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Research article

Novel optical soliton solutions for the generalized integrable (2+1)- dimensional nonlinear Schrödinger system with conformable derivative

  • For the generalized integrable (2+1)-dimensional nonlinear Schrödinger system, new and creative analytical solutions were derived using a novel extended direct algebraic method incorporating conformable derivatives, which could be expressed in terms of elementary functions and yielded a variety of analytical solutions, such as single, optical periodic, and wave solitons. The analytical solutions provided key insights into the effects of conformable derivatives and temporal parameters on the behavior of optical solitons, such as their stability, propagation, and interaction. To further elucidate the dynamics of these solitons, 2D, 3D, and contour plots were created to provide a visual representation of the soliton's form, amplitude, and phase. This helps to better understand the behavior of the soliton and its potential applications in nonlinear equations. Based on the study's demonstration of the extended direct algebraic method's strength and versatility in obtaining analytical solutions for complex non linear systems, it may be a useful tool for solving a variety of nonlinear problems in science and engineering.

    Citation: Muhammad Bilal, Javed Iqbal, Ikram Ullah, Aditi Sharma, Hasim Khan, Sunil Kumar Sharma. Novel optical soliton solutions for the generalized integrable (2+1)- dimensional nonlinear Schrödinger system with conformable derivative[J]. AIMS Mathematics, 2025, 10(5): 10943-10975. doi: 10.3934/math.2025497

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  • For the generalized integrable (2+1)-dimensional nonlinear Schrödinger system, new and creative analytical solutions were derived using a novel extended direct algebraic method incorporating conformable derivatives, which could be expressed in terms of elementary functions and yielded a variety of analytical solutions, such as single, optical periodic, and wave solitons. The analytical solutions provided key insights into the effects of conformable derivatives and temporal parameters on the behavior of optical solitons, such as their stability, propagation, and interaction. To further elucidate the dynamics of these solitons, 2D, 3D, and contour plots were created to provide a visual representation of the soliton's form, amplitude, and phase. This helps to better understand the behavior of the soliton and its potential applications in nonlinear equations. Based on the study's demonstration of the extended direct algebraic method's strength and versatility in obtaining analytical solutions for complex non linear systems, it may be a useful tool for solving a variety of nonlinear problems in science and engineering.



    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:

    eω1μ+ω2μr=τ=0H(r)τ(ω1,ω2)μττ!,rZ+ (1.1)

    and represented by the series

    H(r)τ(ω1,ω2)=τ![τr]κ=0ωτrκ1ωκ2(τrκ)!κ!. (1.2)

    The classical Bell polynomials Belτ(ω) [16,17] are defined by

    eω(eμ1)=τ=0Belτ(ω)μττ!. (1.3)

    The 2-variable Bell polynomials, denoted as Belτ(ω1,ω2), are defined as follows [18,19]:

    eω1μeω2(eμ1)=τ=0Belτ(ω1,ω2)μττ!. (1.4)

    Recently, the Gould-Hopper-Bell polynomials (GHBelP) HBel(r)τ(ω1,ω2,z) were introduced in [20] by the generating function

    eω1μ+ω2μr+z(eμ1)=τ=0HBel(r)τ(ω1,ω2,z)μττ! (1.5)

    and represented by the series

    HBel(r)τ(ω1,ω2,z)=τκ=0(τκ)H(r)τκ(ω1,ω2)Belκ(z). (1.6)

    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) (τN,ωC) is termed quasi-monomial if it is possible to define "multiplicative" (ˆM) and "derivative" (ˆP) operators for which

    ˆM{ρτ(ω)}=ρτ+1(ω), (1.7)
    ˆP{ρτ(ω)}=τρτ1(ω), (1.8)

    for all τN. Moreover, these operators satisfy the relation

    [ˆP,ˆM]=ˆPˆMˆMˆP=ˆ1 (1.9)

    and therefore reveals the Weyl group structure. If the polynomial set {ρτ(ω)}τN 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

    ˆMˆP{ρτ(ω)}=τρτ(ω). (1.10)

    (ⅱ) With the assumption that ρ0(ω)=1, we have an explicit construction for the polynomials ρτ(ω) as

    ρτ(ω)=ˆMτ{ρ0(ω)}=ˆMτ{1}, (1.11)

    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:

    exp(μˆM){1}=τ=0ρτ(ω)μττ!,|μ|<. (1.12)

    The quasi-monomiality of the GHBelP HBel(r)τ(ω1,ω2,z) [20] is established via the following operators:

    ˆMGHBel=ω1+rω2Dr1ω1+zeDω1,(Dω1:=ω1) (1.13)

    and

    ˆPGHBel:=Dω1. (1.14)

    Based on the monomiality principle, the GHBelP HBel(r)τ(ω1,ω2,z) fulfills the following identities:

    ˆMGHBel{HBel(r)τ(ω1,ω2,z)}=HBel(r)τ+1(ω1,ω2,z), (1.15)
    ˆPGHBel{HBel(r)τ(ω1,ω2,z)}=τHBel(r)τ1(ω1,ω2,z), (1.16)
    ˆMGHBelˆPGHBel{HBel(r)τ(ω1,ω2,z)}=τHBel(r)τ(ω1,ω2,z), (1.17)
    exp(ˆMGHBelμ){1}=τ=0HBel(r)τ(ω1,ω2,z)μττ!(|μ|<). (1.18)

    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

    (μζeμ1)σeωμ=τ=0B(σ)τ(ω;ζ)μττ!(|μ+logζ|<2π,1σ:=1), (1.19)
    (2ζeμ+1)σeωμ=τ=0E(σ)τ(ω;ζ)μττ!(|μ+logζ|<π,1σ:=1), (1.20)
    (2μζeμ+1)σeωμ=τ=0G(σ)τ(ω;ζ)μττ!(|μ+logζ|<π,1σ:=1), (1.21)

    where σ and ζ are arbitrary real or complex parameters. When ω=0 in Eqs (1.19)–(1.21), we get

    B(σ)τ(0;ζ)=B(σ)τ(ζ),E(σ)τ(0;ζ)=E(σ)τ(ζ) and G(σ)τ(0;ζ)=G(σ)τ(ζ),

    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

    (21δμδζbeμab)σeωμ=τ=0P(σ)τ,ζ(ω;δ,a,b)μττ!(|μ+blog(ζa)|<2π,δN,a,bR+,σ,ζC), (1.22)

    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

    P(σ)τ,ζ(ω;1,1,1)=B(σ)τ(ω;ζ),P(σ)τ,ζ(ω;0,1,1)=E(σ)τ(ω;ζ)

    and

    P(σ)τ,ζ2(ω;1,12,1)=G(σ)τ(ω;ζ).

    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.

    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

    (21δμδζbeμab)σexp(ˆMGHBelμ)=τ=0P(σ)τ,ζ(ˆMGHBel;δ,a,b)μττ!. (2.1)

    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:

    (21δμδζbeμab)σ(τ=0HBel(r)τ(ω1,ω2,z))=τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!. (2.2)

    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:

    (21δμδζbeμab)σeω1μ+ω2μr+z(eμ1)=τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!, (2.3)
    (|μ+blog(ζa)|<2π,δN,a,bR+,σ,ζC).

