EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6668 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Comprehensive Study of Generalized Bivariate q-Laguerre Polynomials: Structural Properties and Applications Haitham Qawaqneh 1,∗, Waseem Ahmad Khan 2, Hassen Aydi3,4, Ugur Duran5, Cheon Seoung Ryoo6 1 Al-Zaytoonah University of Jordan, Amman 11733, Jordan 2 Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia 3 Institute Supérieur d’Informatique et des Techniques de Communication, Université de Sousse, H. Sousse 4000, Tunisia 4 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa. 5 Department of Basic Sciences of Engineering, Iskenderun Technical University, Hatay 31200, Turkey. 6 Department of Mathematics, Hannam University, Daejeon 34430, South Korea. Abstract. In this paper, utilizing zeroth-order q-Bessel Tricomi functions, we introduce the gen- eralized bivariate q-Laguerre polynomials. Then, we establish the generalized bivariate q-Laguerre polynomials from the context of quasi-monomiality. We examine some of their properties, such as q-multiplicative operator property, q-derivative operator property and two q-integro-differential equations. Additionally, we derive operational representations and three q-partial differential equa- tions for the generalized bivariate q-Laguerre polynomials. Moreover, we draw the zeros of the new polynomials, forming 2D and 3D structures, and provide a table including approximate zeros of the generalized bivariate q-Laguerre polynomials. 2020 Mathematics Subject Classifications: 05A30; 11B83; 11B68, 33C45 Key Words and Phrases: Quantum calculus, q-Laguerre polynomials, generalized 2V q-Laguerre polynomials, Quasi monomiality, Extension of monomiality priciple, q-Dilatation operator, Partial differential equations, Differential equations ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6668 Email addresses: h.alqawaqneh@zuj.edu.jo (H. Qawaqneh), wkhan1@pmu.edu.sa (W. A. Khan), hassen.aydi@isima.rnu.tn (H. Aydi), ugur.duran@iste.edu.tr (U. Duran), ryoocs@hnu.kr (C. S. Ryoo) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 2 of 23 1. Introduction A set of orthogonal polynomials known as Laguerre polynomials is essential to many fields of applied mathematics and mathematical physics. The Laguerre polynomials stand out due to their applications in harmonic oscillator theory, coding theory, and quantum group theory [1]. These polynomials are instrumental in formulating covariant oscillator algebra [2, 3]. For additional information regarding the applications of Laguerre polyno- mials, refer to [4, 5]. Dattoli and Torre [6, 7] demonstrated that the hypothesis of bivari- ate Laguerre polynomials may be applied to ordinary Laguerre polynomials within the framework of quasi-monomials. Bivariate Laguerre polynomials are of great mathematical importance. Laguerre polynomials are used in overcoming radiation physics problems, including quantum beam lifetime in a storage ring and electromagnetic wave propagation [8–11]. They also appear as natural solutions to partial differential equations, such as the heat diffusion equation. To see more detail for the theory of two-variable Laguerre polynomials, one can look at the references [5, 12–16]. Dattoli and Torre [12] introduced the generalized bivariate Laguerre type polynomials, denoted by (2VgLtP) [m]Lω(ξ, η), are considered as follows: exp(ηψm)C0(ξψ) = ∞∑ ω=0 [m]Lω(ξ, η) ψω ω! , (1) where the function C0(ξ) means the 0th order Bessel Tricomi function: [m]Lω(ξ, η) = ω! [ ω m ]∑ θ=0 ηθ(−ξ)ω−mθ θ!