Processing math: 43%
Research article

Clustering quantum Markov chains on trees associated with open quantum random walks

  • In networks, the Markov clustering (MCL) algorithm is one of the most efficient approaches in detecting clustered structures. The MCL algorithm takes as input a stochastic matrix, which depends on the adjacency matrix of the graph network under consideration. Quantum clustering algorithms are proven to be superefficient over the classical ones. Motivated by the idea of a potential clustering algorithm based on quantum Markov chains, we prove a clustering property for quantum Markov chains (QMCs) on Cayley trees associated with open quantum random walks (OQRW).

    Citation: Luigi Accardi, Amenallah Andolsi, Farrukh Mukhamedov, Mohamed Rhaima, Abdessatar Souissi. Clustering quantum Markov chains on trees associated with open quantum random walks[J]. AIMS Mathematics, 2023, 8(10): 23003-23015. doi: 10.3934/math.20231170

    Related Papers:

    [1] Can Kızılateş, Halit Öztürk . On parametric types of Apostol Bernoulli-Fibonacci, Apostol Euler-Fibonacci, and Apostol Genocchi-Fibonacci polynomials via Golden calculus. AIMS Mathematics, 2023, 8(4): 8386-8402. doi: 10.3934/math.2023423
    [2] Rabab Alyusof, Mdi Begum Jeelani . Some families of differential equations associated with the Gould-Hopper-Frobenius-Genocchi polynomials. AIMS Mathematics, 2022, 7(3): 4851-4860. doi: 10.3934/math.2022270
    [3] Letelier Castilla, William Ramírez, Clemente Cesarano, Shahid Ahmad Wani, Maria-Fernanda Heredia-Moyano . A new class of generalized Apostol–type Frobenius–Euler polynomials. AIMS Mathematics, 2025, 10(2): 3623-3641. doi: 10.3934/math.2025167
    [4] Ugur Duran, Can Kızılateş, William Ramírez, Clemente Cesarano . A new class of degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type. AIMS Mathematics, 2025, 10(7): 16117-16138. doi: 10.3934/math.2025722
    [5] Ruma Qamar, Tabinda Nahid, Mumtaz Riyasat, Naresh Kumar, Anish Khan . Gould-Hopper matrix-Bessel and Gould-Hopper matrix-Tricomi functions and related integral representations. AIMS Mathematics, 2020, 5(5): 4613-4623. doi: 10.3934/math.2020296
    [6] William Ramírez, Can Kızılateş, Daniel Bedoya, Clemente Cesarano, Cheon Seoung Ryoo . On certain properties of three parametric kinds of Apostol-type unified Bernoulli-Euler polynomials. AIMS Mathematics, 2025, 10(1): 137-158. doi: 10.3934/math.2025008
    [7] Rabab Alyusof, Mdi Begum Jeelani . Some families of differential equations associated with the Gould-Hopper-Frobenius-Genocchi polynomials. AIMS Mathematics, 2022, 7(11): 20381-20382. doi: 10.3934/math.20221116
    [8] Taekyun Kim, Dae San Kim, Dmitry V. Dolgy, Hye Kyung Kim, Hyunseok Lee . A new approach to Bell and poly-Bell numbers and polynomials. AIMS Mathematics, 2022, 7(3): 4004-4016. doi: 10.3934/math.2022221
    [9] Mohra Zayed, Shahid Ahmad Wani, Georgia Irina Oros, William Ramŕez . A study on extended form of multivariable Hermite-Apostol type Frobenius-Euler polynomials via fractional operators. AIMS Mathematics, 2024, 9(6): 16297-16312. doi: 10.3934/math.2024789
    [10] Aimin Xu . Some identities involving derangement polynomials and r-Bell polynomials. AIMS Mathematics, 2024, 9(1): 2051-2062. doi: 10.3934/math.2024102
  • In networks, the Markov clustering (MCL) algorithm is one of the most efficient approaches in detecting clustered structures. The MCL algorithm takes as input a stochastic matrix, which depends on the adjacency matrix of the graph network under consideration. Quantum clustering algorithms are proven to be superefficient over the classical ones. Motivated by the idea of a potential clustering algorithm based on quantum Markov chains, we prove a clustering property for quantum Markov chains (QMCs) on Cayley trees associated with open quantum random walks (OQRW).



    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 \frac{ \mu^ {\tau }} {\tau!} on both sides of the resultant equation, we arrive at the asserted result, as given by (4.1).

    Remark 13. Setting \omega_{2} = 0 in (4.1), we have

    \begin{equation} \frac{\partial^{\nu}}{\partial \omega_{1}^ {\nu}}\left\lbrace {}_{\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma)}(\omega_{1} , z;\delta,a,b)\right\rbrace = \begin{cases} \frac {\tau !\; {}_{\mathcal{B}el}\mathsf{P}_{\tau-\nu,\zeta}^{(\sigma)}(\omega_{1} , z;\delta,a,b)}{(\tau-\nu)!},\; & \tau\geq \nu; \\ 0, & 0\leq \tau < \nu. \end{cases} \end{equation} (4.3)

    Similarly, upon differentiating relation (2.3) \nu times with respect to \omega_{2} , we can get the following result.

    Theorem 10. For \nu, \tau\in \mathbb{N}_{0}, \delta \in\mathbb{N} , and \sigma, \zeta\in\mathbb{C} , we have

    \begin{equation} \frac{\partial^{\nu}}{\partial \omega_{2}^ {\nu}}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b)\right\rbrace = \frac {\tau !\; {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau-r\nu,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b)}{(\tau-r\nu)!}. \end{equation} (4.4)

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

    \begin{equation} \frac{\partial^{\nu}}{\partial \omega_{2}^ {\nu}}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(2)}(\omega_{1} , \omega_{2} ,z;\delta,a,b)\right\rbrace = \frac {\tau !\; {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau-2\nu,\zeta}^{(2)}(\omega_{1} , \omega_{2} ,z;\delta,a,b)}{(\tau-2\nu)!}. \end{equation} (4.5)