    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:

    (21δμδζbeμab)σeω1μ+ω2μr=τ=0HP(σ,r)τ,ζ(ω1,ω2;δ,a,b)μττ!, (2.4)

    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

    (21δμδζbeμab)σeω1μ+z(eμ1)=τ=0BelP(σ)τ,ζ(ω1,z;δ,a,b)μττ!. (2.5)

    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

    (21δμδζbeμab)eω1μ+ω2μ2+z(eμ1)=τ=0HBelP(2)τ,ζ(ω1,ω2,z;δ,a,b)μττ!. (2.6)

    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

    (21δμδζbeμab)eω1μ+ω2μr+z(eμ1)=τ=0HBelP(r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!. (2.7)

    Taking ω1=ω2=0 and z=1 in (2.3), we get unified Bell-Apostol-type numbers of order σ, which are defined by

    (21δμδζbeμab)σe(eμ1)=τ=0BelP(σ)τ,ζ(δ,a,b)μττ!. (2.8)

    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:

    HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)=τκ=0(τκ)HP(σ,r)τκ,ζ(ω1,ω2;δ,a,b)Belκ(z); (2.9)
    HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)=τκ=0(τκ)P(σ)τκ,ζ(0;δ,a,b)HBel(r)κ(ω1,ω2,z); (2.10)
    HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)=τ![τr]κ=0ωκ2BelP(σ)τrκ,ζ(ω1,z;δ,a,b)κ!(τrκ)!. (2.11)

    Proof. In view of generating relations (1.3), (2.3), and (2.4), we have

    τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!=(21δμδζbeμab)σeω1μ+ω2μr+z(eμ1)=[(21δμδζbeμab)σeω1μ+ω2μr][ez(eμ1)]=[τ=0HP(σ,r)τ,ζ(ω1,ω2;δ,a,b)μττ!][κ=0Belκ(z)μκκ!]=τ=0κ=0HP(σ,r)τ,ζ(ω1,ω2;δ,a,b)Belκ(z)μτ+κκ!τ!, (2.12)

    which, upon substituting ττκ and applying the Cauchy product rule, yields:

    τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!=τ=0τκ=0(τκ)HP(σ,r)τκ,ζ(ω1,ω2;δ,a,b)Belκ(z)μττ!, (2.13)

    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:

    HBelP(2)τ,ζ(ω1,ω2,z;δ,a,b)=τκ=0(τκ)HP(2)τκ,ζ(ω1,ω2;δ,a,b)Belκ(z); (2.14)
    HBelP(2)τ,ζ(ω1,ω2,z;δ,a,b)=τκ=0(τκ)Pτκ,ζ(0;δ,a,b)HBel(2)κ(ω1,ω2,z); (2.15)
    HBelP(2)τ,ζ(ω1,ω2,z;δ,a,b)=τ![τ2]κ=0ωκ2BelPτ2κ,ζ(ω1,z;δ,a,b)κ!(τ2κ)!. (2.16)

    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:

    ˆMHBelP=ω1+rω2Dr1ω1+zeDω1+σδ(ζbeDω1ab)σDω1ζbeDω1Dω1(ζbeDω1ab) (2.17)

    and

    ˆPHBelP=Dω1, (2.18)

    respectively.

    Proof. Differentiating relation (2.3) partially with respect to μ, gives

    (ω1+rω2μr1+zeμ+σδ(ζbeμab)σμζbeμμ(ζbeμab))(21δμδζbeμab)σeω1μ+ω2μr+z(eμ1)=τ=0τHBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μτ1τ!, (2.19)

    which, upon replacing τ by τ+1 in the right-hand side and using relation (2.3) the left-hand side, becomes

    (ω1+rω2μr1+zeμ+σδ(ζbeμab)σμζbeμμ(ζbeμab))τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!=τ=0HBelP(σ,r)τ+1,ζ(ω1,ω2,z;δ,a,b)μττ!. (2.20)

    Matching the coefficients of corresponding powers of μ in Eq (2.20), we obtain

    (ω1+rω2Dr1ω1+zeDω1+σδ(ζbeDω1ab)σDω1ζbeDω1Dω1(ζbeDω1ab))HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)=HBelP(σ,r)τ+1,ζ(ω1,ω2,z;δ,a,b). (2.21)

    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

    Dω1[(21δμδζbeμab)σeω1μ+ω2μr+z(eμ1)]=μ[(21δμδζbeμab)σeω1μ+ω2μr+z(eμ1)]. (2.22)

    Using relation (2.3) in expression (2.22), gives

    Dω1[τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!]=μ[τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!]=τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μτ+1τ!, (2.23)

    which, upon replacing τ by τ1 in the right-hand side and then comparing the coefficients of corresponding powers of μ in the resulting equation, we obtain

    Dω1{HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)}=τHBelP(σ,r)τ1,ζ(ω1,ω2,z;δ,a,b). (2.24)

    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:

    ˆMHP=ω1+rω2Dr1ω1+σδ(ζbeDω1ab)σDω1ζbeDω1Dω1(ζbeDω1ab) (2.25)

    and

    ˆPHP=Dω1, (2.26)

    respectively.

    Theorem 3. The unified polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) satisfy the following differential equation:

    (ω1Dω1+rω2Drω1+zeDω1Dω1+σδ(ζbeDω1ab)σDω1ζbeDω1(ζbeDω1ab)τ)HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)=0. (2.27)

    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:

    (ω1Dω1+rω2Drω1+σδ(ζbeDω1ab)σDω1ζbeDω1(ζbeDω1ab)τ)HP(σ,r)τ,ζ(ω1,ω2;δ,a,b)=0. (2.28)

    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 τN0,δN and σ,ζC, we have

    HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)=12τκ=0(τκ)Eκ(HBelP(σ,r)τκ,ζ(ω1+1,ω2,z;δ,a,b)+HBelP(σ,r)τκ,ζ(ω1,ω2,z;δ,a,b)). (3.1)

    Proof. From generating relation (2.3), it follows that:

    (eμ+1)τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!=τ=0HBelP(σ,r)τ,ζ(ω1+1,ω2,z;δ,a,b)μττ!+τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!, (3.2)

    which can be written as

    τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!=12(τ=0Eτμττ!)(τ=0HBelP(σ,r)τ,ζ(ω1+1,ω2,z;δ,a,b)μττ!+τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!), (3.3)

    where, Eτ represents the Euler numbers defined by [42]:

    2eμ+1=τ=0Eτμττ!. (3.4)

    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:

    HP(σ,r)τ,ζ(ω1,ω2;δ,a,b)=12τκ=0(τκ)Eκ(HP(σ,r)τκ,ζ(ω1+1,ω2;δ,a,b)+HP(σ,r)τκ,ζ(ω1,ω2;δ,a,b)). (3.5)

    Theorem 5. The unified polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) satisfy the following summation formula:

    HBelP(σ,r)τ,ζ(υ,ω2,z;δ,a,b)=τκ=0(τκ)(υω1+θ)τκHBelP(σ,r)κ,ζ(ω1θ,ω2,z;δ,a,b). (3.6)

    Proof. Replacing ω1 by υ in (2.3), we have

    τ=0HBelP(σ,r)τ,ζ(υ,ω2,z;δ,a,b)μττ!=(21δμδζbeμab)σeυμ+ω2μr+z(eμ1)=(21δμδζbeμab)σe(ω1θ)μe(ω1υθ)μeω2μr+z(eμ1)=e(υω1+θ)μτ=0HBelP(σ,r)τ,ζ(ω1θ,ω2,z;δ,a,b)μττ!=(τ=0(υω1+θ)τμττ!)(τ=0HBelP(σ,r)τ,ζ(ω1θ,ω2,z;δ,a,b)μττ!)=τ=0τκ=0(τκ)(υω1+θ)τκHBelP(σ,r)κ,ζ(ω1θ,ω2,z;δ,a,b)μττ!. (3.7)

    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:

    BelP(σ)τ,ζ(υ,z;δ,a,b)=τκ=0(τκ)(υω1+θ)τκBelP(σ)κ,ζ(ω1θ,z;δ,a,b). (3.8)

    Theorem 6. For τN0,δN, and σ,β,ζC, we have

    HBelP(σ+β,r)τ,ζ(ω1+x,ω2+y,z+u;δ,a,b)=τκ=0(τκ)HBelP(σ,r)τκ,ζ(ω1,ω2,z;δ,a,b)HBelP(β,r)κ,ζ(x,y,u;δ,a,b). (3.9)

    Proof. In (2.3), replacing ω1,ω2,z, and σ by ω1+x,ω2+y,z+u, and σ+β, respectively, we have

    τ=0HBelP(σ+β,r)τ,ζ(ω1+x,ω2+y,z+u;δ,a,b)μττ!=(21δμδζbeμab)σ+βe(ω1+x)μ+(ω2+y)μr+(z+u)(eμ1)=((21δμδζbeμab)σeω1μ+ω2μr+z(eμ1))((21δμδζbeμab)βexμ+yμr+u(eμ1))=(τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!)(τ=0HBelP(β,r)τ,ζ(x,y,u;δ,a,b)μττ!)=τ=0τκ=0(τκ)HBelP(σ,r)τκ,ζ(ω1,ω2,z;δ,a,b)HBelP(β,r)κ,ζ(x,y,u;δ,a,b)μττ!. (3.10)

    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:

    HBelP(1+β,2)τ,ζ(ω1+x,ω2+y,z+u;δ,a,b)=τκ=0(τκ)HBelP(2)τκ,ζ(ω1,ω2,z;δ,a,b)HBelP(β,2)κ,ζ(x,y,u;δ,a,b). (3.11)

