhance their evaluability and approximation capabilities, leading to practical applications in areas like numerical analysis and approximation theory.

The study of Hermite–Frobenius–Euler polynomials broadens mathematical applications, strengthening links between different branches. By bridging disciplines, these polynomials facilitate the exchange of techniques and ideas, fostering collaboration, innovation, and practical advancements across various scientific fields.

Beyond these prior studies, the recent advancements in multivariate Hermite polynomials, denoted as ${\sigma }_n^{\left[m\right]}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m)$, have significantly enriched the field of polynomial theory. These polynomials were developed through a generating relation, a powerful tool that facilitates systematic exploration and deeper insights into mathematical structures.

The continued study and advancement of multivariate Hermite polynomials, driven by the generating function technique, have profound implications for polynomial research [10,11]. These polynomials offer a robust framework for addressing complex multivariable systems, paving the way for new avenues of inquiry, enhanced understanding, and innovative applications in both mathematics and the sciences. The generating relation for these polynomials is given by [12]

$$exp({\phi }_1\xi +{\phi }_2{\xi }^2+\cdots +{\phi }_m{\xi }^m)=\sum _{n=0}^{\infty }{\sigma }_n^{\left[m\right]}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m){\displaystyle \frac{t^n}{n!}},$$

(1)

with series representation as follows:

$${\sigma }_n^{\left[m\right]}({\phi }_1,{\phi }_2,\cdots {\phi }_m):=n!\sum _{r=0}^{[n/m]}{\displaystyle \frac{{\phi }_m^r{\sigma }_{n-mr}^{\left[m\right]}({\phi }_1,{\phi }_2,\cdots ,{\phi }_{m-1})}{r!(n-mr)!}}.$$

(2)

A distinct family of polynomials emerges through the application of a convolution operation involving multivariate Hermite polynomials ${\sigma }_n^{\left[m\right]}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m)$ and Frobenius–Euler polynomials (FEPs), represented as ${\mathcal{F}}_n({\phi }_1;\rho )$ [11,13]. This convolution process fuses two different polynomial classes, yielding a novel polynomial sequence with distinctive characteristics.

The structure of the multivariate Hermite polynomials ${\sigma }_n^{\left[m\right]}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m)$ is rooted in classical Hermite polynomials, extending them to multiple variables ${\phi }_1,{\phi }_2,\cdots ,{\phi }_m$. These polynomials have been extensively studied and applied across various branches of mathematics and science.

By integrating the defining properties of both polynomial families, a new class emerges, possessing unique mathematical attributes and interrelations. The primary objective of this work is to establish differential equations and corresponding integral equations governing these polynomials [4,5,8]. We focus on the formal definition of multivariate Hermite–Frobenius–Euler polynomials (MHFEPs), denoted as ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m;\rho )$, where $\rho \in \mathbb{C}$ and $\rho \ne 1$, formulated as follows [14]:

$$\left({\displaystyle \frac{1-\rho }{e^t-\rho }}\right)e^{{\phi }_{1}t+{\phi }_2{t}^2+{\phi }_3{t}^3+\cdots +{\phi }_m{t}^m}=\sum _{n=0}^{\infty }{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m;\rho ){\displaystyle \frac{t^n}{n!}}.$$

(3)

The MHFEP introduces a novel synthesis of classical Hermite and Frobenius–Euler structures within a multivariate framework, offering enhanced flexibility and depth in mathematical modeling and analysis. Unlike their single-variable counterparts, these polynomials accommodate interactions among multiple variables, enabling the representation of more complex systems with inherent multidimensional behavior. Their construction via generating functions and operational techniques reveals new recurrence relations, differential equations, and integral identities that do not arise in the univariate case. This multivariate extension opens new avenues in areas such as quantum field theory, multivariate probability, and systems of differential equations, where traditional polynomial bases fall short in capturing inter-variable dynamics.

Researchers employ various analytical techniques to explore the properties, behavior, and applications of these intricate polynomials. Their study often involves analyzing key aspects such as convergence properties, orthogonality, recurrence relations, and generating functions. These polynomials also establish a fundamental connection between Frobenius–Euler polynomials and multivariate Hermite polynomials, allowing scholars to leverage insights and methodologies from both domains. This integration of distinct polynomial families fosters interdisciplinary collaboration and broadens the application of mathematical concepts across multiple scientific and mathematical fields.

Some specific examples of MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m;\rho )$ and their distinctive properties are given below (

Table 1. Special cases of ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m;\rho )$.

S.No.	Cases	Name of Polynomial	Generating Function
I.	$\rho =-1$	multivariate Hermite–Euler polynomials [10,15]	$\left({\displaystyle \frac{2}{e^t+1}}\right)e^{{\phi }_{1}t+{\phi }_2{t}^2+{\phi }_3{t}^3+\cdots +{\phi }_m{t}^m}={\displaystyle \sum _{n=0}^{\infty }}{}_{\mathcal{H}}\mathcal{E}_n({\phi }_1,{\phi }_2,\cdots ,{\phi }_m){\displaystyle \frac{t^n}{n!}}$
II.	$\rho =-1,m=3$	3-variable Hermite–Euler polynomials [10]	$\left({\displaystyle \frac{2}{e^t+1}}\right)e^{{\phi }_{1}t+{\phi }_2{t}^2+{\phi }_3{t}^3}={\displaystyle \sum _{n=0}^{\infty }}{}_{\mathcal{H}}\mathcal{E}_n({\phi }_1,{\phi }_2,{\phi }_3){\displaystyle \frac{t^n}{n!}}$
III.	$\rho =-1,m=2$,	2-variable Hermite–Euler polynomials [14]	$\left({\displaystyle \frac{2}{e^t+1}}\right)e^{{\phi }_{1}t+{\phi }_2{t}^2}={\displaystyle \sum _{n=0}^{\infty }}{}_{\mathcal{H}}\mathcal{E}_n({\phi }_1,{\phi }_2){\displaystyle \frac{t^n}{n!}}$
IV.	$\rho =-1,{\phi }_1\to 2{\phi }_1$, ${\phi }_2=-1;m=2$	Hermite–Euler polynomials [10]	$\left({\displaystyle \frac{2}{e^t+1}}\right)e^{2{\phi }_{1}t-t^2}={\displaystyle \sum _{n=0}^{\infty }}{}_{\mathcal{H}}\mathcal{E}_n({\phi }_1,{\phi }_2){\displaystyle \frac{t^n}{n!}}$

).

The MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m;\rho )$ are represented by the series [14]

$${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m;\rho )=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\mathcal{E}}^{\mathcal{F}}n-k\left(\rho \right){\mathcal{H}}_k({\phi }_1,{\phi }_2,\cdots ,{\phi }_m),$$

(4)

and for the special case $m=2$, we obtain

$${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2;\rho )=n!\sum _{k=0}^n\sum _{r=0}^{\left[{\displaystyle \frac{k}{2}}\right]}{\displaystyle \frac{{\mathcal{E}}_{n-k}^{\mathcal{F}}\left(\rho \right){{\phi }_1}^r{{\phi }_1}^{k-2r}}{(n-k)!r!(k-2r)}}.$$

(5)

The Hermite–Frobenius–Euler polynomials hold a vital place in both theoretical and applied mathematics, especially in quantum mechanics and probability theory. Their versatility aids in solving diverse mathematical and physical challenges, highlighting their broad relevance.

Differential equations are foundational in disciplines such as mathematics, physics, and engineering, forming the basis for modeling in celestial mechanics, structural analysis, and neural networks. The development of generalized and multivariate special polynomials has provided new tools for investigating these equations. While pure mathematics emphasizes the existence and uniqueness of solutions, applied mathematics prioritizes accurate approximations and simulations critical for complex systems in science and economics.

The factorization approach, as detailed in [16], is a powerful technique employed to address eigenvalue problems. This method involves solving two primary differential equations whose composition yields another differential equation of equal significance. The approach also integrates the computation of transition probabilities, which account for the production or transformation process in the physical system.

The method provides a robust framework for analyzing perturbation problems, utilizing two differential equation classes to derive another critical equation. It extends beyond numerical computation by capturing fluctuations or variations that may affect system precision or stability [17,18,19,20].

