1.
Introduction
Special polynomials are highly important across mathematics, theoretical physics, and engineering due to their fundamental roles and applications, particularly in analyzing the differential equations common to physics and engineering problems. Furthermore, these special polynomials readily yield numerous useful identities and are foundational for defining new polynomial classes. Notably significant are the Gould-Hopper and Bell polynomials, prized for their wide-ranging use across mathematics [1,2,3].
The Apostol-type polynomials, particularly the Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, are significant in pure and applied mathematics, attracting considerable research. Studies by Luo et al. [4,5,6] and Ozarslan [7] explored fundamental properties and explicit series representations for these polynomials. Srivastava [8] focused on explicit representations via a generalized Hurwitz-Lerch; zeta function. Further, various extended forms of Apostol-type polynomials have been explored, for example, parametric extensions [9,10], hybrid classes like truncated-exponential-Apostol-type polynomials [11]), unified formulas connecting to other families [12], and unified frameworks [13,14].
The following notations and definitions will be employed consistently in this study: R refers to the set of real numbers, C refers to the set of complex numbers, Z refers to for the set of integers, N refers to the set of positive integers, and N0=N∪{0} refers to the set of non-negative integers.
The Gould-Hopper polynomials H(r)τ(ω1,ω2) [15] are defined as follows:
and represented by the series
The classical Bell polynomials Belτ(ω) [16,17] are defined by
The 2-variable Bell polynomials, denoted as Belτ(ω1,ω2), are defined as follows [18,19]:
Recently, the Gould-Hopper-Bell polynomials (GHBelP) HBel(r)τ(ω1,ω2,z) were introduced in [20] by the generating function
and represented by the series
Operational methods involving differential operators, derived from the monomiality principle, offer effective tools for studying classical polynomial classes and their diverse extensions. The concept of monomiality originates from the notion of poweroid introduced by Steffensen [21]. This concept was revisited and methodically applied by Dattoli [22]. In line with the monomiality principle [21,22], a polynomial set ρτ(ω) (τ∈N,ω∈C) is termed quasi-monomial if it is possible to define "multiplicative" (ˆM) and "derivative" (ˆP) operators for which
for all τ∈N. Moreover, these operators satisfy the relation
and therefore reveals the Weyl group structure. If the polynomial set {ρτ(ω)}τ∈N under consideration is quasi-monomial, its properties can be readily determined from the properties of the operators ˆM and ˆP. Consequently, we have:
(ⅰ) Differential realizations of ˆM and ˆP imply that ρτ(ω) fulfills the differential equation
(ⅱ) With the assumption that ρ0(ω)=1, we have an explicit construction for the polynomials ρτ(ω) as
from which we derive the series definition of ρτ(ω).
(ⅲ) Based on identity (1.11), we can express the exponential generating function of ρτ(ω) as follows:
The quasi-monomiality of the GHBelP HBel(r)τ(ω1,ω2,z) [20] is established via the following operators:
and
Based on the monomiality principle, the GHBelP HBel(r)τ(ω1,ω2,z) fulfills the following identities:
The Apostol-Bernoulli B(σ)τ(ω;ζ) [5], Apostol-Euler E(σ)τ(ω;ζ) [4], and Apostol-Genocchi G(σ)τ(ω;ζ) [23] polynomials, all of order σ, are respectively defined by
where σ and ζ are arbitrary real or complex parameters. When ω=0 in Eqs (1.19)–(1.21), we get
which denote the Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi numbers of order σ, respectively.
The unified family of generalized Apostol-type polynomials P(σ)τ,ζ(ω;δ,a,b) [7] are given by
where P(σ)τ,ζ(0;δ,a,b):=P(σ)τ,ζ(δ,a,b) denotes the generalized Apostol-type numbers. Also, we note that
and
Special polynomials can be defined through several ways, such as generating functions, series representations, determinant representations, and differential and integral representations. The hybrid special polynomials can be defined mostly by means of the generating functions using several techniques. The choice of the most suitable technique is determined by specific properties inherent to the combined polynomials. Some of these techniques include the operational technique [24,25] and series expansion technique [26,27]. In recent years, there has been growing interest in a novel method concerning special functions, known as the determinant approach, which was introduced by Costabile et al. [26,28,29].