((ω −mθ)!)2 . (2) The origins of quantum calculus can be traced back to the 18th century. This mathe- matical framework, commonly denoted as q-calculus, constitutes a substantial expansion of traditional calculus. Its significance lies in its intricate relationships with various scientific domains, including quantum mechanics, mathematical physics, mathematical analysis, combinatorics, and the theory of orthogonal polynomials. Subsequent to its inception, the q-calculus paradigm underwent extensive investigation and refinement by a multitude of scholars. This mathematical construct enables the scrutiny and examination of q-analogs, which serve as counterparts to fundamental and special functions under q-transformations. Contemporary academic pursuits have predominantly centered on exploring particular families of q-special polynomials. These investigations employ generating functions and their corresponding functional equations to illuminate and broaden the properties and applications of these polynomials across diverse scientific disciplines. The ongoing nature of this research emphasizes the profound influence and enduring relevance of q-calculus in the evolution of modern mathematical theory and its multidisciplinary applications. In a recent study, Fadel and colleagues [17] introduced and examined bivariate q-Hermite polynomials. Furthermore, certain q-special functions were analyzed and studied in the context of q-algebra representations [8, 18, 19]. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 3 of 23 In this study, we assume that 0 < q < 1 and we adhere to the terminology and notions given in [17, 20]. The q-shift factorial (a; q)ω is defined as [21] (a; q)ω = ω−1∏ ϕ=1 (1− qϕa) for ω ∈ N (3) with (a; q)0 := 1. Let ω ∈ C with ω ≥ 1. The q-numbers and q-factorial are provided as follows [ω]q = 1− qw 1− q , . (4) and [ω]q! = ω∏ ϕ=1 [ϕ]q for ω > 0, (5) with [0]q! = 1. The q-extension of (ξ ± a)ω is provided as follows (ξ ± a)ωq = ω∑ ϕ=0 ( ω ϕ ) q ξϕ(±a)ω−ϕq( ω−ϕ 2 ). (6) The two types of q-exponential functions are considered by [22–25] eq(ξ) = ∞∑ ω=0 ξω [ω]q! , (7) and Eq(ξ) = ∞∑ ω=0 q( ω 2)ξω [ω]q! . (8) The following relations are valid: eq(ξ)Eq(η) = ∞∑ ω=0 (ξ ⊕ η)ωq [ω]q! , (9) and eq(ξ)Eq(−ξ) = 1. (10) For ξ ̸= 0, the q-derivative operator is provided as follows D̂q,ξf(ξ) = f(qξ)− f(ξ) qξ − ξ , (11) which satisfies the following operator rules D̂q,ξξ ω = [ω]qξ ω−1, (12) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 4 of 23 D̂q,ξeq(αξ) = αeq(αξ), α ∈ C, (13) and D̂ϕ q,ξeq(αξ) = αϕeq(αξ), ϕ ∈ N, α ∈ C. (14) Here D̂ϕ q,ξ denotes the ϕ th order q-derivative operator with respect to ξ. The product rule is provided as follows [25] D̂q,ξ(f(ξ)g(ξ)) = f(ξ)D̂q,ξg(ξ) + g(qξ)D̂q,ξf(ξ). (15) In recent years, q-Gould-Hopper polynomials qGHPH (m) ω,q (ξ, η) are considered by Khan et al. [23] as follows eq(ξψ)eq(ηψ m) = ∞∑ ω=0 H(m) ω,q (ξ, η) ψω [ω]q! , (16) and also H(m) ω,q (ξ, η) = [ω]q! [ ω m ]∑ ϕ=0 ηϕξω−mϕ [ϕ]q![ω −mϕ]q! . (17) The operational identity of q-Gould-Hopper polynomials H (m) ω,q (ξ, η) is given by (see [23]): H(m) ω,q (ξ, η) = eq ( ηDm q,ξ ) {ξω}. (18) Cao et al. [26] introduced the q-Laguerre polynomials Lω,q(ξ, η) are provided as follows C0,q(ξψ)eq(ηψ) = ∞∑ ω=0 Lω,q(ξ, η) ψω [ω]q! , (19) the series definition, we have Lω,q(ξ, η) = [ω]q! ω∑ ϕ=0 (−1)ϕξϕηω−ϕ ([ϕ]q!)2[ω − ϕ]q! . (20) where the 0th order q-Bessel Tricomi functions C0,q(ξ) are defined by [26]: C0,q(ξψ) = eq(−D−1 q,ξψ){1} (21) and also C0,q(ξ) = ∞∑ ϕ=0 (−1)ϕξϕ ([ϕ]q!)2 , (22) which converges absolutely for all ξ. In view of the following notions [26]: D̂−1 q,ξf(ξ) := ∫ ξ 0 f(ξ)dqξ, (23) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 