In this study, we consider a sequence of polynomials ${\mathcal{P}}_n\left({\eta }_1\right)$ for $n=0,1,2,\cdots$, governed by two differential-type operators ${\mathrm{\Psi }}_n^{-}$ and ${\mathrm{\Psi }}_n^{+}$, satisfying

$${\mathcal{P}}_{n-1}\left({\eta }_1\right)={\psi }_n^{-}\left({\mathcal{P}}_n\left({\eta }_1\right)\right),$$

(6)

$${\mathcal{P}}_{n+1}\left({\eta }_1\right)={\mathrm{\Psi }}_n^{+}\left({\mathcal{P}}_n\left({\eta }_1\right)\right).$$

(7)

A key structural identity is the second-order differential equation

$${\mathcal{P}}_n\left({\eta }_1\right)=\left({\mathrm{\Psi }}_{n+1}^{-}{\mathrm{\Psi }}_n^{+}\right){\mathcal{P}}_n\left({\eta }_1\right).$$

(8)

The operators ${\mathrm{\Psi }}_n^{-}$ and ${\mathrm{\Psi }}_n^{+}$ are critical in generating Equation (8) and form the backbone of the factorization strategy. Equation (8) allows the construction of a hierarchy of differential equations. Here, ${\mathrm{\Psi }}_n^{+}$ is typically a multiplicative-type operator, while ${\mathrm{\Psi }}_n^{-}$ is a derivative-type operator. Choosing these appropriately ensures compatibility with (8), offering insights into the analytic structure of the underlying polynomial family.

Integral and integrodifferential equations frequently arise in advanced applications such as quantum scattering, wave propagation, and fluid dynamics. They add richness to mathematical modeling by embedding unknown functions within integrals and balancing integration with differentiation.

The equations derived for MHFEPs include pure differential, integrodifferential, and integral equations. These reflect the deep mathematical structures encoded in the polynomials and serve as powerful tools for theoretical investigations and applied problem-solving.

Several works—such as [17,18,19,20]—have addressed such equations in different polynomial families, highlighting their versatility and applications in real-world systems. These studies emphasize that beyond mathematical interest, such equations are practical instruments in diverse scientific domains. This paper also explores the computational aspects of MHFEPs. Utilizing Wolfram Mathematica, the zeros of the polynomial family are analyzed graphically. These visualizations enhance understanding of their distribution and behavior. The analytical methods discussed earlier are used to derive the fractional Volterra-type integral equations satisfied by these polynomials. The interplay of symbolic and graphical techniques allows for a comprehensive examination of MHFEPs’ analytic properties.

This study presents a comprehensive exploration of the structural features and analytical behavior of MHFEPs. A central aspect of this work involves employing a factorization-based framework to construct various families of differential equations associated with these polynomials. In Section 2, the focus is on a detailed examination of their generating functions, recurrence structures, and shift operators. Section 3 is dedicated to the systematic formulation of distinct categories of differential equations linked to the polynomial family. Additionally, this research derives the corresponding fractional Volterra-type integral equation satisfied by these polynomials. Section 4 addresses the analytical computation and Wolfram Mathematica-based visualization of their zeros. This paper concludes by summarizing the main contributions and discussing the broader mathematical significance of the findings.

2. Main Results

Recurrence relations, fundamental in modeling dynamic systems, appear in ecology, engineering, and economics, shaping feedback mechanisms and predictive models. In this section, we establish shift operators and recurrence relations for the MHFEP, providing a structured approach to simplify computations and uncover underlying patterns with broad mathematical applications. To derive the recurrence relation for the function ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$, the following result is utilized:

Theorem 1.

The MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$ satisfy the following recurrence relation:

$$\begin{array}{c}{}_{\mathcal{H}}\mathcal{E}_{n+1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right){}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )+2n{\phi }_2\hfill \\ \times {}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )+3n(n-1){\phi }_3{}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )+\cdots \hfill \\ +n(n-1)(n-2)\cdots (n-m+1)m{\phi }_m{}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\hfill \\ +{\displaystyle \frac{1}{1-\rho }}{\displaystyle \sum _{k=0}^{n-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){}_{\mathcal{H}}\mathcal{E}_k^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ){\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right),\hfill \end{array}$$

(9)

where

$${\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right):=-\sum _{i=0}^k{\displaystyle \frac{1}{2^i}}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{i}}\right){\mathcal{E}}_{k-i}^{\mathcal{F}}\left({\displaystyle \frac{1}{2}};\rho \right),{\mathfrak{E}}_0^{\mathcal{F}}=-1,{\mathfrak{E}}_1^{\mathcal{F}}=-1-{\displaystyle \frac{1}{1-\rho }}$$

(10)

are expressed through numerical coefficients and ${\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)$ exhibits a connection with the FEP ${\mathcal{E}}_k^{\mathcal{F}}({\phi }_1;\rho )$.

Proof. 

Using (3) by differentiating it with respect to t, it follows that

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{}_{\mathcal{H}}\mathcal{E}_{n+1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ){\displaystyle \frac{t^n}{n!}}=\left({\phi }_1+2{\phi }_{2}t+3{\phi }_3{t}^2+\cdots +m{\phi }_m{t}^{m-1}\right)\\ \hfill \times \sum _{n=0}^{\infty }{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ){\displaystyle \frac{t^n}{n!}}+{\displaystyle \frac{1}{1-\rho }}\sum _{n=0}^{\infty }\sum _{k=0}^{\infty }{}_{\mathcal{H}}\mathcal{E}_k^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ){\mathfrak{E}}_n^{\mathcal{F}}\left(\rho \right){\displaystyle \frac{t^{n+k}}{n!k!}}.\end{array}$$

(11)

Upon simplification, the right-hand side is evaluated using the Cauchy multiplication principle, which leads to the subsequent outcome.

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{}_{\mathcal{H}}\mathcal{E}_{n+1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }{\phi }_1{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ){\displaystyle \frac{t^n}{n!}}+\sum _{n=0}^{\infty }2n{\phi }_2\\ \hfill \times {}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ){\displaystyle \frac{t^n}{n!}}+\sum _{n=0}^{\infty }3n(n-1){\phi }_3{}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ){\displaystyle \frac{t^n}{n!}}+\cdots +\\ \hfill \sum _{n=0}^{\infty }n(n-1)\cdots (n-m+1)m{\phi }_m\times {}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ){\displaystyle \frac{t^n}{n!}}+{\displaystyle \frac{1}{1-\rho }}\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\\ \hfill \times {}_{\mathcal{H}}\mathcal{E}_k^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ){\mathfrak{E}}_k^{\mathcal{F}}\left(\rho \right){\displaystyle \frac{t^n}{n!}}.\end{array}$$

(12)

Matching the coefficients of t from both sides of the prior equation yields the given expression:

$$\begin{array}{c}\hfill {}_{\mathcal{H}}\mathcal{E}_{n+1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )={\phi }_1{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )+2n{\phi }_2\\ \hfill {}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )+3n(n-1){\phi }_3{}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )+\cdots +\\ \hfill n(n-1)\cdots (n-m+1)m{\phi }_m{}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )+{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\\ \hfill \times {}_{\mathcal{H}}\mathcal{E}_k^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ){\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right).\end{array}$$

(13)

Assertion (9) is obtained by computing the summation at $k=n$ in the previously stated equation and incorporating the substitution ${\mathfrak{E}}_0^{\mathcal{F}}=-1$ into the derived expression. □

In the following analysis, we demonstrate the construction of shift operators for the MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$ by deriving the following result.

Theorem 2.

The MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$ satisfy the listed shift operators:

$${}_{{\phi }_1}\pounds _n^{-}:={\displaystyle \frac{1}{n}}{D}_{{\phi }_1},$$

(14)

$${}_{{\phi }_2}\pounds _n^{-}:={\displaystyle \frac{1}{n}}{D}_{{\phi }_1}^{-1}{D}_{{\phi }_2},$$

(15)

$${}_{{\phi }_3}\pounds _n^{-}:={\displaystyle \frac{1}{n}}{D}_{{\phi }_1}^{-2}{D}_{{\phi }_3},$$

(16)

$$⋮⋮$$

$${}_{{\phi }_m}\pounds _n^{-}:={\displaystyle \frac{1}{n}}{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_m},$$

(17)

$${}_{{\phi }_1}\pounds _n^{+}:=\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)+2{\phi }_2{D}_{{\phi }_1}+3{\phi }_3{D}_{{\phi }_1}^2+\cdots +m{\phi }_m{D}_{{\phi }_1}^{m-1}+{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{(n-k)}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}$$

(18)

$$\begin{array}{c}\hfill {}_{{\phi }_2}\pounds _n^{+}:=\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)+2{\phi }_2{D}_{{\phi }_1}^{-1}{D}_{{\phi }_2}+3{\phi }_3{D}_{{\phi }_1}^{-2}{D}_{{\phi }_2}^2+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_2}^{m-1}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-(n-k)}{D}_{{\phi }_2}^{n-k}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}\end{array}$$

(19)

$$\begin{array}{c}\hfill {}_{{\phi }_3}\pounds _n^{+}:=\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)+2{\phi }_2{D}_{{\phi }_1}^{-2}{D}_{{\phi }_3}+3{\phi }_3{D}_{{\phi }_1}^{-4}{D}_{{\phi }_3}^2+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-2(m-1)}{D}_{{\phi }_3}^{m-1}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-2(n-k)}{D}_{{\phi }_3}^{n-k}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}\end{array}$$

(20)

$$⋮⋮$$

$$\begin{array}{c}\hfill {}_{{\phi }_m}\pounds _n^{+}:=\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)+2{\phi }_2{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_m}+3{\phi }_3{D}_{{\phi }_1}^{-2(m-1)}{D}_{{\phi }_m}^2+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-{(m-1)}^2}\\ \hfill \times D_{{\phi }_m}^{m-1}+{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-(m-1)(n-k)}{D}_{{\phi }_m}^{n-k}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}\end{array}$$

(21)

where

$$D_{{\phi }_1}:={\displaystyle \frac{\partial }{\partial {\phi }_1}},D_{{\phi }_2}:={\displaystyle \frac{\partial }{\partial {\phi }_2}},D_{{\phi }_3}:={\displaystyle \frac{\partial }{\partial {\phi }_3}}\mathit{and}{D}_{{\phi }_1}^{-1}:={\int }_0^{{\phi }_1}f\left(\eta \right)d\eta .$$

Proof. 

Differentiating Equation (3) with respect to ${\phi }_1$ and equating the coefficients of like powers of t on both sides yields the following expression:

$${\displaystyle \frac{\partial }{\partial {\phi }_1}}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}=n{}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ).$$

(22)

Proceeding in accordance with the steps outlined above, we obtain the following expression:

$$\begin{array}{c}{}_{{\phi }_1}\pounds _n^{-}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}={\displaystyle \frac{1}{n}}{D}_{{\phi }_1}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}=\hfill \\ {}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ).\hfill \end{array}$$

(23)

Thus, this confirms the claim presented in (14). Differentiating Equation (3) with respect to ${\phi }_2$ and then matching the coefficients of identical powers of t on both sides yields the following result:

$${\displaystyle \frac{\partial }{\partial {\phi }_2}}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}=n(n-1){}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ).$$

The previous expression can alternatively be written as

$${\displaystyle \frac{\partial }{\partial {\phi }_2}}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}=n{\displaystyle \frac{\partial }{\partial {\phi }_1}}\left\{{}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\},$$

and eventually provides

$$\begin{array}{c}{}_{{\phi }_2}\pounds _n^{-}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}={\displaystyle \frac{1}{n}}{D}_{{\phi }_1}^{-1}{D}_{{\phi }_2}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}=\hfill \\ \hfill {}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ).\end{array}$$

(24)

Thus, the assertion in (15) is confirmed.

Taking the derivative of Equation (3) with respect to ${\phi }_3$ and matching the coefficients of like powers of t on both sides, we derive the following expression:

$${\displaystyle \frac{\partial }{\partial {\phi }_3}}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}=n(n-1)(n-2){}_{\mathcal{H}}\mathcal{E}_{n-3}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ).$$

The previous equation can be rewritten as follows:

$${\displaystyle \frac{\partial }{\partial {\phi }_3}}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}=n{\displaystyle \frac{{\partial }^2}{\partial {\phi }_1^2}}\left\{{}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\},$$

(25)

and thus eventually provides

$$\begin{array}{c}{}_{{\phi }_3}\pounds _n^{-}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}={\displaystyle \frac{1}{n}}{D}_{{\phi }_1}^{-2}{D}_{{\phi }_3}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}\hfill \\ ={}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ),\hfill \end{array}$$

(26)

Hence, the validity of Equation (16) is confirmed.

Finally, by differentiating Equation (3) with respect to ${\phi }_m$ and equating the coefficients of corresponding powers of t on both sides, we derive the following expression:

$${\displaystyle \frac{\partial }{\partial {\phi }_m}}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}=n(n-1)(n-2)(n-m+1){}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$$

(27)

which can be further presented as

$${\displaystyle \frac{\partial }{\partial {\phi }_m}}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}=n{\displaystyle \frac{{\partial }^{m-1}}{\partial {\phi }_1^{m-1}}}\left\{{}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\},$$

(28)

thus eventually giving

$$\begin{array}{c}{}_{{\phi }_m}\pounds _n^{-}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}={\displaystyle \frac{1}{n}}{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_m}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}\hfill \\ \hfill ={}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ).\end{array}$$

(29)

Thus, the validity of Equation (17) is established.

Next, to obtain the equation for the raising operator (18), we utilize the following formulation:

$$\begin{array}{c}{}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=({}_{{\phi }_1}\pounds _{n-m+1}^{-}{}_{{\phi }_1}\pounds _{n-m+2}^{-}\cdots {}_{{\phi }_1}\pounds _{n-1}^{-}{}_{{\phi }_1}\pounds _n^{-})\hfill \\ \hfill \times \left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}.\end{array}$$

(30)

Thus, by referring to expression (23), Equation (30) can be rewritten in a more concise form as follows:

$${}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )={\displaystyle \frac{(n-m)!}{m!}}{D}_{{\phi }_1}^m\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}.$$

(31)

Substituting Equation (31) into the recurrence relation (9), we deduce the following:

$$\begin{array}{c}\hfill {}_{\mathcal{H}}\mathcal{E}_{n+1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=\left(\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)+2{\phi }_2{D}_{{\phi }_1}+3{\phi }_3{D}_{{\phi }_1}^2+\cdots +m{\phi }_m{D}_{{\phi }_1}^{m-1}\right.\\ \hfill \left.+{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{(n-k)}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}\right){}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ).\end{array}$$

(32)

Thus, the legitimacy of the raising operator ${}_{{\phi }_1}\pounds _n^{+}$ in (18) is confirmed. To demonstrate the raising operator in (19), we consider the following relation:

$$\begin{array}{c}{}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=({}_{{\phi }_2}\pounds _{n-m+1}^{-}{}_{{\phi }_2}\pounds _{n-m+2}^{-}\cdots {}_{{\phi }_2}\pounds _{n-2}^{-}{}_{{\phi }_2}\pounds _n^{-})\hfill \\ \hfill \left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\},\end{array}$$

(33)

Taking Equation (24) into account, the preceding expression can be rewritten as follows:

$${}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )={\displaystyle \frac{(n-m)!}{m!}}{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_2}^{(m-1)}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}.$$

(34)

Inserting Equation (34) into the recurrence relation (9) yields

$$\begin{array}{c}\hfill {}_{\mathcal{H}}\mathcal{E}_{n+1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=\left(\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)+2{\phi }_2{D}_{{\phi }_1}^{-1}{D}_{{\phi }_2}+3{\phi }_3{D}_{{\phi }_1}^{-2}{D}_{{\phi }_2}^2+\cdots \right.\\ \hfill \left.+m{\phi }_m{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_2}^{m-1}+{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-(n-k)}{D}_{{\phi }_2}^{n-k}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}\right).\end{array}$$

(35)

We have successfully confirmed the validity of assertion (19) for the raising operator ${}_{{\phi }_2}\pounds _n^{+}$.