The generalized special polynomials enhance the applicability of classical special polynomials by integrating their advantages and offering increased adaptability. This evolution renders them more adaptable and potent in addressing intricate contemporary challenges across diverse fields. These polynomials prove especially effective in tackling multifaceted, cross-disciplinary issues and driving progress in both theoretical frameworks and practical applications within mathematics. By generalizing classical polynomials, researchers can unlock new tools for approximation, interpolation, and solving differential equations, while also gaining deeper insights into the relationships between different polynomial families.
In recent studies, various researchers have utilized operational techniques in combination with the monomiality principle to study classical special polynomials and to develop generalized classes. Notable contributions in this area are presented in several works [11,24,30,31,32], with further advancements and applications reported in several studies [33,34,35], as well as in extended formulations explored in recent literature [36,37,38]. Further, several researchers presented certain results for the hybrid form of special polynomials associated with the Apostol-type polynomials [39,40].
In this work, in Section 2, by combining the Gould-Hopper-Bell polynomials and the unified Apostol-type polynomials, and in view of the monomiality principle, we provide a generalized class of hybrid special polynomials, referred to as the trivariate Gould-Hopper-Bell-Apostol-type polynomials. Next, the series representations, quasi-monomial operators, and differential equations are derived. In Section 3, we establish some summation formulae for the trivariate Gould-Hopper-Bell-Apostol-type polynomials. In Section 4, we investigate some related differential and integral identities. In Section 5, the Gould-Hopper-Bell-Apostol-Bernoulli, Gould-Hopper-Bell-Apostol-Euler, and Gould-Hopper-Bell-Apostol-Genocchi polynomials are introduced as specific cases, and their associated results are also discussed. Finally, the zero distributions and graphical representations are examined.
2.
Family of trivariate Gould-Hopper-Bell-Apostol-type polynomials
In this section, we introduce a novel unified family of hybrid special polynomials, referred to as the trivariate Gould-Hopper-Bell-Apostol-type polynomials (TGHBelATP), through generating functions and series representations. Additionally, based on the principle of monomiality, the generating function is utilized to derive the related multiplicative and derivative operators, as well as the associated differential equation.
In the generating function (1.22), replacing ω by the multiplicative operator ˆMGHBel (1.13) of the GHBelP HBel(r)τ(ω1,ω2,z), gives
Applying Eq (1.18) to the preceding equation, and denoting P(σ)τ,ζ(ˆMGHBel;δ,a,b) by HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) (the trivariate Gould-Hopper-Bell-Apostol-type polynomials), yields:
Now, utilizing Eq (1.5) in the above equation, we arrive at the following definition.
Definition 1. The trivariate Gould-Hopper-Bell-Apostol-type polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) of order σ are defined by the generating function:
Remark 1. Setting z=0 in generating relation (2.3), we get the unified Gould-Hopper-Apostol-type polynomials HP(σ,r)τ,ζ(ω1,ω2;δ,a,b) of order σ which are defined by:
which is a special case of the polynomials defined by the generating function (2.1) in [41, P. 291].
Remark 2. Setting ω2=0 in generating relation (2.3), we get the new 2-variable unified Bell-Apostol-type polynomials BelP(σ)τ,ζ(ω1,z;δ,a,b) of order σ given by the generating function
Remark 3. Setting r=2, σ=1 in the generating relation (2.3), we get new special polynomials, called trivariate Hermite Kampé de Fériet-Bell-Apostol-type polynomials HBelP(2)τ,ζ(ω1,ω2,z;δ,a,b), given by the generating function
Remark 4. Setting σ=1 in the generating relation (2.3), we get the trivariate Gould-Hopper-Bell-Apostol-type polynomials HBelP(r)τ,ζ(ω1,ω2,z;δ,a,b), which are defined by the generating function
Taking ω1=ω2=0 and z=1 in (2.3), we get unified Bell-Apostol-type numbers of order σ, which are defined by
Next, in view of generating function (2.3), we establish certain series representations of the TGHBelATP HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) of order σ.