    Theorem 11. For \tau\in \mathbb{N}_{0}, \delta \in\mathbb{N} , and \sigma, \zeta\in\mathbb{C} , we have

    \begin{align} \frac{\partial}{\partial z}&\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b)\right\rbrace \\& \quad = \sum\limits_{\kappa = 0}^ {\tau } {\tau \choose \kappa} \left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau-\kappa,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b) - {}_{{\mathcal{H}}}\mathsf{P}_{\tau-\kappa,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ;\delta,a,b)\; \mathcal{B}el_{\kappa}( z)\right\rbrace. \end{align} (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

    \begin{align} \; \sum\limits_ {\tau = 0}^{\infty}&\frac{\partial}{\partial z}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b)\right\rbrace \; \frac{ \mu^ {\tau }} {\tau !}\\& = (e^ \mu -1)\left\lbrace \bigg(\frac{2^{1-\delta} \; \mu^{\delta}}{\zeta^{b}e^{\mu}-a^{b}}\bigg)^{\sigma}\; e^{\omega_{1} \mu +\omega_{2} \mu^ {r}+z(e^{ \mu} -1)}\right\rbrace \\& = \left\lbrace\sum\limits_ {\tau = 0}^{\infty}\frac{ \mu^ {\tau}}{ \tau!}\right\rbrace\left\lbrace \sum\limits_ {\tau = 0}^{\infty}{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b)\; \frac{ \mu^ {\tau }} {\tau !}\right\rbrace- \left\lbrace \sum\limits_ {\tau = 0}^{\infty}{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b)\; \frac{ \mu^ {\tau }} {\tau !}\right\rbrace\\& = \sum\limits_ {\tau = 0}^{\infty} \sum\limits_{\kappa = 0}^ {\tau } {\tau \choose \kappa} \left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau-\kappa,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b) - {}_{{\mathcal{H}}}\mathsf{P}_{\tau-\kappa,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ;\delta,a,b)\; \mathcal{B}el_{\kappa}( z)\right\rbrace\; \frac{ \mu^ {\tau }} {\tau !}. \end{align} (4.7)

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

    Remark 15. Setting \omega_{2} = 0 in (4.6), we have

    \begin{align} \frac{\partial}{\partial z} {}_{\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma)}(\omega_{1} ,z;\delta,a,b) = \sum\limits_{\kappa = 0}^ {\tau } {\tau \choose \kappa} \left\lbrace {}_{\mathcal{B}el}\mathsf{P}_{\tau-\kappa,\zeta}^{(\sigma,r)}(\omega_{1} ,z;\delta,a,b) - {}_{}\mathsf{P}_{\tau-\kappa,\zeta}^{(\sigma)}(\omega_{1} ;\delta,a,b)\; \mathcal{B}el_{\kappa}( z)\right\rbrace. \end{align} (4.8)

    Similarly, we can derive the following outcome.

    Corollary 1. For \tau\in \mathbb{N}_{0}, \delta \in\mathbb{N} , and \sigma, \zeta\in\mathbb{C} , we have

    \begin{equation} \frac{\partial}{\partial z}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b)\right\rbrace = {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau-\kappa,\zeta}^{(\sigma,r)}(\omega_{1} +1, \omega_{2} ,z;\delta,a,b) -{}_\mathcal{G}\mathcal{B}el_ {\tau }(\omega_{1} ,\omega_{2} ,z). \end{equation} (4.9)

    Theorem 12. The following formula holds true:

    \begin{align} \int_{u}^{u+\gamma}&{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b) \; \mathrm{d}\omega_{1}\\& = \frac{1} {\tau +1}\bigg[{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau+1,\zeta}^{(\sigma,r)}(u+\gamma , \omega_{2} ,z;\delta,a,b)-{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau+1,\zeta}^{(\sigma,r)}(u, \omega_{2} ,z;\delta,a,b)\bigg]. \end{align} (4.10)

    Proof. By integrating both sides of Eq (2.3) with respect to \omega_{1} , we obtain

    \begin{align} \sum\limits_ {\tau = 0}^{\infty}\int_{u}^{u+\gamma}&{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b) \; \mathrm{d}\omega_{1}\; \frac{ \mu^ {\tau }} {\tau !}\\& = \frac{1}{ \mu} \bigg[\bigg(\frac{2^{1-\delta} \; \mu^{\delta}}{\zeta^{b}e^{\mu}-a^{b}}\bigg)^{\sigma}\; e^{(u+\gamma) \mu +\omega_{2} \mu^ {r}+z(e^{ \mu} -1)}-\bigg(\frac{2^{1-\delta} \; \mu^{\delta}}{\zeta^{b}e^{\mu}-a^{b}}\bigg)^{\sigma}\; e^{u \mu +\omega_{2} \mu^ {r}+z(e^{ \mu} -1)}\bigg]\\& = \frac{1}{ \mu} \bigg[\sum\limits_ {\tau = 0}^{\infty} {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(u+\gamma , \omega_{2} ,z;\delta,a,b) \frac{ \mu^ {\tau }} {\tau !} - \sum\limits_ {\tau = 0}^{\infty} {}_{{\mathcal{H}}}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(u , \omega_{2} ;\delta,a,b)\; \mathcal{B}el_{\kappa}( z)\; \frac{ \mu^ {\tau }} {\tau !}\bigg]. \end{align} (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:

    \begin{align} &\int_{u}^{u+\gamma}{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b) \; \mathrm{d}\omega_{2}\\& \quad = \frac{\tau!} {(\tau+r)! }\bigg[{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau+r,\zeta}^{(\sigma,r)}(\omega_{1}, u+\gamma ,z;\delta,a,b)-{}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau+r,\zeta}^{(\sigma,r)}(\omega_{1}, u ,z;\delta,a,b)\bigg], \end{align} (4.12)
    \begin{align} &\int_{u}^{u+\gamma} {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z;\delta,a,b) \; \mathrm{d}z \\& = \frac{1}{\tau+1} \sum\limits_{k = 0}^{\tau} \binom{\tau+1}{k} \mathfrak{B}_k \left[ {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau+1-k,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,u+\gamma;\delta,a,b) - {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau+1-k,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,u;\delta,a,b) \right]. \end{align} (4.13)

    In the next section, we turn to the consideration of several special cases of the trivariate Gould-Hopper-Bell-Apostol-type polynomials {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z;\delta, a, b) of order \sigma .

    In this section, some applications related to the established polynomials (TGHBelATP {}_{{\mathcal{H}}\mathcal{B}el}\mathsf{P}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z;\delta, a, b) of order \sigma ) are presented. Certain examples are investigated. Further, the zero distributions 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 are examined.

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

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

    with analogous results presented for each.

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

    Since

    \mathsf{P}_{\tau,\zeta}^{(\sigma)}(\omega;1,1,1) = \mathfrak{B}^{(\sigma)}_{\tau}(\omega;\zeta),

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

    \begin{equation} \; \bigg(\frac{\mu}{\zeta e^{\mu}-1}\bigg)^{\sigma}\; e^{\omega_{1} \mu +\omega_{2} \mu^ {r}+z(e^{ \mu} -1)} = \sum\limits_ {\tau = 0}^{\infty} {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z)\; \frac{ \mu^ {\tau }} {\tau !}, \end{equation} (5.1)

    where {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) are referred to as the Gould-Hopper-Bell-Apostol-Bernoulli polynomials (GHBelBP) of order \sigma .