Next, to demonstrate the raising operator ${}_{{\phi }_3}\pounds _n^{+}$, we consider the following expression:

$$\begin{array}{c}{}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=({}_{{\phi }_3}\pounds _{n-m+1}^{-}{}_{{\phi }_3}\pounds _{n-m+2}^{-}\cdots {}_{{\phi }_3}\pounds _{n-1}^{-}{}_{{\phi }_3}\pounds _n^{-})\hfill \\ \hfill \left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\},\end{array}$$

(36)

Taking into account Equation (26), the preceding expression can be developed in the following manner:

$${}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )={\displaystyle \frac{(n-m)!}{m!}}{D}_{{\phi }_1}^{-2(m-1)}{D}_{{\phi }_3}^{(m-1)}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}.$$

(37)

Substituting Equation (37) into the recurrence relation (9) reveals that

$$\begin{array}{c}{}_{\mathcal{H}}\mathcal{E}_{n+1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=\left(\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)+2{\phi }_2{D}_{{\phi }_1}^{-2}{D}_{{\phi }_3}+3{\phi }_3{D}_{{\phi }_1}^{-4}{D}_{{\phi }_3}^2+\cdots \right.\hfill \\ \hfill \left.+m{\phi }_m{D}_{{\phi }_1}^{-2(m-1)}{D}_{{\phi }_3}^{m-1}+{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-2(n-k)}{D}_{{\phi }_3}^{n-k}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}\right).\end{array}$$

(38)

Therefore, we have successfully verified the validity of assertion (20) for the raising operator ${}_{{\phi }_3}\pounds _n^{+}$.

Thus to establish the validity of the raising operator ${}_{{\phi }_m}\pounds _n^{+}$, we examine the following expression:

$$\begin{array}{c}{}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=({}_{{\phi }_m}\pounds _{n-m+1}^{-}{}_{{\phi }_m}\pounds _{n-m+2}^{-}\cdots {}_{{\phi }_m}\pounds _{n-1}^{-}{}_{{\phi }_m}\pounds _n^{-})\hfill \\ \left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\},\hfill \end{array}$$

(39)

Referring to Equation (29), the expression above can be rewritten and expanded in the following manner:

$${}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )={\displaystyle \frac{(n-m)!}{m!}}{D}_{{\phi }_1}^{-{(m-1)}^2}{D}_{{\phi }_m}^{(m-1)}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}.$$

(40)

By substituting Equation (40) into the recurrence relation (9), it follows that

$$\begin{array}{c}\hfill {}_{\mathcal{H}}\mathcal{E}_{n+1}^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=\left(\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)+2{\phi }_2{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_m}+3{\phi }_3{D}_{{\phi }_1}^{-2(m-1)}{D}_{{\phi }_m}^2\right.\\ \hfill \left.+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-{(m-1)}^2}{D}_{{\phi }_m}^{m-1}+{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-(m-1)(n-k)}{D}_{{\phi }_m}^{n-k}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}\right).\end{array}$$

(41)

Hence, the validity of expression (21) for the raising operator ${}_{{\phi }_m}\pounds _n^{+}$ is confirmed. □

In the next section, we examine a class of “differential equations linked to the MHFEP”.

3. Differential and Integral Equations

This section provides a detailed analysis of differential and integral equations associated with MHFEPs, highlighting their structural forms and interconnections. For the MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$, we derive a set of differential, integrodifferential, partial differential, and fractional Volterra integral equations. Thus, the differential equation for the MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$ is formulated below.

Theorem 3.

The MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$ satisfy the following differential equation:

$$\begin{array}{c}\hfill \left(\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_1}+2{\phi }_2{D}_{{\phi }_1}^2+3{\phi }_3{D}_{{\phi }_1}^3+\cdots +m{\phi }_m{D}_{{\phi }_1}^m+{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{(n-k+1)}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-n\right)\\ \hfill \times {}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0.\end{array}$$

(42)

Proof. 

The shift operator expressions given in Equations (14) and (18) are utilized as

$${}_{{\phi }_1}\pounds _{n+1}^{-}{}_{{\phi }_1}\pounds _n^{+}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}={}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ),$$

(43)

After simplifying the mathematical expression, the claim in (42) is proved. □

Theorem 4.

The MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$ satisfy the following integrodifferential equations:

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_2}+2{\phi }_2{D}_{{\phi }_1}^{-1}{D}_{{\phi }_2}^2+3{\phi }_3{D}_{{\phi }_1}^{-2}{D}_{{\phi }_2}^3+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_2}^m\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-(n-k)}{D}_{{\phi }_2}^{n-k+1}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(44)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_3}+2{\phi }_2{D}_{{\phi }_1}^{-1}{D}_{{\phi }_2}{D}_{{\phi }_3}+3{\phi }_3{D}_{{\phi }_1}^{-2}{D}_{{\phi }_2}{D}_{{\phi }_3}+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_2}^{m-1}{D}_{{\phi }_3}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-(n-k)}{D}_{{\phi }_2}^{n-k}{D}_{{\phi }_3}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}^2\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(45)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_m}+2{\phi }_2{D}_{{\phi }_1}^{-1}{D}_{{\phi }_2}{D}_{{\phi }_m}+3{\phi }_3{D}_{{\phi }_1}^{-2}{D}_{{\phi }_2}{D}_{{\phi }_m}+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_2}^{m-1}{D}_{{\phi }_m}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-(n-k)}{D}_{{\phi }_2}^{n-k}{D}_{{\phi }_m}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_m}^{m-1}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(46)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_2}+2{\phi }_2{D}_{{\phi }_1}^{-2}{D}_{{\phi }_2}{D}_{{\phi }_3}+3{\phi }_3{D}_{{\phi }_1}^{-4}{D}_{{\phi }_2}{D}_{{\phi }_3}^2+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-2(m-1)}{D}_{{\phi }_2}{D}_{{\phi }_3}^{m-1}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-2(n-k)}{D}_{{\phi }_2}{D}_{{\phi }_3}^{n-k}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(47)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_3}+2{\phi }_2{D}_{{\phi }_1}^{-2}{D}_{{\phi }_3}^2+3{\phi }_3{D}_{{\phi }_1}^{-4}{D}_{{\phi }_3}^3+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-2(m-1)}{D}_{{\phi }_3}^m\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-2(n-k)}{D}_{{\phi }_3}^{n-k+1}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}^2\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(48)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_m}+2{\phi }_2{D}_{{\phi }_1}^{-2}{D}_{{\phi }_3}{D}_{{\phi }_m}+3{\phi }_3{D}_{{\phi }_1}^{-4}{D}_{{\phi }_3}^2{D}_{{\phi }_m}+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-2(m-1)}{D}_{{\phi }_3}^{m-1}{D}_{{\phi }_m}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-2(n-k)}{D}_{{\phi }_3}^{n-k}{D}_{{\phi }_m}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}^{m-1}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(49)

$$⋮$$

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_2}+2{\phi }_2{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_2}{D}_{{\phi }_m}+3{\phi }_3{D}_{{\phi }_1}^{-2(m-1)}{D}_{{\phi }_2}{D}_{{\phi }_m}^2+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-{(m-1)}^2}{D}_{{\phi }_2}{D}_{{\phi }_m}^{m-1}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-(m-1)(n-k)}{D}_{{\phi }_2}{D}_{{\phi }_m}^{n-k}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(50)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_3}+2{\phi }_2{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_3}{D}_{{\phi }_m}+3{\phi }_3{D}_{{\phi }_1}^{-2(m-1)}{D}_{{\phi }_3}{D}_{{\phi }_m}^2+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-{(m-1)}^2}{D}_{{\phi }_3}{D}_{{\phi }_m}^{m-1}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-(m-1)(n-k)}{D}_{{\phi }_2}{D}_{{\phi }_m}^{n-k}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}^2\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(51)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_m}+2{\phi }_2{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_m}^2+3{\phi }_3{D}_{{\phi }_1}^{-2(m-1)}{D}_{{\phi }_m}^3+\cdots +m{\phi }_m{D}_{{\phi }_1}^{-{(m-1)}^2}{D}_{{\phi }_m}^m\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{-(m-1)(n-k)}{D}_{{\phi }_m}^{n-k+1}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}^{m-1}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0.\end{array}$$

(52)

Proof. 