5 of 23 Particularly, for r ∈ N and choosing D̂−1 q,ξ{1} = ξ, it is seen that( D̂−1 q,ξ )r {1} = ξr [r]q! . (24) From (5) and (21), equation (19) can be written as eq(D̂ −1 q,ξψ m)eq(ηψ){1} = ∞∑ ω=0 [m]Lω,q(ξ, η) ψω [ω]q! , (25) For u being a complex variable, the q-dilation operator Tu is introduced as follows [27]: Tϕuf(u) = f(qϕu), ϕ ∈ R, (26) satisfies the property T−1 u T1 uf(u) = f(u). (27) The following operator rule is valid [26]: D̂q,ψeq(uψ m) = uψm−1T(u;m)eq(uψ m). (28) where T(u;m) = 1− qmTmu 1− qTu = 1 + qTu + · · ·+ qm−1Tm−1 u . (29) Since, in view of equations (20) and (28), we have T r D̂−1 q,ξ C0,q(ξ) = T rξ C0,q(ξ). (30) Therefore, the following operator rule is valid [17]: D̂q,ψC0,q(−ξψm) = D̂−1 q,ξψ m−1T(ξ;m)C0,q(−ξψm), (31) where T(ξ;m) = 1− qmTmξ 1− qTξ = 1 + qTξ + · · ·+ qm−1Tm−1 ξ . (32) Also they proved the q-derivative and q-multiplicative operators of bivariate q-Laguerre polynomials as follows [26] D̂q,ξξD̂q,ξC0,q(αξ) = ∂q ∂qD̂ −1 q,ξ C0,q(αξ) = −αC0,q(αξ). (33) Taking into account equation (15), we have D̂q,ξξD̂q,ξf(ξ) = (qD̂q,ξ + ξD̂2 q,ξ)f(ξ). (34) The monomiality concept is a valuable technique for understanding certain special polynomials and functions in conjunction with relations and properties. This concept has H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 6 of 23 the potential to generate novel sets of the family of q-special polynomials and show the quasi-monomial nature of some previously established q-extension of special polynomials. For further information, one may look at the references [13, 14, 26, 27]. Implementing the monomiality principle to quantum calculus establishes a basis for com- prehending q-special polynomials as specific solutions to expanded versions of q-integro differential equations and q-partial differential equations. Cao and colleagues [26] have recently expanded the idea of the monomiality principle to the field of quantum calculus. Let pω,q(ξ) be a q-polynomial set for ω ∈ N and ξ ∈ C. The q-multiplicative operator M̂q and q-derivative operator P̂q are provided as follows [26] M̂q{pω,q(ξ)} = pω+1,q(ξ), (35) and P̂q{pω,q(ξ)} = ωpω−1,q(ξ), (36) which fulfill the following relation: [M̂q, P̂q] = P̂qM̂q − M̂qP̂q. (37) The characteristics of the polynomials pω,q(ξ) can be deduced from the features of the M̂q and P̂q operators. If the angles M̂q and P̂q have a q-differential realization, then the polynomials pω,q(ξ) must satisfy the q-differential equations: M̂qP̂q{pω,q(ξ)} = [ω]qpω,q(ξ), (38) and P̂qM̂q{pω,q(ξ)} = [ω + 1]qpω,q(ξ). (39) In view of (35) and (36), we have [M̂q, P̂q] = [ω + 1]q − [ω]q. (40) From (35), we have M̂q r {pn,q} = pn+r,q(x). (41) In particular, we have pn,q(x) = M̂q n {p0,q} = M̂q n {1}, (42) where p0,q(ξ) = 1 is the q-sequel of polynomial pω,q(ξ) provided by eq(M̂qψ){1} = ∞∑ ω=0 pω,q(ξ) ψω [ω]q! . (43) Using the idea of the q-monomiality principle, in the present paper, we describe and investigate the unique characteristics of generalized bivariate mth order q-Laguerre poly- nomials, motivated by the possible applications of q-special functions in mathematics and science. Furthermore, we give applications of this recently certain members of generalized mth order q-Laguerre polynomials family to show their graphs. We conclude this research article by computing the zeros of certain members of generalized mth order q-Laguerre polynomials family numerically as well as presenting them graphically. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 7 of 23 2. Generalized Bivariate q-Laguerre Polynomials his part defines the generalized bivariate q-Laguerre polynomials [m]Lω,q(ξ, η) utilizing the function C0,q(ξψ) in (22) and derives explicit formulas, operational identities, q-quasi- monomiality characteristic and q-integro-differential equations for these polynomials. We perform to define q-extension of the polynomials in (1) and we introduce the generalized bivariate q-Laguerre polynomials [m]Lω,q(ξ, η) as follows: C0,q(ξψ)eq(ηψ m) = ∞∑ ω=0 [m]Lω,q(ξ, η) ψω [ω]q! . (44) We readily derive the following explicit formula for the new polynomials in (44): [m]Lω,q(ξ, η) = [ω]q! [ ω m ]∑ θ=0 (−1)ωξω−mθηθ ([ω −mθ]q!)2[θ]q! . (45) We then observe from (24), (25), and (44) that eq(−D−1 q,ξψ)eq(ηψ m){1} = ∞∑ ω=0 [m]Lω,q(ξ, η) ψω [ω]q! . (46) Also we obtain using (18), (45) and (46) that [m]Lω,q(ξ, η) = H(m) ω,q (D −1 q,ξ , η){1}. (47) Now, we provide the following theorem, including q-multiplicative operator and q-derivative operator of the new polynomials [m]Lω,q(ξ, η). Theorem 1. The polynomials [m]Lω,q(ξ, η) are quasi-monomials under the following q- multiplicative operator and q-derivative operator: M̂G2V qLP = ηT(η;m) ( − ∂q ∂qD̂ −1 q,ξ )m−1 − D̂−1 q,ξTqm,η, (48) or, equally M̂G2V qLP = ηT(η;m) ( − ∂q ∂qD̂ −1 q,ξ )m−1 Tq,ξ − D̂−1 q,ξ , (49) and P̂G2V qLP = −D̂q,ξξD̂q,ξ = − ∂q ∂qD̂ −1 q,ξ , (50) respectively. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 8 of 23 Proof. Utilizing (15) and applying q-derivative operator to the both sides of (44) with respect to t, we derive that ∞∑ ω=1 [m]Lω,q(ξ, η)D̂q,ψ ψω [ω]q! = eq(−D̂−1 q,ξψ)D̂q,ψeq(ηψ m) + eq(ηq mψm)D̂q,ψeq(−D̂−1 q,ξψ). Thus, we see from (23), (29) and (44) that ∞∑ ω=1 [m]Lω,q(ξ, η) ψω−1 [ω − 1]q! = ηT(η;m)ψ m−1eq(−D̂−1 q,ξψ)eq(ηψ m)−D̂−1 q,ξTqm,ηeq(−D̂ −1 q,ξψ)eq(ηψ m). (51) Using equations (33) and (44) in the equation (51), we get ∞∑ ω=1 [m]Lω,q(ξ, η) ψω−1 [ω − 1]q! = ∞∑ ω=0 ( ηT(η;m)D̂ m−1 q,η − D̂−1 q,ξTqm,η ) [m]Lω,q(ξ, η) ψω [ω]q! , (52) which means [m]Ln+1,q(x, y) = yT(y;m) ( − ∂q ∂qD̂ −1 q,x )m−1 − D̂−1 q,xTqm,y  [m]Ln,q(x, y), (53) which is the first claimed result (48). In the same way, utilizing (33) and (52) for fq(ψ) = eq(ηψ m) and gq(ψ) = eq(−D̂−1 q,ξψ), and applying q-derivative operator to the both sides of (44) with respect to ψ, we acquire ∞∑ ω=1 [m]Lω,q(ξ, η)D̂q,ψ ψω [ω]q! = ∞∑ ω=0 η(− ∂q ∂qD̂ −1 q,ξ )m−1 T(η;m)Tq,ξ − D̂−1 q,ξ  [m]Lω,q(ξ, η) ψω [ω]q! , (54) which yields [m]Lω+1,q(ξ, η) = η(− ∂q ∂qD̂ −1 q,ξ )m−1 T(η;m)Tq,ξ − D̂−1 q,ξ  [m]Lω,q(ξ, η), (55) which is the second asserted result (49). If we apply the operator D̂q,ξξD̂q,ξ to the both sides of (44) and utilizing (33), we then obtain D̂q,ξξC0,q(ξψ)eq(ηψ m) = − ∂q ∂qD −1 q,ξ eq(−D̂−1 q,ξ t)eq(ηψ) = ∞∑ ω=0 D̂q,ξξD̂q,ξ [m]Lω,q(ξ, η) ψω [ω]q! . (56) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 9 of 23 Using equation (33), we get ψC0,q(ξψ)eq(ηψ m) = ∞∑ ω=0 − ∂q ∂qD −1 q,ξ [m]Lω,q(ξ, η) ψω [ω]q! = − ∞∑ ω=0 D̂q,ξξD̂q,ξ [m]Lω,q(ξ, η) ψω [ω]q! . (57) By (57), we observe that −D̂q,ξξD̂q,ξ [m]Lω,q(ξ, η) = − ∂q ∂qD −1 q,ξ [m]Lω,q(ξ, η) = [ω]q [m]Lω−1,q(ξ, η), (58) which means the third asserted operator formula (50). Remark 1. In view of (33) and Theorem 2.1, the multiplicative operators can also be represented as M̂G2V qLP = ηT(η;m) ( −D̂q,ξξD̂q,ξ )m−1 − D̂−1 q,ξTqm,η, (59) or, equivalently M̂G2V qLP = ηT(η;m) ( −D̂q,ξξD̂q,ξ )m−1 Tq,ξ − D̂−1 q,ξ . (60) Here, we provide the following theorem. Theorem 2. The following q-partial differential equations for [m]Lω,q(ξ, η) hold true:( −D̂q,ξξD̂q,ξ )m [m]Lω,q(ξ, η) = D̂q,η [m]Lω,q(ξ, η), (61) or, equivalently ( − ( qD̂q,ξ + ξD̂2 q,ξ ))m [m]Lω,q(ξ, η) = D̂q,η [m]Lω,q(ξ, η), (62)( − ∂q ∂qD −1 q,ξ )m [m]Lω,q(ξ, η) = D̂q,η [m]Lω,q(ξ, η). (63) Proof. From equation (33) and (44), we have( −D̂q,ξξD̂q,ξ ) C0,q(ξψ)eq(ηψ m) = ψmC0,q(ξψ)eq(ηψ m), (64) and D̂q,ηC0,q(ξψ)eq(ηψ m) = ψmC0,q(ξψ)eq(ηψ m). (65) We readily get the asserted equation (61) utilizing (44) and some series manipulation methods, and also we obtain the claimed equation (62) just by using (61) and (34), equa- tion (44) gives the assertion (62). We promptly acquire the argued equation (63) utilizing (44) and some series manipulation methods. We now provide the q-integro-differential equations for G2V qLP [m]Lω,q(ξ, η). H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 10 of 23 Theorem 3. We have q ∫ ξ 0 Tqm,ηD̂q,u[m]Lω,q(u, η)dqu+ ∫ ξ 0 uTqm,ηD̂ 2 q,u[m]Lω,q(u, η)dqu = ( [ω]q − η(−D̂q,ξξD̂q,ξ) mT(η;m) ) [m]Lω,q(ξ, η), (66) and∫ ξ 0 (D̂q,uuD̂q,u)[m]Lω,q(u, η)dqu = ( [ω]q − ηT(η;m)(−D̂q,ξξD̂q,ξ) m ) [m]Lω,q(ξ, η). (67) Proof. Taking into account (38) and (50) with equations (59) and (60), we get the assertions (66) and (67), respectively. Remark 2. In the special case m = 2 in (44) and (45), we get C0,q(ξψ)eq(ηψ 2) = ∞∑ ω=0 [2]Lω,q(ξ, η) ψω [ω]q! , (68) and [2]Lω,q(ξ, η) = [ω]q! [ω 2 ]∑ θ=0 (−1)ωξω−2θηθ ([ω − 2θ]q!)2[η]q! . (69) Thus, we acquire from (21), (25), and (44) that eq(−D−1 q,ξψ)eq(ηψ 2){1} = ∞∑ ω=0 [2]Lω,q(ξ, η) ψω [ω]q! . (70) We derive from (7) and (46) that [2]Lω,q(ξ, η) = H(2) ω,q(D −1 q,ξ , η){1}. (71) We acquire the following corollary in the special case m = 2 in Theorem 1. Corollary 1. The following operator formulas are valid: M̂G2V qLP = ηT(η;2) ( − ∂q ∂qD̂ −1 q,ξ ) − D̂−1 q,ξTq2,η, (72) or, equally M̂G2V qLP = ηT(η;2) ( − ∂q ∂qD̂ −1 q,ξ ) Tq,x − D̂−1 q,ξ . (73) We acquire the following corollary in the special case m = 2 in Theorem 2. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 11 of 23 Corollary 2. We have( −D̂q,ξξD̂q,ξ )2 [2]Lω,q(ξ, η) = D̂q,η [2]Lω,q(ξ, η), (74) or, equivalently ( − ( qD̂q,ξ + ξD̂2 q,ξ ))2 [2]Lω,q(ξ, η) = D̂q,η [2]Lω,q(ξ, η), (75) and ( − ∂q ∂qD −1 q,ξ )2 [2]Lω,q(ξ, η) = D̂q,η [2]Lω,q(ξ, η). (76) We acquire the following corollary in the special case m = 2 in Theorem 3. Corollary 3. We have q ∫ x 0 Tq2,yD̂q,u[2]Lω,q(u, η)dqu+ ∫ ξ 0 uTq2,ηD̂ 2 q,u[2]Lω,q(u, η)dqu = ( [ω]q − η(−D̂q,ξξD̂q,ξ) 2T(η;2) ) [2]Lω,q(ξ, η), (77) and ∫ ξ 0 (D̂q,uuD̂q,u)[2]Lω,q(u, η)dqu = ( [ω]q − ηT(η;2)(−D̂q,ξξD̂q,ξ) 2 ) [2]Lω,q(ξ, η). (78) 3. Distribution of Zeros and Graphical Representation This section demonstrates how numerical analysis can be employed to confirm theoret- ical predictions and uncover new and interesting patterns in the zeros of certain members of a recently introduced hybrid polynomial family. Specifically, this paper utilizes compu- tational methods to explore the “scattering” of the zeros of the generalized two-variable q-Laguerre polynomials, denoted as [m]Lω,q(ξ, η), within the complex plane a fascinating phenomenon to observe. From (44) and (45), we remember that C0,q(ξψ)eq(ηψ m) = ∞∑ ω=0 [m]Lω,q(ξ, η) ψω [ω]q! , (79) and [m]Lω,q(ξ, η) = [ω]q! [ ω m ]∑ θ=0 (−1)ωξω−mθηθ ([ω −mθ]q!)2[θ]q! . (80) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 12 of 23 The first few generalized bivariate q-Laguerre polynomials are listed as follows: [2]L1,q(ξ, η) = −ξ, [2]L2,q(ξ, η) = ξ2 + η[2]q!, [2]L3,q(ξ, η) = −ξ3 − ηξ[3]q!, [2]L4,q(ξ, η) = ξ4 + η2[4]q! [2]q! + ηξ2[4]q! [2]q! , [2]L5,q(ξ, η) = −ξ5 − η2ξ[5]q! [2]q! − ηξ3[5]q! [3]q! , [2]L6,q(ξ, η) = ξ6 + η2ξ2[6]q! [2]q!2 + η3[6]q! [3]q! + ηξ4[6]q! [4]q! , [2]L7,q(ξ, η) = −ξ7 − η3ξ[7]q! [3]q! − η2ξ3[7]q! [2]q![3]q! − ηξ5[7]q! [5]q! , [2]L8,q(ξ, η) = ξ8 + η3ξ2[8]q! [2]q![3]q! + η4[8]q! [4]q! + η2ξ4[8]q! [2]q![4]q! + η]ξ6[8]q! [6]q! , [2]L9,q(ξ, η) = −ξ9 − η3ξ3[9]q! [3]q!2 − η4ξ[9]q! [4]q! − η2ξ5[9]q! [2]q![5]q! − ηξ7[9]q! [7]q! , [2]L10,q(ξ, η) = −ξ10 + η4ξ2[10]q! [2]q![4]q! + η3ξ4[10]q! [3]q![4]q! + η5[10]q! [5]q! + η2ξ6[10]q! [2]q![6]q! + ηξ8[10]q! [8]q! . H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 13 of 23 We research the solutions of the equality [m]Lω,q(ξ, η) = 0, utilizing a computer pro- gramme. So, we draw these solutions for m = 2, ω = 50 and η = 5 by the following Figure 1: -5 0 5 -5 0 5 Re(ξ) Im(ξ) -5 0 5 -5 0 5 Re(ξ) Im(ξ) -5 0 5 -5 0 5 Re(ξ) Im(ξ) -5 0 5 -5 0 5 Re(ξ) Im(ξ) Figure 1: Zeros of [m]Lω,q(ξ, η) = 0 Especially, we take q = 1 10 (top-left), q = 3 10 (top-right), q = 7 10 (bottom-left) and q = 9 10 (bottom-right) in Figure 1. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 14 of 23 We provide, forming a 3D structure, the stacks of zeros for the equality [m]Lω,q(ξ, η) = 0 for m = 2, η = 5, and 1 ≤ ω ≤ 50 by the following Figure 2: Figure 2: Zeros of [m]Lω,q(ξ, η) = 0 Here, we take q = 1 10 (top-left), q = 3 10 (top-right), q = 7 10 (bottom-left) and q = 9 10 (bottom-right) in Figure 2. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 15 of 23 We give, forming a 2D structure, the stacks of zeros for the equality [m]Lω,q(ξ, η) = 0 for q = 9 10 , η = 5, and 1 ≤ ω ≤ 50 by the following Figure 3: -5 0 5 -10 -5 0 5 10 Re(ξ) Im(ξ) -5 0 5 -10 -5 0 5 10 Re(ξ) Im(ξ) -10 -5 0 5 10 -10 -5 0 5 10 Re(ξ) Im(ξ) -10 -5 0 5 10 -10 -5 0 5 10 Re(ξ) Im(ξ) Figure 3: Zeros of [m]Lω,q(ξ, η) = 0 Here, we take m = 3 (top-left), m = 4 (top-right), m = 5 (bottom-left) and m = 6 (bottom-right) in Figure 3. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 16 of 23 We show, forming a 3D structure, the stacks of zeros for the equality [m]Lω,q(ξ, η) = 0 for q = 9 10 , η = 5, and 1 ≤ ω ≤ 50 by the following Figure 4: Figure 4: Zeros of [m]Lω,q(ξ, η) = 0 Here, we take m = 3 (top-left), m = 4 (top-right), m = 5 (bottom-left) and m = 6 (bottom-right) in Figure 4. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 17 of 23 We present the stacks of real zeros for the equality [m]Lω,q(ξ, η) = 0 for q = 9 10 , η = 5, and 1 ≤ ω ≤ 50 by the following Figure 5: Figure 5: Real zros of [m]Lω,q(ξ, η) = 0 Here, we take m = 2 (top-left), m = 3 (top-right), m = 4 (bottom-left) and m = 5 (bottom-right) in Figure 5. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 18 of 23 We provide the plots of the real zeros for the equality [m]Lω,q(ξ, η) = 0 for m = η = 5, and 1 ≤ ω ≤ 50 by the following Figure 6: Figure 6: Real zros of [m]Lω,q(ξ, η) = 0 Here, we take q = 1 10 (top-left), q = 3 10 (top-right), q = 7 10 (bottom-left) and q = 9 10 (bottom-right) in Figure 6. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 19 of 23 Figure 7: Real zeros of [m]Lω,q(ξ, η) = 0 In Figure 7, we take m = 7, 1 ≤ ω ≤ 50, η = 3 and q = 9 10 . We plot, forming a 3D structure, the stacks of zeros for the equality [m]Lω,q(ξ, η) = 0 in Figure 7 (top-left). We plot, forming a 3D structure, y and x axes but no z axis in the three dimensions in Figure 7 (top-right). We plot, forming a 3D structure, z and y axes but no x axis in Figure 7 (bottom-left). We plot, forming a 3D structure, z and x axes but no y axis in Figure 7 ((bottom-right). H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 20 of 23 Now, we computed an approximate solution fulfilling the equality [m]Lω,q(ξ, η) = 0 for m = 2, η = 5, and q = 9 10 provided by the following Table 1. Table 1. Approximate solutions of [m]Lω,q(ξ, η) = 0 degree ω ξ 1 0 2 −3.0822i, 3.0822i 3 0, −5.0740i, 5.0740i 4 −2.3867i, 