Theorem 1. The trivariate Gould-Hopper-Bell-Apostol-type polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) of order σ satisfy the following series representations:
Proof. In view of generating relations (1.3), (2.3), and (2.4), we have
which, upon substituting τ⟶τ−κ and applying the Cauchy product rule, yields:
from which, by comparing the coefficients of powers of μ, we derive Eq (2.9). Similarly, the assertions in Eqs (2.10) and (2.11) can be proved. □
Remark 5. Setting r=2, σ=1 in series representations (2.9)–(2.11), we find that the trivariate Hermite Kampé de Fériet-Bell-Apostol-type polynomials HBelP(2)τ,ζ(ω1,ω2,z;δ,a,b) satisfy the following series representations:
Theorem 2. For the unified polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b), the associated multiplicative and derivative operators demonstrating their quasi-monomial nature are:
and
respectively.
Proof. Differentiating relation (2.3) partially with respect to μ, gives
which, upon replacing τ by τ+1 in the right-hand side and using relation (2.3) the left-hand side, becomes
Matching the coefficients of corresponding powers of μ in Eq (2.20), we obtain
Using Eq (1.7) (for HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)) in (2.21), we obtain assertion (2.17).
Further, differentiating the left-hand side of (2.3) with respect to ω1, we get
Using relation (2.3) in expression (2.22), gives
which, upon replacing τ by τ−1 in the right-hand side and then comparing the coefficients of corresponding powers of μ in the resulting equation, we obtain
Using Eq (1.8) (for HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)) in (2.24), we obtain assertion (2.18). □
Remark 6. For z=0, Theorem (2) gives the associated multiplicative and derivative operators demonstrating the quasi-monomial nature of the unified Gould-Hopper-Apostol-type polynomials HP(σ,r)τ,ζ(ω1,ω2;δ,a,b) of order σ:
and
respectively.
Theorem 3. The unified polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) satisfy the following differential equation:
Proof. In view of Eq (1.10) (for HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)), utilizing operators (2.17) and (2.18), we get the asserted result (2.27). □
Remark 7. For z=0, Theorem (3) gives the following differential equation that is satisfied by the unified Gould-Hopper-Apostol-type polynomials HP(σ,r)τ,ζ(ω1,ω2;δ,a,b) of order σ:
3.
Summation formulae
In this section, we investigate certain remarkable summation identities for the trivariate Gould-Hopper-Bell-Apostol-type polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) of order σ.
Theorem 4. For τ∈N0,δ∈N and σ,ζ∈C, we have
Proof. From generating relation (2.3), it follows that:
which can be written as
where, Eτ represents the Euler numbers defined by [42]:
By applying the Cauchy product rule to (3.3) and equating the corresponding powers of μ in the resulting equation, we obtain (3.1). □
Remark 8. Setting z=0 in (3.1), we get the following summation formula:
Theorem 5. The unified polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) satisfy the following summation formula:
Proof. Replacing ω1 by υ in (2.3), we have
From (3.7), we get asserted result (3.6). □
Remark 9. For ω2=0 in (3.6), the unified polynomials BelP(σ)τ,ζ(ω1,z;δ,a,b) of order σ satisfy the following summation formula:
Theorem 6. For τ∈N0,δ∈N, and σ,β,ζ∈C, we have
Proof. In (2.3), replacing ω1,ω2,z, and σ by ω1+x,ω2+y,z+u, and σ+β, respectively, we have
From (3.10), we get the asserted result (3.9). □
Remark 10. For r=2 and σ=1 in (3.9), the unified polynomials HBelP(2)τ,ζ(ω1,ω2,z;δ,a,b) satisfy the following summation formula:
The generalized Stirling numbers of the second kind are given as follows [13,43]:
Theorem 7. The unified polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) satisfy the following summation formula:
Proof. In view of (2.3) and (3.12), we can write
Upon simplifying the aforementioned relation and equating the coefficients of μττ! on both sides of the resulting equation, we obtain the stated result (3.13). □
Remark 11. For ω2=0 in (3.13), the unified polynomials BelP(σ)τ,ζ(ω1,z;δ,a,b) of order σ satisfy the following summation formula:
Theorem 8. The unified polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) satisfy the following summation formula:
Proof. In view of (1.5) and (2.3) for σ=1, we can write
which, upon comparing the coefficients of μττ! on both sides, yields the asserted result (3.16). □
Remark 12. For r=2 in (3.16), the unified polynomials HBelP(1,2)τ,ζ(ω1,ω2,z;δ,a,b) satisfy the following summation formula:
4.