    The series representations of the GHBelBP {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) of order \sigma are given as:

    \begin{equation} {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z) = \sum\limits_{\kappa = 0}^ {\tau } {\tau \choose \kappa} {}_{\mathcal{B}el}\mathfrak{B}_{\tau-\kappa,\zeta}^{(\sigma)}(z)\; \mathcal{H}_{\kappa}^{(r)}(\omega_{1},\omega_{2}); \end{equation} (5.2)
    \begin{equation} {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z) = \sum\limits_{\kappa = 0}^ {\tau } {\tau \choose \kappa} {}_{{\mathcal{H}}}\mathfrak{B}_{\tau-\kappa,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} )\; \mathcal{B}el_{\kappa}( z); \end{equation} (5.3)
    \begin{equation} {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z) = \sum\limits_{\kappa = 0}^ {\tau } {\tau \choose \kappa} \mathfrak{B}_{\tau-\kappa,\zeta}^{(\sigma)}(0)\; {}_{\mathcal{H}}\mathcal{B}el^{(r)}_ {\kappa }(\omega_{1} ,\omega_{2} ,z); \end{equation} (5.4)
    \begin{equation} {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z) = \tau!\sum\limits_{\kappa = 0}^ {[\frac{\tau}{r}] } \frac{\omega^{\kappa}_{2}\; {}_{\mathcal{B}el}\mathfrak{B}_{\tau-r\kappa,\zeta}^{(\sigma)}(\omega_{1} ,z)}{\kappa! (\tau-r\kappa)!}. \end{equation} (5.5)

    Certain corresponding results related to the GHBelBP {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) of order \sigma are presented in Table 1.

    Table 1.  Findings for the Gould-Hopper-Bell-Apostol-Bernoulli polynomials {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) .
    Multiplicative and derivative operator \hat{M}_{{}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}}=\omega_{1} +r\omega_{2} D_{\omega_{1}}^{r-1}+z e^ {D_{\omega_{1}}} + \frac{\sigma (\zeta e^{D_{\omega_{1}}}-1)-\sigma D_{\omega_{1}}\zeta e^{D_{\omega_{1}}}}{D_{\omega_{1}}(\zeta e^{D_{\omega_{1}}}-1)}, \hat{P}_{{}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}}=D_{\omega_{1} }
    Differential equation \bigg(\omega_{1} D_{\omega_{1}} +r\omega_{2} D_{\omega_{1}}^{r}+z e^ {D_{\omega_{1}}} D_{\omega_{1}} +\frac{\sigma (\zeta e^{D_{\omega_{1}}}-1)-\sigma D_{\omega_{1}}\zeta e^{D_{\omega_{1}}}}{(\zeta e^{D_{\omega_{1}}}-1)}-\tau\bigg) {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)=0
    Summation {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)=\frac{1}{2}\sum\nolimits_{\kappa=0}^ {\tau } {\tau \choose \kappa} \mathcal{E}_{\kappa}\bigg({}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}+1, \omega_{2}, z)+{}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\bigg)
    Formulae {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\upsilon, \omega_{2}, z)=\sum\limits_{ \kappa=0}^{\tau}\binom{\tau} {\kappa } (\upsilon-\omega_{1}+\theta)^{\tau-\kappa } {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\kappa, \zeta}^{(\sigma, r)}(\omega_{1}-\theta, \omega_{2}, z)
    {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma+\beta, r)}(\omega_{1}+x, \omega_{2}+y, z+u)=\sum\limits_{ \kappa=0}^{\tau}\binom{\tau} {\kappa } {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\kappa, \zeta}^{(\beta, r)}(x, y, u)
    {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau-\varrho, \zeta}^{(\sigma-\varrho, r)}(\omega_{1}, \omega_{2}, z)=\frac{(\tau-\varrho)! \varrho!}{\tau!}\sum\limits_{ \kappa=0}^{\tau}\binom{\tau} {\kappa } {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) S(\kappa, \varrho, \zeta)
    {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau+1, \zeta}^{(1, r)}(\omega_{1}+1, \omega_{2}, z)=\frac{1}{\zeta}\left\lbrace (\tau+1) {}_{\mathcal{H}}\mathcal{B}el^{(r)}_ {\tau }(\omega_{1}, \omega_{2}, z) + {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau+1, \zeta}^{(1, r)}(\omega_{1}, \omega_{2}, z) \right\rbrace
    Differential and \frac{\partial^{\nu}}{\partial \omega_{1}^ {\nu}}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\right\rbrace = \begin{cases} \frac {\tau! {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau-\nu, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)}{(\tau-\nu)!}, & \tau\geq \nu; \\ 0, & 0\leq \tau < \nu. \end{cases}
    \frac{\partial^{\nu}}{\partial \omega_{2}^ {\nu}}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\right\rbrace = \frac {\tau! {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau-r\nu, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)}{(\tau-r\nu)!}
    \frac{\partial}{\partial z}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\right\rbrace =\sum\nolimits_{\kappa=0}^ {\tau } {\tau \choose \kappa} \left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) - {}_{{\mathcal{H}}}\mathfrak{B}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}) \mathcal{B}el_{\kappa}(z)\right\rbrace
    \frac{\partial}{\partial z}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\right\rbrace = {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1} +1, \omega_{2}, z) -{}_\mathcal{G}\mathcal{B}el_ {\tau }(\omega_{1}, \omega_{2}, z)
    Integral Formulae \int_{u}^{u+\gamma}{}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) \mathrm{d}\omega_{1}=\frac{1} {\tau +1}\bigg[{}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau+1, \zeta}^{(\sigma, r)}(u+\gamma, \omega_{2}, z)-{}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau+1, \zeta}^{(\sigma, r)}(u, \omega_{2}, z)\bigg]
    \int_{u}^{u+\gamma} {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z;\delta, a, b) \mathrm{d}z
    = \frac{1}{\tau+1} \sum\nolimits_{k=0}^{\tau} \binom{\tau+1}{k} \mathfrak{B}_k \left[ {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau+1-k, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, u+\gamma; \delta, a, b) - {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau+1-k, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, u;\delta, a, b) \right]

     | Show Table
    DownLoad: CSV

    The first few members of the GHBelBP

    {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z),

    for \sigma = 1 and r = 3 are given as:

    \begin{align} &{}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{0,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = 0,\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{1,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = \frac{1}{\zeta -1},\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{2,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = -\frac{2\zeta }{(\zeta -1)^2}+\frac{2\zeta \omega_{1}}{(\zeta -1)^2}-\frac{2 \omega_{1}}{(\zeta -1)^2}+\frac{2\zeta z}{(\zeta -1)^2}-\frac{2 z}{(\zeta -1)^2},\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{3,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = \frac{3\zeta ^2}{(\zeta -1)^3}+\frac{3\zeta }{(\zeta -1)^3}+\frac{3 \omega_{1}^2}{\zeta -1}-\frac{6\zeta \omega_{1}}{(\zeta -1)^2}+\frac{6 \omega_{1} z}{\zeta -1}+\frac{3 z^2}{\zeta -1}\\& \quad +\frac{3 z}{\zeta -1}-\frac{6\zeta z}{(\zeta -1)^2}, \end{align}
    \begin{align} {}_{{\mathcal{H}}\mathcal{B}el}\mathfrak{B}_{4,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z)& = -\frac{4\zeta ^3}{(\zeta -1)^4}-\frac{16\zeta ^2}{(\zeta -1)^4}-\frac{4\zeta }{(\zeta -1)^4}+\frac{4 \omega_{1}^3}{\zeta -1}-\frac{12\zeta \omega_{1}^2}{(\zeta -1)^2}+\frac{12 \omega_{1}^2 z}{\zeta -1}\\&+\frac{12\zeta ^2 \omega_{1}}{(\zeta -1)^3}+\frac{12\zeta \omega_{1}}{(\zeta -1)^3}+\frac{12 \omega_{1} z^2}{\zeta -1}+\frac{12 \omega_{1} z}{\zeta -1}-\frac{24\zeta \omega_{1} z}{(\zeta -1)^2}+\frac{24 \omega_{2}}{\zeta -1}+\frac{4 z^3}{\zeta -1}\\&+\frac{12 z^2}{\zeta -1}-\frac{12\zeta z^2}{(\zeta -1)^2}+\frac{12\zeta ^2 z}{(\zeta -1)^3}+\frac{12\zeta z}{(\zeta -1)^3}+\frac{4 z}{\zeta -1}-\frac{12\zeta z}{(\zeta -1)^2}. \end{align}

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

    Since

    \mathsf{P}_{\tau,\zeta}^{(\sigma)}(\omega;0,-1,1) = \mathcal{E}^{(\sigma)}_{\tau}(\omega;\zeta),

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

    \begin{equation} \; \bigg(\frac{2}{\zeta e^{\mu}+1}\bigg)^{\sigma}\; e^{\omega_{1} \mu +\omega_{2} \mu^ {r}+z(e^{ \mu} -1)} = \sum\limits_ {\tau = 0}^{\infty} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z)\; \frac{ \mu^ {\tau }} {\tau !}, \end{equation} (5.6)

    where

    {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z)

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

    The series representations of the GHBelEP

    {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z)

    of order \sigma are given as:

    \begin{equation} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z) = \sum\limits_{\kappa = 0}^ {\tau } {\tau \choose \kappa} {}_{\mathcal{B}el}\mathcal{E}_{\tau-\kappa,\zeta}^{(\sigma)}(z)\; \mathcal{H}_{\kappa}^{(r)}(\omega_{1},\omega_{2}); \end{equation} (5.7)
    \begin{equation} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z) = \sum\limits_{\kappa = 0}^ {\tau } {\tau \choose \kappa} {}_{{\mathcal{H}}}\mathcal{E}_{\tau-\kappa,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} )\; \mathcal{B}el_{\kappa}( z); \end{equation} (5.8)
    \begin{equation} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z) = \sum\limits_{\kappa = 0}^ {\tau } {\tau \choose \kappa} \mathcal{E}_{\tau-\kappa,\zeta}^{(\sigma)}(0)\; {}_{\mathcal{H}}\mathcal{B}el^{(r)}_ {\kappa }(\omega_{1} ,\omega_{2} ,z); \end{equation} (5.9)
    \begin{equation} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z) = \tau!\sum\limits_{\kappa = 0}^ {[\frac{\tau}{r}] } \frac{\omega^{\kappa}_{2}\; {}_{\mathcal{B}el}\mathcal{E}_{\tau-r\kappa,\zeta}^{(\sigma)}(\omega_{1} ,z)}{\kappa! (\tau-r\kappa)!}. \end{equation} (5.10)

    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.

    Table 2.  Findings for the Gould-Hopper-Bell-Apostol-Euler polynomials {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) .
    Multiplicative and derivative operators \hat{M}_{{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}}=\omega_{1} +r\omega_{2} D_{\omega_{1}}^{r-1}+z e^ {D_{\omega_{1}}} - \frac{\sigma\zeta e^{D_{\omega_{1}}}}{\zeta e^{D_{\omega_{1}}}+1}, \hat{P}_{{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}}=D_{\omega_{1} }
    Differential equation \bigg(\omega_{1} D_{\omega_{1}} +r\omega_{2} D_{\omega_{1}}^{r}+z e^ {D_{\omega_{1}}} D_{\omega_{1}} -\frac{\sigma D_{\omega_{1}}\zeta e^{D_{\omega_{1}}}}{(\zeta e^{D_{\omega_{1}}}+1)}-\tau\bigg) {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)=0
    Summation {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)=\frac{1}{2}\sum\nolimits_{\kappa=0}^ {\tau } {\tau \choose \kappa} \mathcal{E}_{\kappa}\bigg({}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}+1, \omega_{2}, z)+{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\bigg)
    Formulae {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma, r)}(\upsilon, \omega_{2}, z)=\sum\limits_{ \kappa=0}^{\tau}\binom{\tau} {\kappa } (\upsilon-\omega_{1}+\theta)^{\tau-\kappa } {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\kappa, \zeta}^{(\sigma, r)}(\omega_{1}-\theta, \omega_{2}, z)
    {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma+\beta, r)}(\omega_{1}+x, \omega_{2}+y, z+u)=\sum\limits_{ \kappa=0}^{\tau}\binom{\tau} {\kappa } {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\kappa, \zeta}^{(\beta, r)}(x, y, u)
    {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma-\varrho, r)}(\omega_{1}, \omega_{2}, z)=\frac{\varrho!}{2^{\varrho}}\sum\limits_{ \kappa=0}^{\tau}\binom{\tau} {\kappa } {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) S(\kappa, \varrho, \zeta)
    {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(1, r)}(\omega_{1}+1, \omega_{2}, z)=\frac{1}{\zeta}\left\lbrace 2 {}_{\mathcal{H}}\mathcal{B}el^{(r)}_ {\tau }(\omega_{1}, \omega_{2}, z) - {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau+\delta, \zeta}^{(1, r)}(\omega_{1}, \omega_{2}, z) \right\rbrace
    Differential and \frac{\partial^{\nu}}{\partial \omega_{1}^ {\nu}}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\right\rbrace = \begin{cases} \frac {\tau! {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau-\nu, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)}{(\tau-\nu)!}, & \tau\geq \nu; \\ 0, & 0\leq \tau < \nu. \end{cases}
    \frac{\partial^{\nu}}{\partial \omega_{2}^ {\nu}}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\right\rbrace = \frac {\tau! {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau-r\nu, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)}{(\tau-r\nu)!}
    \frac{\partial}{\partial z}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\right\rbrace =\sum\nolimits_{\kappa=0}^ {\tau } {\tau \choose \kappa} \left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) - {}_{{\mathcal{H}}}\mathcal{E}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}) \mathcal{B}el_{\kappa}(z)\right\rbrace
    \frac{\partial}{\partial z}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\right\rbrace = {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1} +1, \omega_{2}, z) -{}_\mathcal{G}\mathcal{B}el_ {\tau }(\omega_{1}, \omega_{2}, z)
    Integral Formulae \int_{u}^{u+\gamma}{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) \mathrm{d}\omega_{1}=\frac{1} {\tau +1}\bigg[{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau+1, \zeta}^{(\sigma, r)}(u+\gamma, \omega_{2}, z)-{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau+1, \zeta}^{(\sigma, r)}(u, \omega_{2}, z)\bigg]
    \int_{u}^{u+\gamma} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z;\delta, a, b) \mathrm{d}z
    = \frac{1}{\tau+1} \sum\nolimits_{k=0}^{\tau} \binom{\tau+1}{k} \mathfrak{B}_k \left[ {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau+1-k, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, u+\gamma; \delta, a, b) - {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau+1-k, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, u;\delta, a, b) \right]