    The generalized Stirling numbers of the second kind are given as follows [13,43]:

    τ=0S(τ,σ,a,b,ζ)μττ!=(ζbeμab)σσ!. (3.12)

    Theorem 7. The unified polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) satisfy the following summation formula:

    HBelP(σϱ,r)τδϱ,ζ(ω1,ω2,z;δ,a,b)=(τδϱ)!ϱ!2ϱ(1δ)τ!τκ=0(τκ)HBelP(σ,r)τκ,ζ(ω1,ω2,z;δ,a,b)S(κ,ϱ,a,b,ζ). (3.13)

    Proof. In view of (2.3) and (3.12), we can write

    τ=0HBelP(σϱ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!=(21δμδζbeμab)σϱeω1μ+ω2μr+z(eμ1)=ϱ!(21δμδ)ϱ(τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!)(τ=0S(τ,ϱ,a,b,ζ)μττ!)=ϱ!(21δμδ)ϱτ=0τκ=0(τκ)HBelP(σ,r)τκ,ζ(ω1,ω2,z;δ,a,b)S(κ,ϱ,a,b,ζ)μττ!. (3.14)

    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:

    BelP(σϱ)τϱδ,ζ(ω1,z;δ,a,b)=(τϱδ)!ϱ!2ϱ(1δ)τ!τκ=0(τκ)BelP(σ)τκ,ζ(ω1,z;δ,a,b)S(κ,ϱ,a,b,ζ). (3.15)

    Theorem 8. The unified polynomials HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b) satisfy the following summation formula:

    HBelP(1,r)τ+δ,ζ(ω1+1,ω2,z;δ,a,b)=1ζb{21δ(τ+δ)!τ!HBel(r)τ(ω1,ω2,z)+abHBelP(1,r)τ+δ,ζ(ω1,ω2,z;δ,a,b)}. (3.16)

    Proof. In view of (1.5) and (2.3) for σ=1, we can write

    τ=0HBel(r)τ(ω1,ω2,z)μττ!=eω1μ+ω2μr+z(eμ1)=(ζbeμab21δμδ)τ=0HBelP(1,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!=1(21δμδ){ζbτ=0HBelP(1,r)τ,ζ(ω1+1,ω2,z;δ,a,b)μττ!abτ=0HBelP(1,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!}=121δτ=0τ!(τ+δ)!{ζbHBelP(1,r)τ+δ,ζ(ω1+1,ω2,z;δ,a,b)abHBelP(1,r)τ+δ,ζ(ω1,ω2,z;δ,a,b)}μττ!, (3.17)

    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:

    HBelP(1,2)τ+δ,ζ(ω1+1,ω2,z;δ,a,b)=1ζb{21δ(τ+δ)!τ!HBel(2)τ(ω1,ω2,z)+abHBelP(1,2)τ+δ,ζ(ω1,ω2,z;δ,a,b)}. (3.18)

    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,τN0,δN, and σ,ζC, we have

    νων1{HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)}={τ!HBelP(σ,r)τν,ζ(ω1,ω2,z;δ,a,b)(τν)!,τν;0,0τ<ν. (4.1)

    Proof. Differentiating the generating relation (2.3) ν times with respect to ω1, we obtain

    τ=0νων1{HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)}μττ!=μν{(21δμδζbeμab)σeω1μ+ω2μr+z(eμ1)}=τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μτ+ντ!=τ=νHBelP(σ,r)τν,ζ(ω1,ω2,z;δ,a,b)μτ(τν)!. (4.2)

    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

    νων1{BelP(σ)τ,ζ(ω1,z;δ,a,b)}={τ!BelP(σ)τν,ζ(ω1,z;δ,a,b)(τν)!,τν;0,0τ<ν. (4.3)

    Similarly, upon differentiating relation (2.3) ν times with respect to ω2, we can get the following result.

    Theorem 10. For ν,τN0,δN, and σ,ζC, we have

    νων2{HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)}=τ!HBelP(σ,r)τrν,ζ(ω1,ω2,z;δ,a,b)(τrν)!. (4.4)

    Remark 14. Setting r=2 and σ=1 in (4.4), we have

    νων2{HBelP(2)τ,ζ(ω1,ω2,z;δ,a,b)}=τ!HBelP(2)τ2ν,ζ(ω1,ω2,z;δ,a,b)(τ2ν)!. (4.5)

    Theorem 11. For τN0,δN, and σ,ζC, we have

    z{HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)}=τκ=0(τκ){HBelP(σ,r)τκ,ζ(ω1,ω2,z;δ,a,b)HP(σ,r)τκ,ζ(ω1,ω2;δ,a,b)Belκ(z)}. (4.6)

    Proof. We start with generating relation (2.3). Differentiating it with respect to z, followed by simplification using equation (2.9), results in

    τ=0z{HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)}μττ!=(eμ1){(21δμδζbeμab)σeω1μ+ω2μr+z(eμ1)}={τ=0μττ!}{τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!}{τ=0HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)μττ!}=τ=0τκ=0(τκ){HBelP(σ,r)τκ,ζ(ω1,ω2,z;δ,a,b)HP(σ,r)τκ,ζ(ω1,ω2;δ,a,b)Belκ(z)}μττ!. (4.7)

    From (4.7), we get the asserted result (4.6).

    Remark 15. Setting ω2=0 in (4.6), we have

    zBelP(σ)τ,ζ(ω1,z;δ,a,b)=τκ=0(τκ){BelP(σ,r)τκ,ζ(ω1,z;δ,a,b)P(σ)τκ,ζ(ω1;δ,a,b)Belκ(z)}. (4.8)

    Similarly, we can derive the following outcome.

    Corollary 1. For τN0,δN, and σ,ζC, we have

    z{HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)}=HBelP(σ,r)τκ,ζ(ω1+1,ω2,z;δ,a,b)GBelτ(ω1,ω2,z). (4.9)

    Theorem 12. The following formula holds true:

    u+γuHBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)dω1=1τ+1[HBelP(σ,r)τ+1,ζ(u+γ,ω2,z;δ,a,b)HBelP(σ,r)τ+1,ζ(u,ω2,z;δ,a,b)]. (4.10)

    Proof. By integrating both sides of Eq (2.3) with respect to ω1, we obtain

    τ=0u+γuHBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)dω1μττ!=1μ[(21δμδζbeμab)σe(u+γ)μ+ω2μr+z(eμ1)(21δμδζbeμab)σeuμ+ω2μr+z(eμ1)]=1μ[τ=0HBelP(σ,r)τ,ζ(u+γ,ω2,z;δ,a,b)μττ!τ=0HP(σ,r)τ,ζ(u,ω2;δ,a,b)Belκ(z)μττ!]. (4.11)

    From (4.11), we get asserted result (4.10).

    Similarly, the following results can be proved.

    Theorem 13. The following formulas hold true:

    u+γuHBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)dω2=τ!(τ+r)![HBelP(σ,r)τ+r,ζ(ω1,u+γ,z;δ,a,b)HBelP(σ,r)τ+r,ζ(ω1,u,z;δ,a,b)], (4.12)
    u+γuHBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)dz=1τ+1τk=0(τ+1k)Bk[HBelP(σ,r)τ+1k,ζ(ω1,ω2,u+γ;δ,a,b)HBelP(σ,r)τ+1k,ζ(ω1,ω2,u;δ,a,b)]. (4.13)

    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.

    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.

    Here, we introduce certain special members belonging to the unified family

    HBelP(σ,r)τ,ζ(ω1,ω2,z;δ,a,b),

    with analogous results presented for each.

    Example 1. Gould-Hopper-Bell-Apostol-Bernoulli polynomials

    Since

    P(σ)τ,ζ(ω;1,1,1)=B(σ)τ(ω;ζ),

    therefore, taking δ=a=b=1 in generating function (2.3), gives

    (μζeμ1)σeω1μ+ω2μr+z(eμ1)=τ=0HBelB(σ,r)τ,ζ(ω1,ω2,z)μττ!, (5.1)

    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:

    HBelB(σ,r)τ,ζ(ω1,ω2,z)=τκ=0(τκ)BelB(σ)τκ,ζ(z)H(r)κ(ω1,ω2); (5.2)
    HBelB(σ,r)τ,ζ(ω1,ω2,z)=τκ=0(τκ)HB(σ,r)τκ,ζ(ω1,ω2)Belκ(z); (5.3)
    HBelB(σ,r)τ,ζ(ω1,ω2,z)=τκ=0(τκ)B(σ)τκ,ζ(0)HBel(r)κ(ω1,ω2,z); (5.4)
    HBelB(σ,r)τ,ζ(ω1,ω2,z)=τ![τr]κ=0ωκ2BelB(σ)τrκ,ζ(ω1,z)κ!(τrκ)!. (5.5)

    Certain corresponding results related to the GHBelBP HBelB(σ,r)τ,ζ(ω1,ω2,z) of order σ are presented in Table 1.