Differential and integral formulae
In this section, we establish some differential and integral formulae associated with the trivariate Gould-Hopper-Bell-Apostol-type polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) of order σ.
Theorem 9. For ν,τ∈N0,δ∈N, and σ,ζ∈C, we have
Proof. Differentiating the generating relation (2.3) ν times with respect to ω1, we obtain
By simplifying Eq (4.2) and subsequently comparing the coefficients of μττ! on both sides of the resultant equation, we arrive at the asserted result, as given by (4.1). □
Remark 13. Setting ω2=0 in (4.1), we have
Similarly, upon differentiating relation (2.3) ν times with respect to ω2, we can get the following result.
Theorem 10. For ν,τ∈N0,δ∈N, and σ,ζ∈C, we have
Remark 14. Setting r=2 and σ=1 in (4.4), we have
Theorem 11. For τ∈N0,δ∈N, and σ,ζ∈C, we have
Proof. We start with generating relation (2.3). Differentiating it with respect to z, followed by simplification using equation (2.9), results in
From (4.7), we get the asserted result (4.6). □
Remark 15. Setting ω2=0 in (4.6), we have
Similarly, we can derive the following outcome.
Corollary 1. For τ∈N0,δ∈N, and σ,ζ∈C, we have
Theorem 12. The following formula holds true:
Proof. By integrating both sides of Eq (2.3) with respect to ω1, we obtain
From (4.11), we get asserted result (4.10). □
Similarly, the following results can be proved.
Theorem 13. The following formulas hold true:
In the next section, we turn to the consideration of several special cases of the trivariate Gould-Hopper-Bell-Apostol-type polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) of order σ.
5.
Applications
In this section, some applications related to the established polynomials (TGHBelATP HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) of order σ) are presented. Certain examples are investigated. Further, the zero distributions of the TGHBelATP HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) of order σ are examined.
5.1. Examples
Here, we introduce certain special members belonging to the unified family
with analogous results presented for each.
Example 1. Gould-Hopper-Bell-Apostol-Bernoulli polynomials
Since
therefore, taking δ=a=b=1 in generating function (2.3), gives
where HBelB(σ,r)τ,ζ(ω1,ω2,z) are referred to as the Gould-Hopper-Bell-Apostol-Bernoulli polynomials (GHBelBP) of order σ.
The series representations of the GHBelBP HBelB(σ,r)τ,ζ(ω1,ω2,z) of order σ are given as:
Certain corresponding results related to the GHBelBP HBelB(σ,r)τ,ζ(ω1,ω2,z) of order σ are presented in Table 1.
The first few members of the GHBelBP
for \sigma = 1 and r = 3 are given as:
Example 2. Gould-Hopper-Bell-Apostol-Euler polynomials
Since
therefore, taking \delta = 0, a = -1 and b = 1 in generating function (2.3), gives
where
are referred to as the Gould-Hopper-Bell-Apostol-Euler polynomials (GHBelEP) of order \sigma .
The series representations of the GHBelEP
of order \sigma are given as:
Certain corresponding results related to the GHBelEP {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) of order \sigma are presented in Table 2.
The first few members of the GHBelEP
for \sigma = 2 and r = 3 are given as:
Example 3. Gould-Hopper-Bell-Apostol-Genocchi polynomials
Since
therefore, taking \delta = 1, a = -\frac{1}{2}, b = 1 and \zeta\longrightarrow\frac{\zeta}{2} in generating function (2.3), gives
where {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) are referred to as the Gould-Hopper-Bell-Apostol-Genocchi polynomials (GHBelGP) of order \sigma .
The series representations of the GHBelGP {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) of order \sigma are given as:
Certain corresponding results related to the GHBelGP {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) of order \sigma are presented in Table 3.