Consider the factorization relation

$${\pounds }_{n+1}^{-}{\pounds }_n^{+}\left\{{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )\right\}={}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho ).$$

(53)

By inserting Equations (15) and (19) into expression (53), assertion (44) is verified.

When Equations (16) and (19) are substituted into expression (52), the correctness of assertion (45) is established.

The application of Equations (17) and (19) in expression (52) ensures the validity of assertion (46).

By combining Equations (15), (16), and (17) with Equation (20), the individual verification of assertions (47), (48), and (49) are established.

Moreover, the use of Equations (15), (16), and (17) along with Equation (21) facilitates the proof of assertions (50), (51), and (52). □

Theorem 5.

The MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$ satisfy the following partial differential equations:

$$\begin{array}{cc}\hfill & \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_1}^n{D}_{{\phi }_2}+2{\phi }_2{D}_{{\phi }_1}^{n-1}{D}_{{\phi }_2}^2+3{\phi }_3{D}_{{\phi }_1}^{n-2}{D}_{{\phi }_2}^3+\cdots +m{\phi }_m{D}_{{\phi }_1}^{n-(m-1)}{D}_{{\phi }_2}^m\hfill \\ \hfill & +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^k{D}_{{\phi }_2}^{n-k+1}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D{n+1}_{{\phi }_1}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\hfill \end{array}$$

(54)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_1}^n{D}_{{\phi }_3}+2{\phi }_2{D}_{{\phi }_1}^{n-1}{D}_{{\phi }_2}{D}_{{\phi }_3}+3{\phi }_3{D}_{{\phi }_1}^{n-2}{D}_{{\phi }_2}{D}_{{\phi }_3}+\cdots +m{\phi }_m{D}_{{\phi }_1}^{n-(m-1)}{D}_{{\phi }_2}^{m-1}{D}_{{\phi }_3}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^k{D}_{{\phi }_2}^{n-k}{D}_{{\phi }_3}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}^{n+2}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(55)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_1}^{2n}{D}_{{\phi }_m}+2{\phi }_2{D}_{{\phi }_1}^{2n-1}{D}_{{\phi }_2}{D}_{{\phi }_m}+3{\phi }_3{D}_{{\phi }_1}^{2n-2}{D}_{{\phi }_2}{D}_{{\phi }_m}+\cdots +m{\phi }_m{D}_{{\phi }_1}^{2n-(m-1)}{D}_{{\phi }_2}^{m-1}{D}_{{\phi }_m}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{n+k}{D}_{{\phi }_2}^{n-k}{D}_{{\phi }_m}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_m}^{2n+m-1}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(56)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_1}^{2n+2}{D}_{{\phi }_2}+2{\phi }_2{D}_{{\phi }_1}^{2n}{D}_{{\phi }_2}{D}_{{\phi }_3}+3{\phi }_3{D}_{{\phi }_1}^{2n-2}{D}_{{\phi }_2}{D}_{{\phi }_3}^2+\cdots +m{\phi }_m{D}_{{\phi }_1}^{2n-2(m-2)}{D}_{{\phi }_2}{D}_{{\phi }_3}^{m-1}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{2(k+1)}{D}_{{\phi }_2}{D}_{{\phi }_3}^{n-k}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D^{2n+3}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(57)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D^{2n+2}{D}_{{\phi }_3}+2{\phi }_2{D}_{{\phi }_1}^{2n}{D}_{{\phi }_3}^2+3{\phi }_3{D}_{{\phi }_1}^{2n-2}{D}_{{\phi }_3}^3+\cdots +m{\phi }_m{D}_{{\phi }_1}^{2n-2(m-2)}{D}_{{\phi }_3}^m\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{2(k+1)}{D}_{{\phi }_3}^{n-k+1}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D^{2n+4}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(58)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_1}^{n^2+1}{D}_{{\phi }_m}+2{\phi }_2{D}_{{\phi }_1}^{n^2-1}{D}_{{\phi }_3}{D}_{{\phi }_m}+3{\phi }_3{D}_{{\phi }_1}^{n^2-3}{D}_{{\phi }_3}^2{D}_{{\phi }_m}+\cdots +m{\phi }_m{D}_{{\phi }_1}^{n^2+3-2m}{D}_{{\phi }_3}^{m-1}{D}_{{\phi }_m}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{n^2+1-2(n-k)}{D}_{{\phi }_3}^{n-k}{D}_{{\phi }_m}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}^{n^2+m}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(59)

$$⋮$$

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_1}^{n^2+1}{D}_{{\phi }_2}+2{\phi }_2{D}_{{\phi }_1}^{-(m-1)}{D}_{{\phi }_2}{D}_{{\phi }_m}+3{\phi }_3{D}_{{\phi }_1}^{n^2+3-2m}{D}_{{\phi }_2}{D}_{{\phi }_m}^2+\cdots +m{\phi }_m{D}_{{\phi }_1}^{n^2+1-{(m-1)}^2}{D}_{{\phi }_2}{D}_{{\phi }_m}^{m-1}\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{n^2+1-(m-1)(n-k)}{D}_{{\phi }_2}{D}_{{\phi }_m}^{n-k}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}^{n^2+2}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(60)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_1}^{n^2+2}{D}_{{\phi }_3}+2{\phi }_2{D}_{{\phi }_1}^{n^2+2-(m-1)}{D}_{{\phi }_3}{D}_{{\phi }_m}+3{\phi }_3{D}_{{\phi }_1}^{-2(m-1)}{D}_{{\phi }_3}{D}_{{\phi }_m}^2+\cdots +m{\phi }_m{D}_{{\phi }_1}^{n^2+2-{(m-1)}^2}\\ \hfill \times D_{{\phi }_3}{D}_{{\phi }_m}^{m-1}+{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{n^2+2-(m-1)(n-k)}{D}_{{\phi }_2}{D}_{{\phi }_m}^{n-k}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}^{n^2+4}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0,\end{array}$$

(61)

$$\begin{array}{c}\hfill \{\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_1}^{n^2+2}{D}_{{\phi }_m}+2{\phi }_2{D}_{{\phi }_1}^{n^2+1-m}{D}_{{\phi }_m}^2+3{\phi }_3{D}_{{\phi }_1}^{n^2-2m}{D}_{{\phi }_m}^3+\cdots +m{\phi }_m{D}_{{\phi }_1}^{n^2+2-{(m-1)}^2}{D}_{{\phi }_m}^m\\ \hfill +{\displaystyle \frac{1}{1-\rho }}\sum _{k=0}^{n-1}{D}_{{\phi }_1}^{n^2+2-(m-1)(n-k)}{D}_{{\phi }_m}^{n-k+1}{\displaystyle \frac{{\mathfrak{E}}_{n-k}^{\mathcal{F}}\left(\rho \right)}{(n-k)!}}-(n+1)D_{{\phi }_1}^{n^2+m+1}\}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0.\end{array}$$

(62)

Proof. 

By differentiating the integrodifferential expressions (44) and (45) with respect to ${\phi }_1$ up to n times, the validity of assertions (54) and (55) is established.

Similarly, differentiating the integrodifferential expression (46) with respect to ${\phi }_1$ for $2n$ times confirms the correctness of assertion (56).

In the same way, by applying partial derivatives $2n+2$ times to the integrodifferential expressions (47) through (49) with respect to ${\phi }_1$, the validity of assertions (57) through (59) is verified.

Furthermore, differentiating the integrodifferential expressions (50) and (51) $n^2+1$ times with respect to ${\phi }_1$ confirms the truth of assertions (60) and (61).

Finally, taking partial derivatives $n^2+2$ times with respect to ${\phi }_1$ for the integrodifferential expression (52) proves assertion (62). □

Fractional Volterra integral equations are pivotal in special functions, particularly in integral transforms and functional analysis. They provide integral representations, aiding in solving differential equations and analyzing orthogonal polynomials. Essential in Laplace, Fourier, and Mellin transforms, they extend to applications in physics, engineering, and dynamic systems, offering a powerful framework for modeling and analysis.