    Table 1.  Findings for the Gould-Hopper-Bell-Apostol-Bernoulli polynomials HBelB(σ,r)τ,ζ(ω1,ω2,z).
    Multiplicative and derivative operator ˆMHBelB=ω1+rω2Dr1ω1+zeDω1+σ(ζeDω11)σDω1ζeDω1Dω1(ζeDω11),ˆPHBelB=Dω1
    Differential equation (ω1Dω1+rω2Drω1+zeDω1Dω1+σ(ζeDω11)σDω1ζeDω1(ζeDω11)τ)HBelB(σ,r)τ,ζ(ω1,ω2,z)=0
    Summation HBelB(σ,r)τ,ζ(ω1,ω2,z)=12τκ=0(τκ)Eκ(HBelB(σ,r)τκ,ζ(ω1+1,ω2,z)+HBelB(σ,r)τκ,ζ(ω1,ω2,z))
    Formulae HBelB(σ,r)τ,ζ(υ,ω2,z)=τκ=0(τκ)(υω1+θ)τκHBelB(σ,r)κ,ζ(ω1θ,ω2,z)
    HBelB(σ+β,r)τ,ζ(ω1+x,ω2+y,z+u)=τκ=0(τκ)HBelB(σ,r)τκ,ζ(ω1,ω2,z)HBelB(β,r)κ,ζ(x,y,u)
    HBelB(σϱ,r)τϱ,ζ(ω1,ω2,z)=(τϱ)!ϱ!τ!τκ=0(τκ)HBelB(σ,r)τκ,ζ(ω1,ω2,z)S(κ,ϱ,ζ)
    HBelB(1,r)τ+1,ζ(ω1+1,ω2,z)=1ζ{(τ+1)HBel(r)τ(ω1,ω2,z)+HBelB(1,r)τ+1,ζ(ω1,ω2,z)}
    Differential and νων1{HBelB(σ,r)τ,ζ(ω1,ω2,z)}={τ!HBelB(σ,r)τν,ζ(ω1,ω2,z)(τν)!,τν;0,0τ<ν.
    νων2{HBelB(σ,r)τ,ζ(ω1,ω2,z)}=τ!HBelB(σ,r)τrν,ζ(ω1,ω2,z)(τrν)!
    z{HBelB(σ,r)τ,ζ(ω1,ω2,z)}=τκ=0(τκ){HBelB(σ,r)τκ,ζ(ω1,ω2,z)HB(σ,r)τκ,ζ(ω1,ω2)Belκ(z)}
    z{HBelB(σ,r)τ,ζ(ω1,ω2,z)}=HBelB(σ,r)τκ,ζ(ω1+1,ω2,z)GBelτ(ω1,ω2,z)
    Integral Formulae u+γuHBelB(σ,r)τ,ζ(ω1,ω2,z)dω1=1τ+1[HBelB(σ,r)τ+1,ζ(u+γ,ω2,z)HBelB(σ,r)τ+1,ζ(u,ω2,z)]
    u+γuHBelB(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)dz
    =1τ+1τk=0(τ+1k)Bk[HBelB(σ,r)τ+1k,ζ(ω1,ω2,u+γ;δ,a,b)HBelB(σ,r)τ+1k,ζ(ω1,ω2,u;δ,a,b)]

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    The first few members of the GHBelBP

    HBelB(σ,r)τ,ζ(ω1,ω2,z),

    for σ=1 and r=3 are given as:

    HBelB(1,3)0,ζ(ω1,ω2,z)=0,HBelB(1,3)1,ζ(ω1,ω2,z)=1ζ1,HBelB(1,3)2,ζ(ω1,ω2,z)=2ζ(ζ1)2+2ζω1(ζ1)22ω1(ζ1)2+2ζz(ζ1)22z(ζ1)2,HBelB(1,3)3,ζ(ω1,ω2,z)=3ζ2(ζ1)3+3ζ(ζ1)3+3ω21ζ16ζω1(ζ1)2+6ω1zζ1+3z2ζ1+3zζ16ζz(ζ1)2,
    HBelB(1,3)4,ζ(ω1,ω2,z)=4ζ3(ζ1)416ζ2(ζ1)44ζ(ζ1)4+4ω31ζ112ζω21(ζ1)2+12ω21zζ1+12ζ2ω1(ζ1)3+12ζω1(ζ1)3+12ω1z2ζ1+12ω1zζ124ζω1z(ζ1)2+24ω2ζ1+4z3ζ1+12z2ζ112ζz2(ζ1)2+12ζ2z(ζ1)3+12ζz(ζ1)3+4zζ112ζz(ζ1)2.

    Example 2. Gould-Hopper-Bell-Apostol-Euler polynomials

    Since

    P(σ)τ,ζ(ω;0,1,1)=E(σ)τ(ω;ζ),

    therefore, taking δ=0,a=1 and b=1 in generating function (2.3), gives

    (2ζeμ+1)σeω1μ+ω2μr+z(eμ1)=τ=0HBelE(σ,r)τ,ζ(ω1,ω2,z)μττ!, (5.6)

    where

    HBelE(σ,r)τ,ζ(ω1,ω2,z)

    are referred to as the Gould-Hopper-Bell-Apostol-Euler polynomials (GHBelEP) of order σ.

    The series representations of the GHBelEP

    HBelE(σ,r)τ,ζ(ω1,ω2,z)

    of order σ are given as:

    HBelE(σ,r)τ,ζ(ω1,ω2,z)=τκ=0(τκ)BelE(σ)τκ,ζ(z)H(r)κ(ω1,ω2); (5.7)
    HBelE(σ,r)τ,ζ(ω1,ω2,z)=τκ=0(τκ)HE(σ,r)τκ,ζ(ω1,ω2)Belκ(z); (5.8)
    HBelE(σ,r)τ,ζ(ω1,ω2,z)=τκ=0(τκ)E(σ)τκ,ζ(0)HBel(r)κ(ω1,ω2,z); (5.9)
    HBelE(σ,r)τ,ζ(ω1,ω2,z)=τ![τr]κ=0ωκ2BelE(σ)τrκ,ζ(ω1,z)κ!(τrκ)!. (5.10)

    Certain corresponding results related to the GHBelEP HBelE(σ,r)τ,ζ(ω1,ω2,z) of order σ are presented in Table 2.

    Table 2.  Findings for the Gould-Hopper-Bell-Apostol-Euler polynomials HBelE(σ,r)τ,ζ(ω1,ω2,z).
    Multiplicative and derivative operators ˆMHBelE=ω1+rω2Dr1ω1+zeDω1σζeDω1ζeDω1+1,ˆPHBelE=Dω1
    Differential equation (ω1Dω1+rω2Drω1+zeDω1Dω1σDω1ζeDω1(ζeDω1+1)τ)HBelE(σ,r)τ,ζ(ω1,ω2,z)=0
    Summation HBelE(σ,r)τ,ζ(ω1,ω2,z)=12τκ=0(τκ)Eκ(HBelE(σ,r)τκ,ζ(ω1+1,ω2,z)+HBelE(σ,r)τκ,ζ(ω1,ω2,z))
    Formulae HBelE(σ,r)τ,ζ(υ,ω2,z)=τκ=0(τκ)(υω1+θ)τκHBelE(σ,r)κ,ζ(ω1θ,ω2,z)
    HBelE(σ+β,r)τ,ζ(ω1+x,ω2+y,z+u)=τκ=0(τκ)HBelE(σ,r)τκ,ζ(ω1,ω2,z)HBelE(β,r)κ,ζ(x,y,u)
    HBelE(σϱ,r)τ,ζ(ω1,ω2,z)=ϱ!2ϱτκ=0(τκ)HBelE(σ,r)τκ,ζ(ω1,ω2,z)S(κ,ϱ,ζ)
    HBelE(1,r)τ,ζ(ω1+1,ω2,z)=1ζ{2HBel(r)τ(ω1,ω2,z)HBelE(1,r)τ+δ,ζ(ω1,ω2,z)}
    Differential and νων1{HBelE(σ,r)τ,ζ(ω1,ω2,z)}={τ!HBelE(σ,r)τν,ζ(ω1,ω2,z)(τν)!,τν;0,0τ<ν.
    νων2{HBelE(σ,r)τ,ζ(ω1,ω2,z)}=τ!HBelE(σ,r)τrν,ζ(ω1,ω2,z)(τrν)!
    z{HBelE(σ,r)τ,ζ(ω1,ω2,z)}=τκ=0(τκ){HBelE(σ,r)τκ,ζ(ω1,ω2,z)HE(σ,r)τκ,ζ(ω1,ω2)Belκ(z)}
    z{HBelE(σ,r)τ,ζ(ω1,ω2,z)}=HBelE(σ,r)τκ,ζ(ω1+1,ω2,z)GBelτ(ω1,ω2,z)
    Integral Formulae u+γuHBelE(σ,r)τ,ζ(ω1,ω2,z)dω1=1τ+1[HBelE(σ,r)τ+1,ζ(u+γ,ω2,z)HBelE(σ,r)τ+1,ζ(u,ω2,z)]
    u+γuHBelE(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)dz
    =1τ+1τk=0(τ+1k)Bk[HBelE(σ,r)τ+1k,ζ(ω1,ω2,u+γ;δ,a,b)HBelE(σ,r)τ+1k,ζ(ω1,ω2,u;δ,a,b)]