     | Show Table
    DownLoad: CSV

    The first few members of the GHBelEP

    {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z),

    for \sigma = 2 and r = 3 are given as:

    \begin{align} &{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{0,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = \frac{4}{(\zeta +1)^2},\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{1,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = -\frac{8\zeta }{(\zeta +1)^3}+\frac{4\zeta \omega_{1}}{(\zeta +1)^3}+\frac{4 \omega_{1}}{(\zeta +1)^3}+\frac{4 z}{(\zeta +1)^3}+\frac{4\zeta z}{(\zeta +1)^3},\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{2,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = \frac{16\zeta ^2}{(\zeta +1)^4}-\frac{8\zeta }{(\zeta +1)^4}+\frac{4 \omega_{1}^2}{(\zeta +1)^2}-\frac{16\zeta \omega_{1}}{(\zeta +1)^3}+\frac{8 \omega_{1} z}{(\zeta +1)^2}+\frac{4 z^2}{(\zeta +1)^2}\\& \quad +\frac{4 z}{(\zeta +1)^2}-\frac{16\zeta z}{(\zeta +1)^3},\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{3,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = -\frac{32\zeta ^3}{(\zeta +1)^5}+\frac{56\zeta ^2}{(\zeta +1)^5}-\frac{8\zeta }{(\zeta +1)^5}+\frac{4 \omega_{1}^3}{(\zeta +1)^2}-\frac{24\zeta \omega_{1}^2}{(\zeta +1)^3}+\frac{12 \omega_{1}^2 z}{(\zeta +1)^2}\\& \quad +\frac{48\zeta ^2 \omega_{1}}{(\zeta +1)^4}-\frac{24\zeta \omega_{1}}{(\zeta +1)^4}+\frac{12 \omega_{1} z^2}{(\zeta +1)^2}+\frac{12 \omega_{1} z}{(\zeta +1)^2}-\frac{48\zeta \omega_{1} z}{(\zeta +1)^3}+\frac{24 \omega_{2}}{(\zeta +1)^2}+\frac{4 z^3}{(\zeta +1)^2}\\& \quad +\frac{12 z^2}{(\zeta +1)^2}-\frac{24\zeta z^2}{(\zeta +1)^3}+\frac{48\zeta ^2 z}{(\zeta +1)^4}+\frac{4 z}{(\zeta +1)^2}-\frac{24\zeta z}{(\zeta +1)^3}-\frac{24\zeta z}{(\zeta +1)^4}, \end{align}
    \begin{align} &{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{E}_{4,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = \frac{64\zeta ^4}{(\zeta +1)^6}-\frac{264\zeta ^3}{(\zeta +1)^6}+\frac{144\zeta ^2}{(\zeta +1)^6}-\frac{8\zeta }{(\zeta +1)^6}+\frac{4 \omega_{1}^4}{(\zeta +1)^2}-\frac{32\zeta \omega_{1}^3}{(\zeta +1)^3}\\& \quad +\frac{16 \omega_{1}^3 z}{(\zeta +1)^2}+\frac{96\zeta ^2 \omega_{1}^2}{(\zeta +1)^4}-\frac{48\zeta \omega_{1}^2}{(\zeta +1)^4}+\frac{24 \omega_{1}^2 z^2}{(\zeta +1)^2}+\frac{24 \omega_{1}^2 z}{(\zeta +1)^2}-\frac{96\zeta \omega_{1}^2 z}{(\zeta +1)^3}-\frac{128\zeta ^3 \omega_{1}}{(\zeta +1)^5}\\& \quad +\frac{224\zeta ^2 \omega_{1}}{(\zeta +1)^5}-\frac{32\zeta \omega_{1}}{(\zeta +1)^5}+\frac{96 \omega_{1} \omega_{2}}{(\zeta +1)^2}+\frac{16 \omega_{1} z^3}{(\zeta +1)^2}+\frac{48 \omega_{1} z^2}{(\zeta +1)^2}-\frac{96\zeta \omega_{1} z^2}{(\zeta +1)^3}+\frac{192\zeta ^2 \omega_{1} z}{(\zeta +1)^4}\\& \quad +\frac{16 \omega_{1} z}{(\zeta +1)^2}-\frac{96\zeta \omega_{1} z}{(\zeta +1)^3}-\frac{96\zeta \omega_{1} z}{(\zeta +1)^4}-\frac{192\zeta \omega_{2}}{(\zeta +1)^3}+\frac{96 \omega_{2} z}{(\zeta +1)^2}+\frac{4 z^4}{(\zeta +1)^2}+\frac{24 z^3}{(\zeta +1)^2}\\& \quad -\frac{32\zeta z^3}{(\zeta +1)^3}+\frac{96\zeta ^2 z^2}{(\zeta +1)^4}+\frac{28 z^2}{(\zeta +1)^2}-\frac{96\zeta z^2}{(\zeta +1)^3}-\frac{48\zeta z^2}{(\zeta +1)^4}-\frac{128\zeta ^3 z}{(\zeta +1)^5}+\frac{96\zeta ^2 z}{(\zeta +1)^4}\\& \quad +\frac{224\zeta ^2 z}{(\zeta +1)^5}+\frac{4 z}{(\zeta +1)^2}-\frac{32\zeta z}{(\zeta +1)^3}-\frac{48\zeta z}{(\zeta +1)^4}-\frac{32\zeta z}{(\zeta +1)^5}. \end{align}