For the MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$, the integral equation is derived by proving the following result.

Theorem 6.

The MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$ satisfy the following homogeneous Volterra integral equation:

$$\begin{array}{cc}\hfill & \mathrm{\Psi }\left({\phi }_1\right)=-{\displaystyle \frac{m!(1-\rho )}{{\mathfrak{E}}_m^{\mathbb{F}}\left(\rho \right)}}(m{\phi }_{m}n(n-1)(n-2)\cdots (n-m+1){}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}(\nu ,\sigma ,\rho )+\cdots +3{\phi }_{3}n(n-1)(n-2)\hfill \\ \hfill & {}_{\mathcal{H}}\mathcal{E}_{n-3}^{\mathcal{F}}(\nu ,\sigma ,\rho )+2{\phi }_{2}n(n-1)(n-2){}_{\mathcal{H}}\mathcal{E}_{n-3}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\phi }_1\hfill \\ \hfill & +2{\phi }_{1}n(n-1){}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}(\nu ,\sigma ,\rho )+\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)(n(n-1)\cdots (n-m+1){}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\displaystyle \frac{{\phi }_1^m}{m!}}+\cdots +n(n-1)\hfill \\ \hfill & {}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}(\nu ,\sigma ,\rho )x+n{}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}(\nu ,\sigma ,\rho ))-n(n-1)\cdots (n-m+1){}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\displaystyle \frac{{\phi }_1^m}{2!m!}}-\cdots \hfill \\ \hfill & -n(n-1){}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\displaystyle \frac{{\phi }_1^2}{2!}}-n{}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\phi }_1-{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}(\nu ,\sigma ,\rho ))+\underset{0}{\overset{{\phi }_1}{\int }}(-{\displaystyle \frac{m!(1-\rho )}{{\mathfrak{E}}_m^{\mathbb{F}}\left(\rho \right)}}(3{\phi }_3+2{\phi }_2\hfill \\ \hfill & ({\phi }_1-\xi )+\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right){\displaystyle \frac{{({\phi }_1-\xi )}^2}{2!}})-n{\displaystyle \frac{{({\phi }_1-\xi )}^3}{3!}})\mathrm{\Psi }\left(\xi \right)d\xi .\hfill \end{array}$$

(63)

Proof. 

Consider the fourth-order differential equation of MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )$ of the following form:

$$\begin{array}{c}\left(D_{{\phi }_1}^m+\cdots +{\displaystyle \frac{m!(1-\rho )}{{\mathfrak{E}}_m^{\mathbb{F}}\left(\rho \right)}}\left(3{\phi }_3{D}_{{\phi }_1}^3+2{\phi }_2{D}_{{\phi }_1}^2+\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)D_{{\phi }_1}-n\right)\right){}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3,\cdots ,{\phi }_m;\rho )=0.\hfill \end{array}$$

(64)

For initial conditions, we find

$$\begin{array}{cc}\hfill & {}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,0,\cdots ,0;\rho )={}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2;\rho )=n!{\displaystyle \sum _{k=0}^n}{\displaystyle \sum _{r=0}^{\left[{\displaystyle \frac{k}{2}}\right]}}{\displaystyle \frac{{\mathbb{E}}_{n-k}^{\mathbb{F}}\left(\rho \right){{\phi }_1}^r{{\phi }_2}^{k-2r}}{(n-k)!r!(k-2r)}}\hfill \\ \hfill & :={}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}(\nu ,\sigma ,\rho ),\hfill \\ \\ \hfill & {\displaystyle \frac{d}{d{\phi }_1}}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,0,\cdots ,0;\rho )=n{}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,0,\cdots ,0;\rho )=n(n-1)!{\displaystyle \sum _{k=0}^{n-1}}{\displaystyle \sum _{r=0}^{\left[{\displaystyle \frac{k}{2}}\right]}}{\displaystyle \frac{{\mathbb{E}}_{n-1-k}^{\mathbb{F}}\left(\rho \right){{\phi }_1}^r{{\phi }_2}^{k-2r}}{(n-1-k)!r!(k-2r)}}\hfill \\ \hfill & :=n{}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}(\nu ,\sigma ,\rho ),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d^2}{d_{{\phi }_1}^2}}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,0,\cdots ,0;\rho )=n(n-1){}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}({\phi }_1,{\phi }_2,0,\cdots ,0;\rho )=n(n-1)(n-2)!\hfill \\ \hfill & \times {\displaystyle \sum _{k=0}^{n-2}}{\displaystyle \sum _{r=0}^{\left[{\displaystyle \frac{k}{2}}\right]}}{\displaystyle \frac{{\mathbb{E}}_{n-2-k}^{\mathbb{F}}\left(\rho \right){{\phi }_1}^r{{\phi }_2}^{k-2r}}{(n-2-k)!r!(k-2r)}}:=n(n-1){}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}(\nu ,\sigma ,\rho ),\hfill \\ \hfill & {\displaystyle \frac{d^3}{d_{{\phi }_1}^3}}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,0,\cdots ,0;\rho )=n(n-1)(n-2){}_{\mathcal{H}}\mathcal{E}_{n-3}^{\mathcal{F}}({\phi }_1,{\phi }_2,0,\cdots ,0;\rho )=n(n-1)(n-2)(n-3)!\hfill \\ \hfill & {\displaystyle \sum _{k=0}^{n-3}}{\displaystyle \sum _{r=0}^{\left[{\displaystyle \frac{k}{2}}\right]}}{\displaystyle \frac{{\mathbb{E}}_{n-3-k}^{\mathbb{F}}\left(\rho \right){{\phi }_1}^r{{\phi }_2}^{k-2r}}{(n-3-k)!r!(k-2r)}}\hfill \\ \hfill & :=n(n-1)(n-2){}_{\mathcal{H}}\mathcal{E}_{n-3}^{\mathcal{F}}(\nu ,\sigma ,\rho ),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ⋮\hfill \\ \hfill & {\displaystyle \frac{d^m}{d_{{\phi }_1}^m}}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,0,\cdots ,0;\rho )=n(n-1)(n-2)\cdots (n-m+1){}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}({\phi }_1,{\phi }_2,0,\cdots ,0;\rho )\hfill \\ \hfill & =n(n-1)(n-2)\cdots (n-m+1)!{\displaystyle \sum _{k=0}^{n-m+1}}{\displaystyle \sum _{r=0}^{\left[{\displaystyle \frac{k}{2}}\right]}}{\displaystyle \frac{{\mathbb{E}}_{n-m-k}^{\mathbb{F}}\left(\rho \right){{\phi }_1}^r{{\phi }_2}^{k-2r}}{(n-m-k)!r!(k-2r)}}\hfill \\ \hfill & :=n(n-1)(n-2)\cdots (n-m+1){}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}(\nu ,\sigma ,\rho ),\hfill \end{array}$$

(65)

respectively, where

$${}_{\mathcal{H}}\mathcal{E}_s^{\mathcal{F}}(\nu ,\sigma ,\rho ):=s!\sum _{k=0}^s\sum _{r=0}^{\left[{\displaystyle \frac{k}{2}}\right]}{\displaystyle \frac{{\mathbb{E}}_{s-k}^{\mathbb{F}}\left(\rho \right){{\phi }_1}^r{{\phi }_2}^{k-2r}}{(s-k)!r!(k-2r)}},s=n,n-1,n-2,n-3\cdots n-m+1.$$

(66)

Consider

$$D_{{\phi }_1}^m{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m;\rho )=\mathrm{\Psi }\left({\phi }_1\right).$$

(67)