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    The first few members of the GHBelEP

    HBelE(σ,r)τ,ζ(ω1,ω2,z),

    for σ=2 and r=3 are given as:

    HBelE(1,3)0,ζ(ω1,ω2,z)=4(ζ+1)2,HBelE(1,3)1,ζ(ω1,ω2,z)=8ζ(ζ+1)3+4ζω1(ζ+1)3+4ω1(ζ+1)3+4z(ζ+1)3+4ζz(ζ+1)3,HBelE(1,3)2,ζ(ω1,ω2,z)=16ζ2(ζ+1)48ζ(ζ+1)4+4ω21(ζ+1)216ζω1(ζ+1)3+8ω1z(ζ+1)2+4z2(ζ+1)2+4z(ζ+1)216ζz(ζ+1)3,HBelE(1,3)3,ζ(ω1,ω2,z)=32ζ3(ζ+1)5+56ζ2(ζ+1)58ζ(ζ+1)5+4ω31(ζ+1)224ζω21(ζ+1)3+12ω21z(ζ+1)2+48ζ2ω1(ζ+1)424ζω1(ζ+1)4+12ω1z2(ζ+1)2+12ω1z(ζ+1)248ζω1z(ζ+1)3+24ω2(ζ+1)2+4z3(ζ+1)2+12z2(ζ+1)224ζz2(ζ+1)3+48ζ2z(ζ+1)4+4z(ζ+1)224ζz(ζ+1)324ζz(ζ+1)4,
    HBelE(1,3)4,ζ(ω1,ω2,z)=64ζ4(ζ+1)6264ζ3(ζ+1)6+144ζ2(ζ+1)68ζ(ζ+1)6+4ω41(ζ+1)232ζω31(ζ+1)3+16ω31z(ζ+1)2+96ζ2ω21(ζ+1)448ζω21(ζ+1)4+24ω21z2(ζ+1)2+24ω21z(ζ+1)296ζω21z(ζ+1)3128ζ3ω1(ζ+1)5+224ζ2ω1(ζ+1)532ζω1(ζ+1)5+96ω1ω2(ζ+1)2+16ω1z3(ζ+1)2+48ω1z2(ζ+1)296ζω1z2(ζ+1)3+192ζ2ω1z(ζ+1)4+16ω1z(ζ+1)296ζω1z(ζ+1)396ζω1z(ζ+1)4192ζω2(ζ+1)3+96ω2z(ζ+1)2+4z4(ζ+1)2+24z3(ζ+1)232ζz3(ζ+1)3+96ζ2z2(ζ+1)4+28z2(ζ+1)296ζz2(ζ+1)348ζz2(ζ+1)4128ζ3z(ζ+1)5+96ζ2z(ζ+1)4+224ζ2z(ζ+1)5+4z(ζ+1)232ζz(ζ+1)348ζz(ζ+1)432ζz(ζ+1)5.

    Example 3. Gould-Hopper-Bell-Apostol-Genocchi polynomials

    Since

    P(σ)τ,ζ2(ω;1,12,1)=G(σ)τ(ω;ζ),

    therefore, taking δ=1,a=12,b=1 and ζζ2 in generating function (2.3), gives

    (2μζeμ+1)σeω1μ+ω2μr+z(eμ1)=τ=0HBelG(σ,r)τ,ζ(ω1,ω2,z)μττ!, (5.11)

    where HBelG(σ,r)τ,ζ(ω1,ω2,z) are referred to as the Gould-Hopper-Bell-Apostol-Genocchi polynomials (GHBelGP) of order σ.

    The series representations of the GHBelGP HBelG(σ,r)τ,ζ(ω1,ω2,z) of order σ are given as:

    HBelG(σ,r)τ,ζ(ω1,ω2,z)=τκ=0(τκ)BelG(σ)τκ,ζ(z)H(r)κ(ω1,ω2); (5.12)
    HBelG(σ,r)τ,ζ(ω1,ω2,z)=τκ=0(τκ)HG(σ,r)τκ,ζ(ω1,ω2)Belκ(z); (5.13)
    HBelG(σ,r)τ,ζ(ω1,ω2,z)=τκ=0(τκ)G(σ)τκ,ζ(0)HBel(r)κ(ω1,ω2,z); (5.14)
    HBelG(σ,r)τ,ζ(ω1,ω2,z)=τ![τr]κ=0ωκ2BelG(σ)τrκ,ζ(ω1,z)κ!(τrκ)!. (5.15)

    Certain corresponding results related to the GHBelGP HBelG(σ,r)τ,ζ(ω1,ω2,z) of order σ are presented in Table 3.

    Table 3.  Findings for the Gould-Hopper-Bell-Apostol-Genocchi polynomials HBelG(σ,r)τ,ζ(ω1,ω2,z).
    Multiplicative and derivative operators ˆMHBelG=ω1+rω2Dr1ω1+zeDω1+σ(ζeDω1+1)σDω1ζeDω1Dω1(ζeDω1+1),ˆPHBelG=Dω1
    Differential equation (ω1Dω1+rω2Drω1+zeDω1Dω1+σ(ζeDω1+1)σDω1ζeDω1(ζeDω1+1)τ)HBelG(σ,r)τ,ζ(ω1,ω2,z)=0
    Summation HBelG(σ,r)τ,ζ(ω1,ω2,z)=12τκ=0(τκ)Eκ(HBelG(σ,r)τκ,ζ(ω1+1,ω2,z)+HBelG(σ,r)τκ,ζ(ω1,ω2,z))
    Formulae HBelG(σ,r)τ,ζ(υ,ω2,z)=τκ=0(τκ)(υω1+θ)τκHBelG(σ,r)κ,ζ(ω1θ,ω2,z)
    HBelG(σ+β,r)τ,ζ(ω1+x,ω2+y,z+u)=τκ=0(τκ)HBelG(σ,r)τκ,ζ(ω1,ω2,z)HBelG(β,r)κ,ζ(x,y,u)
    HBelG(σϱ,r)τϱ,ζ(ω1,ω2,z)=(τϱ)!ϱ!τ!τκ=0(τκ)HBelG(σ,r)τκ,ζ(ω1,ω2,z)S(κ,ϱ,ζ)
    HBelG(1,r)τ+1,ζ(ω1+1,ω2,z)=2ζ{(τ+1)HBel(r)τ(ω1,ω2,z)12HBelG(1,r)τ+1,ζ(ω1,ω2,z)}
    Differential and νων1{HBelG(σ,r)τ,ζ(ω1,ω2,z)}={τ!HBelG(σ,r)τν,ζ(ω1,ω2,z)(τν)!,τν;0,0τ<ν.
    νων2{HBelG(σ,r)τ,ζ(ω1,ω2,z)}=τ!HBelG(σ,r)τrν,ζ(ω1,ω2,z)(τrν)!
    z{HBelG(σ,r)τ,ζ(ω1,ω2,z)}=τκ=0(τκ){HBelG(σ,r)τκ,ζ(ω1,ω2,z)HG(σ,r)τκ,ζ(ω1,ω2)Belκ(z)}
    z{HBelG(σ,r)τ,ζ(ω1,ω2,z)}=HBelG(σ,r)τκ,ζ(ω1+1,ω2,z)GBelτ(ω1,ω2,z)
    Integral Formulae u+γuHBelG(σ,r)τ,ζ(ω1,ω2,z)dω1=1τ+1[HBelG(σ,r)τ+1,ζ(u+γ,ω2,z)HBelG(σ,r)τ+1,ζ(u,ω2,z)]
    u+γuHBelG(σ,r)τ,ζ(ω1,ω2,z;δ,a,b)dz
    =1τ+1τk=0(τ+1k)Bk[HBelG(σ,r)τ+1k,ζ(ω1,ω2,u+γ;δ,a,b)HBelG(σ,r)τ+1k,ζ(ω1,ω2,u;δ,a,b)]