The first few members of the GHBelGP {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) , for \sigma = 1 and r = 3 are given as:
The established family in this study, the TGHBelATP {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z; \delta, a, b) of order \sigma , can be considered as a generalization of the Gould-Hopper, Bell, unified Apostol-type, Gould-Hopper-Bell, Apostol Bernoulli, Apostol Euler, Apostol Genocchi, Gould-Hopper-Apostol-type, Apostol-type-Hermite, Bell-Apostol-type, Gould-Hopper-Bernoulli, Gould-Hopper-Euler, Gould-Hopper-Genocchi, Bell-Bernoulli, Bell-Euler, and Bell-Genocchi polynomials [39,44,45].
5.2. Zeros and graphical representations
In this subsection, we explore the distributions of zeros and present graphical illustrations of the TGHBelATP {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z; \delta, a, b) of order \sigma for specific parameter values and indices.
In view of (2.3), we list the first six terms of the TGHBelATP
for \delta = 2, \sigma = 1 , and r = 3 as:
To show the shapes of the TGHBelATP {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z;\delta, a, b) for \tau = 2; 3; 4; 5; 6; 7, -100 \leq \omega_{1} \leq 100, \omega_{2} = \frac{1}{2}, z = \frac{1}{3}, \zeta = 7, a = 5, b = 2, \sigma = 1, r = 3 , and \delta = 2 , Figure 1 is given.
Certain interesting zeros of the TGHBelATP {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z; \delta, a, b) = 0 , for \sigma = 1, r = 3, \delta = 2 , and \tau = 60 are shown in Figure 2.
Remark 16. We observed that zeros the TGHBelATP {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z; \delta, a, b) , for \sigma = 1, r = 3, \delta = 2 , and \tau = 60 have the following properties:
(1) When \tau is assigned a non-negative value m\geq2 , the TGHBelATP _{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z, a, b) possesses m-2 zeros.
(2) Altering the variables, parameters, or indices generates distinct zero distributions and varied graphical configurations.
(3) The zeros (complex zeros) of _{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z, a, b) = 0 exhibit symmetry about the real axis.
The stacking structures of approximation zeros of the TGHBelATP
for \sigma = 1, r = 3, \delta = 2 , and 3\leq \tau\leq 60 , give 3D structures, which are presented in Figure 3.
6.
Conclusions
The hybrid form of special polynomials has attracted significant attention from numerous researchers. In this work, we presented and explored a novel hybrid class of special polynomials, referred to as the trivariate Gould-Hopper-Bell-Apostol-type polynomials. By employing the monomiality principle, we constructed the associated generating function, series representations, quasi-monomial operators, and differential equation. Additionally, summation formulae, differential representations, and integral representations were derived, providing a comprehensive framework for the study of these polynomials.
Special examples of this unified family-such as the Gould-Hopper-Bell-Apostol-Bernoulli, Gould-Hopper-Bell-Apostol-Euler, and Gould-Hopper-Bell-Apostol-Genocchi polynomials were examined, revealing analogous results for each. These results underscore the adaptability and relevance of the unified family across diverse mathematical contexts. Furthermore, computational investigations using Mathematica were conducted to explore the zero distributions and graphical representations of the trivariate Gould-Hopper-Bell-Apostol-type polynomials. The visual and numerical analyses offer a more profound understanding of the behavior and characteristics of these polynomials.
In summary, this work not only establishes a new class of polynomials, but also lays the groundwork for further research into their theoretical and practical applications. The results presented here contribute to the broader field of special functions and polynomial theory, offering a unified approach to studying diverse polynomial families. Future research could explore the degenerate forms of the established special polynomials in this study, along with their associated applications.
Author contributions
Rabeb Sidaoui: Conceptualization, Methodology, Investigation; Abdulghani Muhyi: Conceptualization, Formal analysis, Writing-oiginal draft preparation; Khaled Aldwoah: Writing-review & editing, Supervision, Project administration; Ayman Alahmade: Methodology, Investigation, Data curation; Mohammed Rabih: Validation, Visualization; Amer Alsulami: Software, Data curation; Khidir Mohamed: Validation, Resources, Visualization. All authors have read and approved the final version of the manuscript for publication.
Use of Generative-AI tools declaration
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
Acknowledgments
This research work was funded by the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2025).
Conflict of interest
The authors declare no conflict of interest.