Applying the initial conditions from Equation (3) and integrating the given equation, we derive the following result:

$$\begin{array}{c}{\displaystyle \frac{d^m}{d{{\phi }_1}^m}}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m;\rho )=\underset{0}{\overset{{\phi }_1}{\int }}\mathrm{\Psi }\left(\xi \right)d\xi +n(n-1)\cdots (n-m+1){}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}(\nu ,\sigma ,\rho ),\hfill \\ ⋮\hfill \\ \\ {\displaystyle \frac{d^3}{d{{\phi }_1}^3}}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m;\rho )=\underset{0}{\overset{{\phi }_1}{\int }}\mathrm{\Psi }\left(\xi \right)d\xi +n(n-1)(n-2){}_{\mathcal{H}}\mathcal{E}_{n-3}^{\mathcal{F}}(\nu ,\sigma ,\rho ),\hfill \\ \\ {\displaystyle \frac{d^2}{d{q-1}^2}}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m;\rho )=\underset{0}{\overset{{\phi }_1}{\int }}\mathrm{\Psi }\left(\xi \right)d{\xi }^2+n(n-1)(n-2){}_{\mathcal{H}}\mathcal{E}_{n-3}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\phi }_1+n(n-1){}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}(\nu ,\sigma ,\rho ),\hfill \\ \\ {\displaystyle \frac{d}{d_{{\phi }_1}}}{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m;\rho )=\underset{0}{\overset{{\phi }_1}{\int }}\mathrm{\Psi }\left(\xi \right)d{\xi }^3+n(n-1)(n-2){}_{\mathcal{H}}\mathcal{E}_{n-3}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\displaystyle \frac{{{\phi }_1}^2}{2!}}+n(n-1){}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\phi }_1\hfill \\ +n{}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}(\nu ,\sigma ,\rho ),\hfill \\ \\ {}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,\cdots ,{\phi }_m;\rho )=\underset{0}{\overset{{\phi }_1}{\int }}\mathrm{\Psi }\left(\xi \right)d{\xi }^4+n(n-1)(n-2){}_{\mathcal{H}}\mathcal{E}_{n-3}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\displaystyle \frac{{{\phi }_1}^3}{2!3!}}+n(n-1){}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\displaystyle \frac{{q-1}^2}{2!}}\hfill \\ +n{}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\phi }_1+{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}(\nu ,\sigma ,\rho ).\hfill \end{array}$$

(68)

Based on the preceding expression in (64), we obtain

$$\begin{array}{cc}\hfill & \mathrm{\Psi }\left({\phi }_1\right)=-{\displaystyle \frac{m!(1-\rho )}{{\mathfrak{E}}_m^{\mathbb{F}}\left(\rho \right)}}(m{\phi }_m\left(\underset{0}{\overset{{\phi }_1}{\int }}\mathrm{\Psi }\left(\xi \right)d\xi +n(n-1)\cdots (n-m+1){}_{\mathcal{H}}\mathcal{E}_{n-m}^{\mathcal{F}}(\nu ,\sigma ,\rho )\right)+\cdots \hfill \\ \hfill & +2{\phi }_2\left(\underset{0}{\overset{{\phi }_1}{\int }}\mathrm{\Psi }\left(\xi \right)d{\xi }^3+n(n-1)(n-2){}_{\mathcal{H}}\mathcal{E}_{n-3}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\phi }_1+n(n-1){}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}(\nu ,\sigma ,\rho )\right)+\left({\phi }_1-{\displaystyle \frac{1}{1-\rho }}\right)\hfill \\ \hfill & \left(\underset{0}{\overset{{\phi }_1}{\int }}\mathrm{\Psi }\left(\xi \right)d{\xi }^3+n(n-1)(n-2){}_{\mathcal{H}}\mathcal{E}_{n-3}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\displaystyle \frac{{{\phi }_1}^2}{2!}}+n(n-1){}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\phi }_1+n{}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}(\nu ,\sigma ,\rho )\right))\hfill \\ \hfill & +{\displaystyle \frac{6n(1-\rho )}{{\mathfrak{E}}_3^{\mathbb{F}}\left(\rho \right)}}(\underset{0}{\overset{{\phi }_1}{\int }}\mathrm{\Psi }\left(\xi \right)d{\xi }^4+n(n-1)(n-2){}_{\mathcal{H}}\mathcal{E}_{n-3}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\displaystyle \frac{{{\phi }_1}^3}{2!3!}}+n(n-1){}_{\mathcal{H}}\mathcal{E}_{n-2}^{\mathcal{F}}(\nu ,\sigma ,\rho ){\displaystyle \frac{{{\phi }_1}^2}{2!}}\hfill \\ \hfill & +n{}_{\mathcal{H}}\mathcal{E}_{n-1}^{\mathcal{F}}(\nu ,\sigma ,\rho )x+{}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}(\nu ,\sigma ,\rho )),\hfill \end{array}$$

(69)

Thus, using the following method after simplifying and integrating the obtained equation,

$$\underset{b}{\overset{q_1}{\int }}f\left(\eta \right)d{\eta }^n=\underset{b}{\overset{q_1}{\int }}{\displaystyle \frac{{(q_1-\eta )}^{n-1}}{(n-1)!}}f\left(\eta \right)d\eta ,$$

(70)

assertion (63) is proved. □

4. Approximate Roots of MHFEPs

The investigation of approximate roots of MHFEPs is essential for uncovering their structural and analytical characteristics. These roots offer deep insights into the distribution of zeros, stability behavior, and asymptotic properties of the polynomials. Understanding them is pivotal for solving complex polynomial equations that frequently arise in mathematical frameworks such as differential, integrodifferential, and functional equations.

Furthermore, the numerical computation of approximate roots holds significant relevance in computational mathematics, optimization theory, and applied sciences. Through advanced simulation and visualization techniques, researchers can analyze root configurations, examine their mutual interactions, and assess their influence on system dynamics. This knowledge not only enhances the practical applicability of MHFEPs in fields like physics, engineering, and mathematical modeling but also facilitates the development of robust numerical algorithms for tackling large-scale polynomial systems. Thus, studying approximate roots broadens the theoretical depth and expands the real-world utility of MHFEPs across diverse scientific disciplines.

For any $n\in {\mathbb{N}}_0$ and $m=3$, the first few MHFEPs ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3;\rho )$ are given as

$$\begin{array}{ccc}\hfill {}_{\mathcal{H}}\mathcal{E}_0^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3;\rho )& =& 1,\hfill \\ \hfill {}_{\mathcal{H}}\mathcal{E}_1^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3;\rho )& =& {\rho }_1+{\displaystyle \frac{1}{-1+\rho }},\hfill \\ \hfill {}_{\mathcal{H}}\mathcal{E}_2^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3;\rho )& =& {\rho }_1^2+2{\rho }_2+{\displaystyle \frac{2{\rho }_1}{-1+\rho }}+{\displaystyle \frac{1+\rho }{{(-1+\rho )}^2}},\hfill \\ \hfill {}_{\mathcal{H}}\mathcal{E}_3^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3;\rho )& =& {\displaystyle \frac{1-6{\rho }_3+3{\rho }_1^2{(-1+\rho )}^2+6{\rho }_2{(-1+\rho )}^2+{\rho }_1^3{(-1+\rho )}^3+4\rho +18{\rho }_3\rho +{\rho }^2}{{(-1+\rho )}^3}}\hfill \\ & & -{\displaystyle \frac{-18{\rho }_3{\rho }^2+6{\rho }_3{\rho }^3+3{\rho }_1(-1+\rho )\left(1+2{\rho }_2{(-1+\rho )}^2+\rho \right)}{{(-1+\rho )}^3}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {}_{\mathcal{H}}\mathcal{E}_4^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3;\rho )& =& {\displaystyle \frac{1}{{(-1+\rho )}^4}}\left(1-24{\rho }_3+4{\rho }_1^3{(-1+\rho )}^3+{\rho }_1^4{(-1+\rho )}^4+12{\rho }_2^2{(-1+\rho )}^4\right)\hfill \\ & & +{\displaystyle \frac{1}{{(-1+\rho )}^4}}\left(11\rho +72{\rho }_3\rho +11{\rho }^2-72{\rho }_3{\rho }^2+{\rho }^3+24{\rho }_3{\rho }^3+12{\rho }_2{(-1+\rho )}^2(1+\rho )\right)\hfill \\ & & +{\displaystyle \frac{1}{{(-1+\rho )}^4}}\left(6{\rho }_1^2{(-1+\rho )}^2\left(1+2{\rho }_2{(-1+\rho )}^2+\rho \right)\right)\hfill \\ & & +{\displaystyle \frac{1}{{(-1+\rho )}^4}}\left(4{\rho }_1(-1+\rho )\left(1+6{\rho }_2{(-1+\rho )}^2+6{\rho }_3{(-1+\rho )}^3+4\rho +{\rho }^2\right)\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {}_{\mathcal{H}}\mathcal{E}_5^{\mathcal{F}}({\phi }_1,{\phi }_2,{\phi }_3;\rho )& =& {\displaystyle \frac{1-\rho }{{(\rho -1)}^6}}\left(120+240(-1+\rho )+150{(-1+\rho )}^2+30{(-1+\rho )}^3+{(-1+\rho )}^4\right)\hfill \\ & & +{\displaystyle \frac{1-\rho }{{(\rho -1)}^6}}\left(5\left({\rho }_1^4+12{\rho }_1^2{\rho }_2+12{\rho }_2^2+24{\rho }_1{\rho }_3\right){(-1+\rho )}^4\right)\hfill \\ & & +{\displaystyle \frac{1-\rho }{{(\rho -1)}^6}}\left(\left({\rho }_1^5+20{\rho }_1^3{\rho }_2+60{\rho }_1{\rho }_2^2+60{\rho }_1^2{\rho }_3+120{\rho }_2{\rho }_3\right){(-1+\rho )}^5\right)\hfill \\ & & +{\displaystyle \frac{1-\rho }{{(\rho -1)}^6}}\left(10\left({\rho }_1^3+6{\rho }_1{\rho }_2+6{\rho }_3\right){(-1+\rho )}^3(1+\rho )+10\left({\rho }_1^2+2{\rho }_2\right){(-1+\rho )}^2\left(1+4\rho +{\rho }^2\right)\right)\hfill \\ & & +{\displaystyle \frac{1-\rho }{{(\rho -1)}^6}}\left(5\times (-1+\rho )\left(1+11\rho +11{\rho }^2+{\rho }^3\right)\right).\hfill \end{array}$$