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    The first few members of the GHBelGP HBelG(σ,r)τ,ζ(ω1,ω2,z), for σ=1 and r=3 are given as:

    HBelG(1,3)0,ζ(ω1,ω2,z)=0,HBelG(1,3)1,ζ(ω1,ω2,z)=2ζ+1,HBelG(1,3)2,ζ(ω1,ω2,z)=4ζ(ζ+1)2+4ζω1(ζ+1)2+4ω1(ζ+1)2+4z(ζ+1)2+4ζz(ζ+1)2,HBelG(1,3)3,ζ(ω1,ω2,z)=6ζ2(ζ+1)36ζ(ζ+1)3+6ω21ζ+112ζω1(ζ+1)2+12ω1zζ+1+6z2ζ+1+6zζ+112ζz(ζ+1)2,
    \begin{align} &{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{4,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = -\frac{8\zeta ^3}{(\zeta +1)^4}+\frac{32\zeta ^2}{(\zeta +1)^4}-\frac{8\zeta }{(\zeta +1)^4}+\frac{8 \omega_{1}^3}{\zeta +1}-\frac{24\zeta \omega_{1}^2}{(\zeta +1)^2}+\frac{24 \omega_{1}^2 z}{\zeta +1}\\& \quad +\frac{24\zeta ^2 \omega_{1}}{(\zeta +1)^3}-\frac{24\zeta \omega_{1}}{(\zeta +1)^3}+\frac{24 \omega_{1} z^2}{\zeta +1}+\frac{24 \omega_{1} z}{\zeta +1}-\frac{48\zeta \omega_{1} z}{(\zeta +1)^2}+\frac{48 \omega_{2}}{\zeta +1}+\frac{8 z^3}{\zeta +1}+\frac{24 z^2}{\zeta +1}\\& \quad -\frac{24\zeta z^2}{(\zeta +1)^2}+\frac{24\zeta ^2 z}{(\zeta +1)^3}+\frac{8 z}{\zeta +1}-\frac{24\zeta z}{(\zeta +1)^2}-\frac{24\zeta z}{(\zeta +1)^3}. \end{align}

    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].

    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

    {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1}, \omega_{2},z;\delta, a, b)

    for \delta = 2, \sigma = 1 , and r = 3 as:

    \begin{align} &{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{0,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z;2,a,b) = 0,\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{1,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z;2,a,b) = 0,\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{2,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z;2,a,b) = \frac{1}{\zeta ^b-a^b},\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{3,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z;2,a,b) = -\frac{3\zeta ^b}{\left(a^b-\zeta ^b\right)^2}-\frac{3 \omega_{1} a^b}{\left(a^b-\zeta ^b\right)^2}+\frac{3 \omega_{1}\zeta ^b}{\left(a^b-\zeta ^b\right)^2}-\frac{3 z a^b}{\left(a^b-\zeta ^b\right)^2}+\frac{3 z\zeta ^b}{\left(a^b-\zeta ^b\right)^2},\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{4,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z;2,a,b) = \frac{6 a^b\zeta ^b}{\left(\zeta ^b-a^b\right)^3}+\frac{6\zeta ^{2 b}}{\left(\zeta ^b-a^b\right)^3}+\frac{6 \omega_{1}^2}{\zeta ^b-a^b}-\frac{12 \omega_{1}\zeta ^b}{\left(\zeta ^b-a^b\right)^2}+\frac{12 \omega_{1} z}{\zeta ^b-a^b}\\& \quad +\frac{6 z^2}{\zeta ^b-a^b}-\frac{12 z\zeta ^b}{\left(\zeta ^b-a^b\right)^2}+\frac{6 z}{\zeta ^b-a^b},\\ &{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{5,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z;2,a,b) = -\frac{40 a^b\zeta ^{2 b}}{\left(a^b-\zeta ^b\right)^4}-\frac{10\zeta ^{3 b}}{\left(a^b-\zeta ^b\right)^4}+\frac{10 \omega_{1}^3}{\zeta ^b-a^b}-\frac{30 \omega_{1}^2\zeta ^b}{\left(\zeta ^b-a^b\right)^2}+\frac{30 \omega_{1}^2 z}{\zeta ^b-a^b}\\& \quad -\frac{30 \omega_{1} a^b\zeta ^b}{\left(a^b-\zeta ^b\right)^3}-\frac{30 \omega_{1}\zeta ^{2 b}}{\left(a^b-\zeta ^b\right)^3}+\frac{30 \omega_{1} z^2}{\zeta ^b-a^b}+\frac{30 \omega_{1} z}{\zeta ^b-a^b}-\frac{60 \omega_{1} z\zeta ^b}{\left(\zeta ^b-a^b\right)^2}\\& \quad +\frac{60 \omega_{2}}{\zeta ^b-a^b}+\frac{10 z^3}{\zeta ^b-a^b}+\frac{30 z^2}{\zeta ^b-a^b}-\frac{30 z^2\zeta ^b}{\left(\zeta ^b-a^b\right)^2}-\frac{30 z a^b\zeta ^b}{\left(a^b-\zeta ^b\right)^3}\\& \quad +\frac{10 z}{\zeta ^b-a^b}-\frac{30 z\zeta ^b}{\left(\zeta ^b-a^b\right)^2}-\frac{30 z\zeta ^{2 b}}{\left(a^b-\zeta ^b\right)^3}-\frac{10 a^{2 b}\zeta ^b}{\left(a^b-\zeta ^b\right)^4}. \end{align}

    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.

    Figure 1.  Curves of {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z; \delta, a, b) .

    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.

    Figure 2.  Zeros of {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z; \delta, a, b) = 0 .

    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

    {}_{{\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 3\leq \tau\leq 60 , give 3D structures, which are presented in Figure 3.

    Figure 3.  Stacking structure zeros of {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z; \delta, a, b) = 0 . This figure shows the 3D plot of the 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 3\leq \tau\leq 60 .

    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.

    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.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This research work was funded by the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2025).

    The authors declare no conflict of interest.