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

    Since

    \mathsf{P}_{\tau,\frac{\zeta}{2}}^{(\sigma)}\bigg(\omega;1,-\frac{1}{2},1\bigg) = \mathcal{G}^{(\sigma)}_{\tau}(\omega;\zeta),

    therefore, taking \delta = 1, a = -\frac{1}{2}, b = 1 and \zeta\longrightarrow\frac{\zeta}{2} in generating function (2.3), gives

    \begin{equation} \; \bigg(\frac{2\mu}{\zeta e^{\mu}+1}\bigg)^{\sigma}\; e^{\omega_{1} \mu +\omega_{2} \mu^ {r}+z(e^{ \mu} -1)} = \sum\limits_ {\tau = 0}^{\infty} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z)\; \frac{ \mu^ {\tau }} {\tau !}, \end{equation} (5.11)

    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:

    \begin{equation} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z) = \sum\limits_{\kappa = 0}^ {\tau } {\tau \choose \kappa} {}_{\mathcal{B}el}\mathcal{G}_{\tau-\kappa,\zeta}^{(\sigma)}(z)\; \mathcal{H}_{\kappa}^{(r)}(\omega_{1},\omega_{2}); \end{equation} (5.12)
    \begin{equation} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z) = \sum\limits_{\kappa = 0}^ {\tau } {\tau \choose \kappa} {}_{{\mathcal{H}}}\mathcal{G}_{\tau-\kappa,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} )\; \mathcal{B}el_{\kappa}( z); \end{equation} (5.13)
    \begin{equation} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z) = \sum\limits_{\kappa = 0}^ {\tau } {\tau \choose \kappa} \mathcal{G}_{\tau-\kappa,\zeta}^{(\sigma)}(0)\; {}_{\mathcal{H}}\mathcal{B}el^{(r)}_ {\kappa }(\omega_{1} ,\omega_{2} ,z); \end{equation} (5.14)
    \begin{equation} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau,\zeta}^{(\sigma,r)}(\omega_{1} , \omega_{2} ,z) = \tau!\sum\limits_{\kappa = 0}^ {[\frac{\tau}{r}] } \frac{\omega^{\kappa}_{2}\; {}_{\mathcal{B}el}\mathcal{G}_{\tau-r\kappa,\zeta}^{(\sigma)}(\omega_{1} ,z)}{\kappa! (\tau-r\kappa)!}. \end{equation} (5.15)

    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.

    Table 3.  Findings for the Gould-Hopper-Bell-Apostol-Genocchi polynomials {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) .
    Multiplicative and derivative operators \hat{M}_{{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}}=\omega_{1} +r\omega_{2} D_{\omega_{1}}^{r-1}+z e^ {D_{\omega_{1}}} + \frac{\sigma (\zeta e^{D_{\omega_{1}}}+1)-\sigma D_{\omega_{1}}\zeta e^{D_{\omega_{1}}}}{D_{\omega_{1}}(\zeta e^{D_{\omega_{1}}}+1)}, \hat{P}_{{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}}=D_{\omega_{1} }
    Differential equation \bigg(\omega_{1} D_{\omega_{1}} +r\omega_{2} D_{\omega_{1}}^{r}+z e^ {D_{\omega_{1}}} D_{\omega_{1}} +\frac{\sigma (\zeta e^{D_{\omega_{1}}}+1)-\sigma D_{\omega_{1}}\zeta e^{D_{\omega_{1}}}}{(\zeta e^{D_{\omega_{1}}}+1)}-\tau\bigg) {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)=0
    Summation {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)=\frac{1}{2}\sum\nolimits_{\kappa=0}^ {\tau } {\tau \choose \kappa} \mathcal{E}_{\kappa}\bigg({}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}+1, \omega_{2}, z)+{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\bigg)
    Formulae {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\upsilon, \omega_{2}, z)=\sum\limits_{ \kappa=0}^{\tau}\binom{\tau} {\kappa } (\upsilon-\omega_{1}+\theta)^{\tau-\kappa } {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\kappa, \zeta}^{(\sigma, r)}(\omega_{1}-\theta, \omega_{2}, z)
    {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma+\beta, r)}(\omega_{1}+x, \omega_{2}+y, z+u)=\sum\limits_{ \kappa=0}^{\tau}\binom{\tau} {\kappa } {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\kappa, \zeta}^{(\beta, r)}(x, y, u)
    {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau-\varrho, \zeta}^{(\sigma-\varrho, r)}(\omega_{1}, \omega_{2}, z)=\frac{(\tau-\varrho)! \varrho!}{\tau!}\sum\limits_{ \kappa=0}^{\tau}\binom{\tau} {\kappa } {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) S(\kappa, \varrho, \zeta)
    {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau+1, \zeta}^{(1, r)}(\omega_{1}+1, \omega_{2}, z)=\frac{2}{\zeta}\left\lbrace (\tau+1) {}_{\mathcal{H}}\mathcal{B}el^{(r)}_ {\tau }(\omega_{1}, \omega_{2}, z) -\frac{1}{2} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau+1, \zeta}^{(1, r)}(\omega_{1}, \omega_{2}, z) \right\rbrace
    Differential and \frac{\partial^{\nu}}{\partial \omega_{1}^ {\nu}}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\right\rbrace = \begin{cases} \frac {\tau! {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau-\nu, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)}{(\tau-\nu)!}, & \tau\geq \nu; \\ 0, & 0\leq \tau < \nu. \end{cases}
    \frac{\partial^{\nu}}{\partial \omega_{2}^ {\nu}}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\right\rbrace = \frac {\tau! {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau-r\nu, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)}{(\tau-r\nu)!}
    \frac{\partial}{\partial z}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\right\rbrace =\sum\nolimits_{\kappa=0}^ {\tau } {\tau \choose \kappa} \left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) - {}_{{\mathcal{H}}}\mathcal{G}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}) \mathcal{B}el_{\kappa}(z)\right\rbrace
    \frac{\partial}{\partial z}\left\lbrace {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z)\right\rbrace = {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau-\kappa, \zeta}^{(\sigma, r)}(\omega_{1} +1, \omega_{2}, z) -{}_\mathcal{G}\mathcal{B}el_ {\tau }(\omega_{1}, \omega_{2}, z)
    Integral Formulae \int_{u}^{u+\gamma}{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z) \mathrm{d}\omega_{1}=\frac{1} {\tau +1}\bigg[{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau+1, \zeta}^{(\sigma, r)}(u+\gamma, \omega_{2}, z)-{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau+1, \zeta}^{(\sigma, r)}(u, \omega_{2}, z)\bigg]
    \int_{u}^{u+\gamma} {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, z;\delta, a, b) \mathrm{d}z
    = \frac{1}{\tau+1} \sum\nolimits_{k=0}^{\tau} \binom{\tau+1}{k} \mathfrak{B}_k \left[ {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau+1-k, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, u+\gamma; \delta, a, b) - {}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{\tau+1-k, \zeta}^{(\sigma, r)}(\omega_{1}, \omega_{2}, u;\delta, a, b) \right]