To analyze the zero distributions of the MHFEP polynomials, we employ the computational software Wolfram Mathematica. Graphical representations of these zero distributions are generated by assigning specific values to the polynomial parameters: ${\rho }_2\to 2,{\rho }_3\to 3,\rho \to -1$. The generating function used for this analysis is given by

$$\left({\displaystyle \frac{2}{e^t+1}}\right)e^{{\phi }_{1}t+2t^2+3t^3}.$$

(71)

We illustrate the zero distributions for the cases $n=5,10,15,20$ through corresponding plots, see Figure 1.

These visualizations provide meaningful insights into the behavior and spatial arrangement of the zeros, see Figure 2, highlighting how the structure of MHFEP polynomials evolves with increasing degree.

Using advanced computational techniques, we numerically approximated the zeros of the hybrid multivariate Hermite–Frobenius–Euler polynomials ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,2,3;-1)$ for degrees ranging from $n=1$ to $n=12$. These computations were carried out by fixing the parameters ${\phi }_2=2$, ${\phi }_3=3$, and $\rho =-1$, allowing us to isolate the behavior of the polynomials with respect to the primary variable ${\phi }_1$. Root-finding algorithms were implemented in symbolic computation software (e.g., Wolfram Mathematica https://www.wolfram.com/mathematica/), enabling the identification of real and complex zeros with high numerical precision. The results, which are systematically tabulated in

Table 2. Approximate solutions of ${}_{\mathcal{H}}\mathcal{E}_n^{\mathcal{F}}({\phi }_1,2,3;-1)=0$.

Degree (n)	Zeros
1	$0.5$
2	$0.5-1.93649i$,    $0.5+1.93649i$
3	$-0.870956i$,    $1.18548-3.55804i$,    $1.18548+3.55804i$
4	$-1.40559$,    $-0.286074$,    $1.84583\pm 5.14158i$
5	$-3.006$,    $0.274234\pm 1.99996i$,    $2.47876\pm 6.60399i$
6	$-4.20824$,    $-0.587872$,    $0.807897\pm 3.33397i$,    $3.09016\pm 7.97583i$
7	$-5.40258$,    $-0.601532\pm 0.955592i$,    $1.37165\pm 4.61844i$,    $3.68118\pm 9.2747i$
8	$-6.5522$,    $-1.96148$,    $0.0832452\pm 2.10631i$,    $1.91983\pm 5.84918i$,    $4.25377\pm 10.5132i$
9	$-7.66744$,    $-2.88547$,    $-0.646923$,    $0.583922\pm 3.28674i$,    $2.45622\pm 7.03152i$,    $4.80978\pm 11.7006i$
10	$-8.7515$,    $-3.9782$,    $-0.557851\pm 1.22206i$,    $1.09079\pm 4.4138i$,    $2.98106\pm 8.17181i$,    $5.35085\pm 12.8442i$
11	$-9.80765$,    $-5.00856$,    $-1.4906$,    $-0.059321\pm 2.22853i$,    $1.58938\pm 5.50867i$,    $3.49494\pm 9.27496i$,    $5.87841\pm 13.9494i$
12	$-10.8386$,    $-6.02182$,    $-2.00533$,    $-0.917208$,    $0.419532\pm 3.29027i$,    $2.07977\pm 6.57233i$,    $3.99847\pm 10.3451i$,    $6.39368\pm 15.0209i$

, reveal distinct patterns and symmetries in the zero distributions as the degree increases, reflecting the underlying algebraic structure and stability properties of the polynomials. This numerical investigation provides valuable insight into the oscillatory behavior and convergence characteristics of these polynomials, thereby reinforcing their theoretical significance and supporting their applicability in spectral methods, approximation theory, and mathematical modeling of multidimensional systems.

5. Conclusions

In this study, we introduced and systematically developed a new class of hybrid multidimensional polynomials, constructed through a convolution of Frobenius–Euler and multivariate Hermite polynomials. This novel formulation bridges two important families of special functions, resulting in a rich polynomial structure that captures both the discrete exponential-type behavior of Frobenius–Euler polynomials and the Gaussian-type behavior of Hermite polynomials. The multidimensional nature of the construction allows for the modeling of systems with multiple interacting variables, significantly expanding the analytical and applicational potential of these polynomials.

We derived recurrence relations that elucidate the stepwise generation of higher-order terms, highlighting internal algebraic consistency and facilitating algorithmic computation. The formulation of shift operators and structural identities further provided insights into their operational behavior, enabling a deeper understanding of how these polynomials transform under various differential and algebraic operations. These results established a solid foundation for exploring the polynomials’ behavior in dynamic systems governed by symmetry and recurrence.

A key advancement of this work lies in the application of the factorization method, which allowed us to generate associated differential, partial differential, and integrodifferential equations. These equations not only characterize the polynomials analytically but also serve as gateways to applying them in mathematical physics, particularly in quantum mechanics and heat conduction models. The derivation of a fractional Volterra-type integral equation represents a novel contribution, offering a nonlocal operator-based perspective that connects the polynomials with fractional calculus and memory-dependent systems—domains increasingly relevant in modeling viscoelasticity, anomalous diffusion, and complex networks.

Furthermore, computational simulations were conducted to visualize and analyze the distribution of polynomial zeros, providing empirical validation of their structural properties. The approximated roots exhibited symmetry and regularity patterns consistent with the theoretical framework, indicating their potential use in spectral methods, approximation theory, and numerical quadrature. The computational analysis thus complements the symbolic derivations and illustrates how these polynomials behave under parameter variation, especially in high-dimensional settings.

The results of this work establish a foundational analytical framework for multivariate Hermite–Frobenius–Euler polynomials. The derived equations and identities not only demonstrate their internal consistency and algebraic richness but also point toward a wide array of future research opportunities. For example, a deeper investigation into their symmetry properties, determinantal structures, and orthogonality relations could yield important connections to representation theory and matrix analysis. Moreover, exploring their generating functions in the multivariate q-calculus framework could reveal new avenues in quantum algebra and combinatorics.

From an application standpoint, these polynomials hold promise for solving complex differential systems in computational physics, engineering models, and statistical mechanics, particularly where multidimensional interactions and memory effects are prominent. Future studies could also extend this class by incorporating deformations, weight functions, or parameterized symmetries and by developing efficient symbolic algorithms for their computation, contributing to advancements in both analytical theory and computational implementation.