    [1] H. Yépez-Martínez, J. F. Gómez-Aguilar, A. Atangana, First integral method for non-linear differential equations with conformable derivative, Math. Model. Nat. Phenom., 13 (2018), 14. https://doi.org/10.1051/mmnp/2018012 doi: 10.1051/mmnp/2018012
    [2] H. Tajadodi, Z. A. Khan, A. ur R. Irshad, J. F. Gómez-Aguilar, A. Khan, H. Khan, Exact solutions of conformable fractional differential equations, Results Phys., 22 (2021), 103916. https://doi.org/10.1016/j.rinp.2021.103916 doi: 10.1016/j.rinp.2021.103916
    [3] S. R. Aderyani, R. Saadati, J. Vahidi, J. F. Gómez-Aguilar, The exact solutions of conformable time-fractional modified nonlinear Schrödinger equation by first integral method and functional variable method, Opt. Quant. Electron., 54 (2022), 218. https://doi.org/10.1007/s11082-022-03605-y doi: 10.1007/s11082-022-03605-y
    [4] H. Yépez-Martínez, A. Pashrashid, J. F. Gómez-Aguilar, L. Akinyemi, H. Rezazadeh, The novel soliton solutions for the conformable perturbed nonlinear Schrödinger equation, Mod. Phys. Lett. B, 36 (2022), 2150597. https://doi.org/10.1142/S0217984921505977 doi: 10.1142/S0217984921505977
    [5] K. Zhang, J. P. Cao, J. J. Lyu, Dynamic behavior and modulation instability for a generalized nonlinear Schrödinger equation with nonlocal nonlinearity, Phys. Scr., 100 (2024), 015262. https://doi.org/ 10.1088/1402-4896/ad9cfa doi: 10.1088/1402-4896/ad9cfa
    [6] Z. Li, J. J. Lyu, E. Hussain, Bifurcation, chaotic behaviors and solitary wave solutions for the fractional Twin-Core couplers with Kerr law non-linearity, Sci. Rep., 14 (2024), 22616. https://doi.org/10.1038/s41598-024-74044-w doi: 10.1038/s41598-024-74044-w
    [7] D. Chen, D. Shi, F. Chen, Qualitative analysis and new traveling wave solutions for the stochastic Biswas-Milovic equation, AIMS Mathematics, 10 (2025), 4092–4119. https://doi.org/10.3934/math.2025190 doi: 10.3934/math.2025190
    [8] Z. Li, S. Zhao, Bifurcation, chaotic behavior and solitary wave solutions for the Akbota equation, AIMS Mathematics, 9 (2024), 22590–22601. https://doi.org/10.3934/math.20241100 doi: 10.3934/math.20241100
    [9] A. R. Seadawy, M. Iqbal, Dispersive propagation of optical solitions and solitary wave solutions of Kundu-Eckhaus dynamical equation via modified mathematical method, Appl. Math. J. Chin. Univ., 38 (2023), 16–26. https://doi.org/10.1007/s11766-023-3861-2 doi: 10.1007/s11766-023-3861-2
    [10] W. A. Faridi, S. A. AlQahtani, The explicit power series solution formation and computationof Lie point infinitesimals generators: Lie symmetry approach, Phys. Scr., 98 (2023), 125249. https://doi.org/10.1088/1402-4896/ad0948 doi: 10.1088/1402-4896/ad0948
    [11] K. K. Ahmed, H. M. Ahmed, W. B. Rabie, M. F. Shehab, Effect of noise on wave solitons for (3+1)-dimensional nonlinear Schrödinger equation in optical fiber, Indian J. Phys., 98 (2024), 4863–4882. https://doi.org/10.1007/s12648-024-03222-3 doi: 10.1007/s12648-024-03222-3
    [12] A. M. Elsherbeny, A. Bekir, A. H. Arnous, M. Sadaf, G. Akram, Solitons to the time-fractional Radhakrishnan–Kundu–Lakshmanan equation with \beta and M-truncated fractional derivatives: a comparative analysis, Opt. Quant. Electron., 55 (2023), 1112. https://doi.org/10.1007/s11082-023-05414-3 doi: 10.1007/s11082-023-05414-3
    [13] M. A. S. Murad, Analysis of time-fractional Schrödinger equation with group velocity dispersion coefficients and second-order spatiotemporal effects: a new Kudryashov approach, Opt. Quant. Electron., 56 (2024), 908. https://doi.org/10.1007/s11082-024-06661-8 doi: 10.1007/s11082-024-06661-8
    [14] J. Vega-Guzman, M. F. Mahmood, Q. Zhou, H. Triki, A. H. Arnous, A. Biswas, et al., Solitons in nonlinear directional couplers with optical metamaterials, Nonlinear Dyn., 87 (2017), 427–458. https://doi.org/10.1007/s11071-016-3052-2 doi: 10.1007/s11071-016-3052-2
    [15] M. A. S. Murad, Perturbation of optical solutions and conservation laws in the presence of a dual form of generalized nonlocal nonlinearity and Kudryashov's refractive index having quadrupled power-law, Opt. Quant. Electron., 56 (2024), 864. https://doi.org/10.1007/s11082-024-06676-1 doi: 10.1007/s11082-024-06676-1
    [16] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A. S. Alshomrani, M. Z. Ullah, et al., Optical soliton perturbation with full nonlinearity for Gerdjikov–Ivanov equation by trial equation method, Optik, 157 (2018), 1214–1218. https://doi.org/10.1016/j.ijleo.2017.12.099 doi: 10.1016/j.ijleo.2017.12.099
    [17] E. Parasuraman, Evolution of dark optical soliton in birefringent fiber of Kundu-Eckhaus equation with four wave mixing and inter-modal dispersion, Optik, 243 (2021), 167380. https://doi.org/10.1016/j.ijleo.2021.167380 doi: 10.1016/j.ijleo.2021.167380
    [18] M. A. S. Murad, H. F. Ismael, F. K. Hamasalh, N. A. Shah, S. M. Eldin, Optical soliton solutions for time-fractional Ginzburg–Landau equation by a modified sub-equation method, Results Phys., 53 (2023), 106950. https://doi.org/10.1016/j.rinp.2023.106950 doi: 10.1016/j.rinp.2023.106950
    [19] S. Z. Majid, M. I. Asjad, W. A. Faridi, Solitary travelling wave profiles to the nonlinear generalized Calogero-Bogoyavlenskii-Schiff equation and dynamical assessment, Eur. Phys. J. Plus, 138 (2023), 1040. https://doi.org/10.1140/epjp/s13360-023-04681-z doi: 10.1140/epjp/s13360-023-04681-z
    [20] M. A. S. Murad, Formation of optical soliton wave profiles of nonlinear conformable Schrödinger equation in weakly non-local media: Kudryashov auxiliary equation method, J. Opt., 2024 (2024), 1–14. https://doi.org/10.1007/s12596-024-02110-7 doi: 10.1007/s12596-024-02110-7
    [21] K. Hosseini, E. Hincal, K. Sadri, F. Rabiei, M. Ilie, A. Akgül, et al., The positive multi-complexiton solution to a generalized Kadomtsev-Petviashvili equation, Partial Differential Equations in Applied Mathematics, 9 (2024), 100647. https://doi.org/10.1016/j.padiff.2024.100647 doi: 10.1016/j.padiff.2024.100647
    [22] M. Iqbal, A. R. Seadawy, D. C. Lu, Z. D. Zhang, Multiple optical soliton solutions for wave propagation in nonlinear low-pass electrical transmission lines under analytical approach, Opt. Quant. Electron., 56 (2024), 35. https://doi.org/10.1007/s11082-023-05611-0 doi: 10.1007/s11082-023-05611-0
    [23] A. H. Arnous, A. Biswas, A. H. Kara, Y. Yıldırım, L. Moraru, C. Iticescu, et al., Optical solitons and conservation laws for the concatenation model with spatio-temporal dispersion (internet traffic regulation), J. Eur. Opt. Society-Rapid Publ., 19 (2023), 35. https://doi.org/10.1051/jeos/2023031 doi: 10.1051/jeos/2023031
    [24] E. M. E. Zayed, A. G. Al-Nowehy, A. H. Arnous, M. S. Hashemi, M. A. S. Murad, M. Bayram, Investigating the generalized Kudryashov's equation in magneto-optic waveguide through the use of a couple integration techniques, J. Opt., 2024 (2024), 1–17. https://doi.org/10.1007/s12596-024-01857-3 doi: 10.1007/s12596-024-01857-3
    [25] M. A. S. Murad, Optical solutions with Kudryashov's arbitrary type of generalized non-local nonlinearity and refractive index via the new Kudryashov approach, Opt. Quant. Electron., 56 (2024), 999. https://doi.org/10.1007/s11082-024-06820-x doi: 10.1007/s11082-024-06820-x
    [26] C. Bhan, R. Karwasra, S. Malik, S. Kumar, A. H. Arnous, N. A. Shah, et al., Bifurcation, chaotic behavior, and soliton solutions to the KP-BBM equation through new Kudryashov and generalized Arnous methods, AIMS Mathematics, 9 (2024), 8749–8767. https://doi.org/10.3934/math.2024424 doi: 10.3934/math.2024424