     | Show Table
    DownLoad: CSV

    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:

    \begin{align} &{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{0,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = 0,\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{1,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = \frac{2}{\zeta +1},\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{2,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = -\frac{4\zeta }{(\zeta +1)^2}+\frac{4\zeta \omega_{1}}{(\zeta +1)^2}+\frac{4 \omega_{1}}{(\zeta +1)^2}+\frac{4 z}{(\zeta +1)^2}+\frac{4\zeta z}{(\zeta +1)^2},\\&{}_{{\mathcal{H}}\mathcal{B}el}\mathcal{G}_{3,\zeta}^{(1,3)}(\omega_{1} , \omega_{2} ,z) = \frac{6\zeta ^2}{(\zeta +1)^3}-\frac{6\zeta }{(\zeta +1)^3}+\frac{6 \omega_{1}^2}{\zeta +1}-\frac{12\zeta \omega_{1}}{(\zeta +1)^2}+\frac{12 \omega_{1} z}{\zeta +1}+\frac{6 z^2}{\zeta +1}+\frac{6 z}{\zeta +1}\\& \quad -\frac{12\zeta z}{(\zeta +1)^2}, \end{align}
    \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] L. Accardi, Non-commutative Markov chains, Proc. Int. Sch. Math. Phys., 1974,268–295.
    [2] L. Accardi, A. Frigerio, Markovian cocycles, Math. Proc. R. Ir. Acad., 83 (1983), 251–263.
    [3] L. Accardi, F. Mukhamedov, A. Souissi, Construction of a new class of quantum Markov fields, Adv. Oper. Theory, 1 (2016), 206–218. https://doi.org/10.22034/aot.1610.1031 doi: 10.22034/aot.1610.1031
    [4] L. Accardi, F. Mukhamedov, M. Saburov, On quantum Markov chains on Cayley tree I: Uniqueness of the associated chain with XY-model on the Cayley tree of order two, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 14 (2011), 443–463. https://doi.org/10.1142/S021902571100447X doi: 10.1142/S021902571100447X
    [5] L. Accardi, F. Mukhamedov, M. Saburov, On quantum Markov chains on Cayley tree II: phase transitions for the associated chain with XY-model on the Cayley tree of order three, Ann. Henri Poincaré, 12 (2011), 1109–1144. https://doi.org/10.1007/s00023-011-0107-2 doi: 10.1007/s00023-011-0107-2
    [6] L. Accardi, A. Souissi, E. G. Soueidy, Quantum Markov chains: A unification approach, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 23 (2020), 2050016. https://doi.org/10.1142/S0219025720500162 doi: 10.1142/S0219025720500162
    [7] L. Accardi, Y. G. Lu, A. Souissi, A Markov-Dobrushin inequality for quantum channels, Open Syst. Inf. Dyn., 28 (2021), 2150018. https://doi.org/10.1142/S1230161221500189 doi: 10.1142/S1230161221500189
    [8] L. Accardi, G. S. Watson, Quantum random walks, In: Lecture notes in mathematics, Heidelberg: Springer, 1989. https://doi.org/10.1007/BFb0083545
    [9] S. Attal, F. Petruccione, C. Sabot, I. Sinayskiy, Open quantum random walks, J. Stat. Phys., 147 (2012), 832–852. https://doi.org/10.1007/s10955-012-0491-0
    [10] O. Bratteli, D. W. Robinson, Operator algebras and quantum statistical mechanics, Bull. Amer. Math. Soc., 7 (1982), 425.
    [11] A. Barhoumi, A. Souissi, Recurrence of a class of quantum Markov chains on trees, Chaos Solitons Fract., 164 (2022), 112644. https://doi.org/10.1016/j.chaos.2022.112644 doi: 10.1016/j.chaos.2022.112644
    [12] A. Dhahri, F. Mukhamedov, Open quantum random walks, quantum Markov chains and recurrence, Rev. Math. Phys., 31 (2019), 1950020. https://doi.org/10.1142/S0129055X1950020X doi: 10.1142/S0129055X1950020X
    [13] B. D. McKay, A. Piperno, Practical graph isomorphism, II, J. Symb. Comput., 60 (2014), 94–112. https://doi.org/10.1016/j.jsc.2013.09.003
    [14] M. Fannes, B. Nachtergaele, R. F. Werner, Finitely correlated states on quantum spin chains, Commun. Math. Phys., 144 (1992), 443–490. https://doi.org/10.1007/BF02099178 doi: 10.1007/BF02099178
    [15] M. Fannes, B. Nachtergaele, R. F. Werner, Ground states of VBS models on Cayley trees, J. Stat. Phys., 66 (1992), 939–973. https://doi.org/10.1007/BF01055710 doi: 10.1007/BF01055710
    [16] Y. Feng, N. K. Yu, M. S. Ying, Model checking quantum Markov chains, J. Comput. Sys. Sci., 79 (2013), 1181–1198. https://doi.org/10.1016/j.jcss.2013.04.002 doi: 10.1016/j.jcss.2013.04.002
    [17] D. Kastler, D. W. Robinson, Invariant states in statistical mechanics, Commun. Math. Phys., 3 (1966), 151–180. https://doi.org/10.1007/BF01645409 doi: 10.1007/BF01645409
    [18] C. K. Ko, H. J. Yoo, Quantum Markov chains associated with unitary quantum walks, J. Stoch. Anal., 1 (2020), 4. https://doi.org/10.31390/josa.1.4.04 doi: 10.31390/josa.1.4.04
    [19] F. Mukhamedov, S. El Gheteb, Uniqueness of quantum Markov chain associated with XY -Ising model on the Cayley tree of order two, Open Syst. Inf. Dyn., 24 (2017), 175010. https://doi.org/10.1142/S123016121750010X doi: 10.1142/S123016121750010X
    [20] F. Mukhamedov, S. El Gheteb, Clustering property of quantum Markov chain associated to XY-model with competing Ising interactions on the Cayley tree of order two, Math. Phys. Anal. Geom., 22 (2019), 10. https://doi.org/10.1007/s11040-019-9308-6 doi: 10.1007/s11040-019-9308-6