2.3867i, −6.3955i, 6.3955i 5 0, −4.2793i, 4.2793i, −7.2182i, 7.2182i 6 −2.0470i, 2.0470i, −5.8493i, 5.8493i, −7.5852i, 7.5852i 7 −0.5543− 7.3600i, −0.5543 + 7.3600i, 0, −3.8079i, 3.8079i, 0.5543− 7.3600i, 0.5543 + 7.3600i 8 −1.0763− 7.7067i, −1.0763 + 7.7067i, −1.8399i, 1.8399i, −5.3580i, 5.3580i, 1.0763− 7.7067i, 1.0763 + 7.7067i 9 −1.4703− 7.7823i, −1.4703 + 7.7823i, 0, −3.4948i, 3.4948i, −6.7398i, 6.7398i, 1.4703− 7.7823i, 1.4703 + 7.7823i 10 −1.9450− 7.8041i, −1.9450 + 7.8041i, −1.6989i, 1.6989i, −5.0096i, 5.0096i, −7.5676i, 7.5676i, 1.9450− 7.8041i, 1.9450 + 7.8041i 4. Conclusion In conclusion, this study has systematically investigated the unique characteristics and applications of generalized bivariate q-Laguerre polynomials. It has derived some of their properties, such as explicit formulas, operational identities, q-quasi-monomiality charac- teristics, and q-integro-differential equations for these polynomials. Moreover, we have provided the stacks of the zeros of the new polynomials, forming 2D and 3D structures, H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 21 of 23 and provided a table including approximate zeros of the generalized bivariate q-Laguerre polynomials. Motivated by the potential applications of q-special functions in various sci- entific and mathematical fields, we have utilized the q-analog of the monomiality principle to describe the mentioned polynomials. The distribution of non-coherent or coherent radi- ation areas in quantum optics, electromagnetic radiation problems, and wave propagation phenomena are some practical applications that inspired this work. Furthermore, we have presented the graphical representations and numerical computations of the zeros of certain members of the mth-order q-Laguerre polynomials family. The insights gained from these investigations provide a foundation for further exploration of q-special functions and their applications in more complex multidimensional systems. We anticipate that the results and methodologies discussed in this paper will stimulate future research in this promising area of mathematical analysis and its applications. Availability of data and materials Not applicable. Competing interests The authors declare no competing interests. Acknowledgements The authors acknowledge the financial support from Al-Zaytoonah University of Jor- dan, Amman 11733, Jordan. References [1] Waseem Ahmad Khan, Khidir Shaib Mohamed, Francesco Aldo Costabile, Shahid Ah- mad Wani, and Alawia Adam. A new generalization of m th-order laguerre-based appell polynomials associated with two-variable general polynomials. Mathematics, 13(13):2179, 2025. [2] Shahid Ahmad Wani, Mumtaz Riyasat, Ramı́rez William, and Waseem Ahmad Khan. Multivariate q-hermite-based appell polynomials: structural properties and applica- tions. Afrika Matematika, 36(2):97, 2025. [3] Naeem Ahmad and Waseem Ahmad Khan. A new generalization of q-laguerre-based appell polynomials and quasi-monomiality. Symmetry, 17(3):439, 2025. [4] Richard Askey. Limits of some q-laguerre polynomials. Journal of Approximation Theory, 46(3):213–216, 1986. [5] G Dattoli, Hari M Srivastava, and C Cesarano. On a new family of laguerre polyno- mials. 1999. [6] G Dattoli and A Torre. Operational methods and two variable laguerre polynomials. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur, 132:3–9, 1998. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 22 of 23 [7] G Dattoli and A Torre. Exponential operators, quasi-monomials and generalized polynomials. Radiation Physics and Chemistry, 57(1):21–26, 2000. [8] Da-Wei Niu. Generalized q-laguerre type polynomials and q-partial differential equa- tions. Filomat, 33(5):1403–1415, 2019. [9] D. Judeh and M. Abu Hammad. Applications of conformable fractional pareto prob- ability distribution. International Journal of Advances in Soft Computing and Its Applications, 14:116–124, 2022. [10] T. Kanan, M. Elbes, K. Abu Maria, and M. Alia. Exploring the potential of iot- based learning environments in education. , International Journal of Advances in Soft Computing and its Applications, 15:116–124, 2023. [11] H. Qawaqneh, M. S. Noorani, H. Aydi, A. Zraiqat, and A. H. Ansari. On fixed pointresults in partial b-metric spaces. Journal of Function Spaces, 8769190:9 pages, 2021. [12] Giuseppe Dattoli and Amalia Torre. Theory and applications of generalized bessel functions. Aracne Rome, 1996. [13] Mohra Zayed, Waseem Ahmad Khan, Cheon Seoung Ryoo, and Ugur Duran. An exploratory study on bivariate extended q-laguerre-based appell polynomials with some applications. AIMS Mathematics, 10(6):12841–12867, 2025. [14] Naeem Ahmad and Waseem Ahmad Khan. Insights into new generalization of q- legendre-based appell polynomials: Properties and quasi monomiality. Mathematics, 13(6):955, 2025. [15] H. Qawaqneh, M. S. Noorani, and H. Aydi. Some new characterizations and results for fuzzy contractions in fuzzy b-metric spaces and applications. AIMS Mathematics, 8:6682–6696, 2023. [16] H. Qawaqneh, H. A. Hammad, and H. Aydi. Exploring new geometric contraction mappings and their applications in fractional metric spaces. AIMS Mathematics, 9:521–541, 2024. [17] Mohammed Fadel, Maryam Salem Alatawi, and Waseem Ahmad Khan. Two-variable q-hermite-based appell polynomials and their applications. Mathematics, 12(9):1358, 2024. [18] M. Nazam, H. Aydi, M.S. Noorani, and H. Qawaqneh. Existence of fixed points of four maps for a new generalized f−contraction and an application. Journal of Function Spaces, 5980312:8 pages, 2019. [19] H. Qawaqneh, M. S. Noorani, H. Aydi, and W. Shatanawi. , on common fixed point results for new contractions with applications to graph and integral equations. Mathematics, 7:1082, 2019. [20] Jung Yoog Kang and Waseem A Khan. A new class of q-hermite-based apostol type frobenius genocchi polynomials. Communications of the Korean Mathematical Society, 35(3):759–771, 2020. [21] George Gasper and Mizan Rahman. Basic hypergeometric series, volume 96. Cam- bridge university press, 2004. [22] Maryam Salem Alatawi, Waseem Ahmad Khan, and Cheon Seoung Ryoo. Explicit properties of q-cosine and q-sine array-type polynomials containing symmetric struc- H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6668 23 of 23 tures. Symmetry, 14(8):1675, 2022. [23] Waseem Ahmad Khan, Khidir Shaib Mohamed, Francesco Aldo Costabile, Can Kızılateş, and Cheon Seoung Ryoo. Finding the q-appell convolution of certain poly- nomials within the context of quantum calculus. Mathematics, 13(13):2073, 2025. [24] Noor Alam, Waseem Ahmad Khan, Can Kızılateş, and Cheon Seoung Ryoo. Two- variable q-general-appell polynomials within the context of the monomiality principle. Mathematics, 13(5):765, 2025. [25] Waseem Ahmad Khan, Mofareh Alhazmi, and Tabinda Nahid. A novel family of q-mittag-leffler-based bessel and tricomi functions via umbral approach. Symmetry, 16(12):1580, 2024. [26] Jian Cao, Nusrat Raza, and Mohammed Fadel. Two-variable q-laguerre polynomi- als from the context of quasi-monomiality. Journal of Mathematical Analysis and Applications, 535(2):128126, 2024. [27] Giuseppe Dattoli and Amalia Torre. Symmetric q-bessel functions. Le Matematiche, 51(1):153–167, 1996.