    [27] M. A. S. Murad, Optical solutions for perturbed conformable Fokas–Lenells equation via Kudryashov auxiliary equation method, Mod. Phys. Lett. B, 39 (2025), 2450418. https://doi.org/10.1142/S0217984924504189 doi: 10.1142/S0217984924504189
    [28] M. A. S. Murad, Analyzing the time-fractional (3+1)-dimensional nonlinear Schrödinger equation: a new Kudryashov approach and optical solutions, Int. J. Comput. Math., 101 (2024), 524–537. https://doi.org/10.1080/00207160.2024.2351110 doi: 10.1080/00207160.2024.2351110
    [29] E. H. M. Zahran, M. S. M. Shehata, S. M. Mirhosseini-Alizamini, M. N. Alam, L. Akinyemi, Exact propagation of the isolated waves model described by the three coupled nonlinear Maccari's system with complex structure, Int. J. Mod. Phys. B, 35 (2021), 2150193. https://doi.org/10.1142/S0217979221501939 doi: 10.1142/S0217979221501939
    [30] L. Akinyemi, F. Erebholo, V. Palamara, K. Oluwasegun, A Study of nonlinear Riccati equation and its applications to multi-dimensional nonlinear evolution equations, Qual. Theory Dyn. Syst., 23 (2024), 296. https://doi.org/10.1007/s12346-024-01137-2 doi: 10.1007/s12346-024-01137-2
    [31] P. L. Li, S. Shi, C. J. Xu, M. ur Rahman, Bifurcations, chaotic behavior, sensitivity analysis and new optical solitons solutions of Sasa-Satsuma equation, Nonlinear Dyn., 112 (2024), 7405–7415. https://doi.org/10.1007/s11071-024-09438-6 doi: 10.1007/s11071-024-09438-6
    [32] X. H. Zhu, P. F. Xia, Q. Z. He, Z. W. Ni, L. P. Ni, Ensemble classifier design based on perturbation binary salp swarm algorithm for classification, CMES-Comp. Model. Eng., 135 (2023), 653–671. https://doi.org/10.32604/cmes.2022.022985 doi: 10.32604/cmes.2022.022985
    [33] H. Khan, S. Barak, P. Kumam, M. Arif, Analytical solutions of fractional Klein-Gordon and gas dynamics equations, via the (G'/G)-expansion method, Symmetry, 11 (2019), 566. https://doi.org/10.3390/sym11040566 doi: 10.3390/sym11040566
    [34] B. Li, Y. Zhang, X. L. Li, Z. Eskandari, Q. Z. He, Bifurcation analysis and complex dynamics of a Kopel triopoly model, J. Comput. Appl. Math., 426 (2023), 115089. https://doi.org/10.1016/j.cam.2023.115089 doi: 10.1016/j.cam.2023.115089
    [35] X. Zhang, X. Yang, Q. Z. He, Multi-scale systemic risk and spillover networks of commodity markets in the bullish and bearish regimes, N. Am. J. Econ. Financ., 62 (2022), 101766. https://doi.org/10.1016/j.najef.2022.101766 doi: 10.1016/j.najef.2022.101766
    [36] T. Y. Han, H. Rezazadeh, M. ur Rahman, High-order solitary waves, fission, hybrid waves and interaction solutions in the nonlinear dissipative (2+1)-dimensional Zabolotskaya-Khokhlov model, Phys. Scr., 99 (2024), 115212. https://doi.org/10.1088/1402-4896/ad7f04 doi: 10.1088/1402-4896/ad7f04
    [37] A. H. Arnous, M. Z. Ullah, S. P. Moshokoa, Q. Zhou, H. Triki, M. Mirzazadeh, et al., Optical solitons in birefringent fibers with modified simple equation method, Optik, 130 (2017), 996–1003. https://doi.org/10.1016/j.ijleo.2016.11.101 doi: 10.1016/j.ijleo.2016.11.101
    [38] Z. Eskandari, Z. Avazzadeh, R. K. Ghaziani, B. Li, Dynamics and bifurcations of a discrete‐time Lotka–Volterra model using nonstandard finite difference discretization method, Math. Method. Appl. Sci., 48 (2022), 7197–7212. https://doi.org/10.1002/mma.8859 doi: 10.1002/mma.8859
    [39] A. H. Arnous, M. Mirzazadeh, L. Akinyemi, A. Akbulut, New solitary waves and exact solutions for the fifth-order nonlinear wave equation using two integration techniques, J. Ocean Eng. Sci., 8 (2023), 475–480. https://doi.org/10.1016/j.joes.2022.02.012 doi: 10.1016/j.joes.2022.02.012
    [40] A. M. Elsherbeny, M. Mirzazadeh, A. Akbulut, A. H. Arnous, Optical solitons of the perturbation Fokas–Lenells equation by two different integration procedures, Optik, 273 (2023), 170382. https://doi.org/10.1016/j.ijleo.2022.170382 doi: 10.1016/j.ijleo.2022.170382
    [41] Y. Yildirim, E. Yasar, A. R. Adem, A multiple exp-function method for the three model equations of shallow water waves, Nonlinear Dyn., 89 (2017), 2291–2297. https://doi.org/10.1007/s11071-017-3588-9 doi: 10.1007/s11071-017-3588-9
    [42] Y. Yıldırım, Optical solitons to Kundu–Mukherjee–Naskar model in birefringent fibers with trial equation approach, Optik, 183 (2019), 1026–1031. https://doi.org/10.1016/j.ijleo.2019.02.141 doi: 10.1016/j.ijleo.2019.02.141
    [43] R. Radha, M. Lakshmanan, Singularity structure analysis and bilinear form of a (2+1) dimensional non-linear Schrodinger (NLS) equation, Inverse Probl., 10 (1994), L29. https://doi.org/10.1088/0266-5611/10/4/002 doi: 10.1088/0266-5611/10/4/002
    [44] Z. M. Yan, J. B. Li, S. Barak, S. Haque, N. Mlaiki, Delving into quasi-periodic type optical solitons in fully nonlinear complex structured perturbed Gerdjikov–Ivanov equation, Sci. Rep., 15 (2025), 8818. https://doi.org/10.1038/s41598-025-91978-x doi: 10.1038/s41598-025-91978-x
    [45] M. A. S. Murad, F. M. Omar, Optical solitons, dynamics of bifurcation, and chaos in the generalized integrable (2+1)-dimensional nonlinear conformable Schrödinger equations using a new Kudryashov technique, J. Comput. Appl. Math., 457 (2025), 116298. https://doi.org/10.1016/j.cam.2024.116298 doi: 10.1016/j.cam.2024.116298
    [46] A. R. Seadawy, N. Cheemaa, A. Biswas, Optical dromions and domain walls in (2+1)-dimensional coupled system, Optik, 227 (2021), 165669. https://doi.org/10.1016/j.ijleo.2020.165669 doi: 10.1016/j.ijleo.2020.165669
    [47] K. Hosseini, K. Sadri, M. Mirzazadeh, S. Salahshour, An integrable (2+1)-dimensional nonlinear Schrödinger system and its optical soliton solutions, Optik, 229 (2021), 166247. https://doi.org/10.1016/j.ijleo.2020.166247 doi: 10.1016/j.ijleo.2020.166247
    [48] Y. L. Xiao, S. Barak, M. Hleili, K. Shah, Exploring the dynamical behaviour of optical solitons in integrable kairat-Ⅱ and kairat-X equations, Phys. Scr., 99 (2024), 095261. https://doi.org/10.1088/1402-4896/ad6e34 doi: 10.1088/1402-4896/ad6e34
    [49] M. A. S. Murad, M. Iqbal, A. H. Arnous, A. Biswas, Y. Yildirim, A. S. Alshomrani, Optical dromions with fractional temporal evolution by enhanced modified tanh expansion approach, J. Opt., 2024 (2024), 1–10. https://doi.org/10.1007/s12596-024-01979-8 doi: 10.1007/s12596-024-01979-8
    [50] D. Z. Zhao, M. K. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54 (2017), 903–917. https://doi.org/10.1007/s10092-017-0213-8 doi: 10.1007/s10092-017-0213-8
    [51] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
    [52] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
    [53] M. Bilal, A. Khan, I. Ullah, H. Khan, J. Alzabut, H. M. Alkhawar, Application of modified extended direct algebraic method to nonlinear fractional diffusion reaction equation with cubic nonlinearity, Bound. Value Probl., 2025 (2025), 16. https://doi.org/10.1186/s13661-025-01997-w doi: 10.1186/s13661-025-01997-w
    [54] M. Bilal, J. Iqbal, I. Ullah, K. Shah, T. Abdeljawad, Using extended direct algebraic method to investigate families of solitary wave solutions for the space-time fractional modified benjamin bona mahony equation, Phys. Scr., 100 (2025), 015283. https://doi.org/10.1088/1402-4896/ad96e9 doi: 10.1088/1402-4896/ad96e9
    [55] I. Ullah, M. Bilal, J. Iqbal, H. Bulut, F. Turk, Single wave solutions of the fractional Landau-Ginzburg-Higgs equation in space-time with accuracy via the beta derivative and mEDAM approach, AIMS Mathematics, 10 (2025), 672–693. https://doi.org/10.3934/math.2025030 doi: 10.3934/math.2025030
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