    [21] F. Mukhamedov, S. El Gheteb, Factors generated by XY-model with competing Ising interactions on the Cayley tree, Ann. Henri Poincaré, 21 (2020), 241–253. https://doi.org/10.1007/s00023-019-00853-9 doi: 10.1007/s00023-019-00853-9
    [22] F. Mukhamedov, A. Barhoumi, A. Souissi, Phase transitions for quantum Markov chains associated with Ising type models on a Cayley tree, J. Stat. Phys., 163 (2016), 544–567. https://doi.org/10.1007/s10955-016-1495-y doi: 10.1007/s10955-016-1495-y
    [23] F. Mukhamedov, A. Barhoumi, A. Souissi, On an algebraic property of the disordered phase of the Ising model with competing interactions on a Cayley tree, Math. Phys. Anal. Geom., 19 (2016), 21. https://doi.org/10.1007/s11040-016-9225-x doi: 10.1007/s11040-016-9225-x
    [24] F. Mukhamedov, A. Barhoumi, A. Souissi, S. El Gheteb, A quantum Markov chain approach to phase transitions for quantum Ising model with competing XY-interactions on a Cayley tree, J. Math. Phys., 61 (2020), 093505. https://doi.org/10.1063/5.0004889 doi: 10.1063/5.0004889
    [25] F. Mukhamedov, A. Souissi, Types of factors generated by quantum Markov states of Ising model with competing interactions on the Cayley tree, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 23 (2020), 2050019. https://doi.org/10.1142/S0219025720500198 doi: 10.1142/S0219025720500198
    [26] F. Mukhamedov, A. Souissi, Quantum Markov states on Cayley trees, J. Math. Anal. Appl., 473 (2019), 313–333. https://doi.org/10.1016/j.jmaa.2018.12.050 doi: 10.1016/j.jmaa.2018.12.050
    [27] F. Mukhamedov, A. Souissi, Diagonalizability of quantum Markov states on trees, J. Stat. Phys., 182 (2021), 9. https://doi.org/10.1007/s10955-020-02674-1 doi: 10.1007/s10955-020-02674-1
    [28] F. Mukhamedov, A. Souissi, Refinement of quantum Markov states on trees, J. Stat. Mech. Theory Exp., 2021 (2021), 083103. https://doi.org/10.1088/1742-5468/ac150b doi: 10.1088/1742-5468/ac150b
    [29] F. Mukhamedov, A. Souissi, Entropy for quantum Markov states on Cayley trees, J. Stat. Mech. Theory Exp., 2022 (2022), 093101. https://doi.org/10.1088/1742-5468/ac8740 doi: 10.1088/1742-5468/ac8740
    [30] F. Mukhamedov, A. Souissi, T. Hamdi, Quantum Markov chains on comb graphs: Ising model, Proc. Steklov Inst. Math., 313 (2021), 178–192. https://doi.org/10.1134/S0081543821020176 doi: 10.1134/S0081543821020176
    [31] F. Mukhamedov, A. Souissi, T. Hamdi, Open quantum random walks and quantum Markov chains on trees I: Phase transitions, Open Syst. Inf. Dyn., 29 (2022), 2250003. https://doi.org/10.1142/S1230161222500032 doi: 10.1142/S1230161222500032
    [32] F. Mukhamedov, A. Souissi, T. Hamdi, A. Andolsi, Open quantum random walks and quantum Markov Chains on trees II: The recurrence, Quantum Inf. Process., 22 (2023), 232. https://doi.org/10.1007/s11128-023-03980-9 doi: 10.1007/s11128-023-03980-9
    [33] N. Masuda, M. A. Porter, R. Lambiotte, Random walks and diffusion on networks, Phys. Rep., 716 (2017), 1–58. https://doi.org/10.1016/j.physrep.2017.07.007 doi: 10.1016/j.physrep.2017.07.007
    [34] R. Orus, A practical introduction of tensor networks: Matrix product states and projected entangled pair states, Ann Phys., 349 (2014), 117–158. https://doi.org/10.1016/j.aop.2014.06.013 doi: 10.1016/j.aop.2014.06.013
    [35] D. Ruelle, Statistical mechanics: Rigorous results, 1969.
    [36] A. Souissi, A class of quantum Markov fields on tree-like graphs: Ising-type model on a Husimi tree, Open Syst. Inf. Dyn., 28 (2021), 2150004. https://doi.org/10.1142/S1230161221500049 doi: 10.1142/S1230161221500049
    [37] A. Souissi, On stopping rules for tree-indexed quantum Markov chains, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2023. https://doi.org/10.1142/S0219025722500308
    [38] A. Souissi, F. Mukhamedov, A. Barhoumi, Tree-homogeneous quantum Markov chains, Int. J. Theor. Phys., 62 (2023), 19. https://doi.org/10.1007/s10773-023-05276-1 doi: 10.1007/s10773-023-05276-1
    [39] A. Souissi, E. G. Soueidy, M. Rhaima, Clustering property for quantum Markov chains on the comb graph, AIMS Mathematics, 8 (2023), 7865–7880. https://doi.org/10.3934/math.2023396 doi: 10.3934/math.2023396
    [40] A. Souissi, El G. Soueidy, A. Barhoumi, On a \psi-mixing property for entangled Markov chains, Phys. A, 613 (2023), 128533, https://doi.org/10.1016/j.physa.2023.128533 doi: 10.1016/j.physa.2023.128533
    [41] S. M. Van Dongen, Graph clustering by flow simulation, 2000.
    [42] S. Van Dongen, Graph clustering via a discrete uncoupling process, SIAM J. Matrix Anal. Appl., 30 (2008), 121–141. https://doi.org/10.1137/040608635 doi: 10.1137/040608635
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1727) PDF downloads(65) Cited by(0)

Figures and Tables

Figures(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog