EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6658 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bivariate Kind of Generalized Laguerre-Based Appell Polynomials with Applications to Special Polynomials Waseem Ahmad Khan1, Haitham Qawaqneh2, Hassen Aydi3,4,∗ 1 Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia 2 Department of Mathematics, Al-Zaytoonah University of Jordan, Amman 11733, Jordan. 3 Institut 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 Abstract. In this paper, we introduce a new generalization of Laguerre and Laguerre-based Ap- pell polynomials and investigate their fundamental properties. We derive a recurrence relation, multiplicative and derivative operators, and differential equation by verifying quasi-monomiality. Also, the series representation and determinant representation for this novel polynomial family are established. Furthermore, we define subpolynomials within this framework, namely general- ized Laguerre-Hermite Appell polynomials and establish their corresponding results. Addition- ally, Laguerre-Hermite-Bernoulli, Euler and Genocchi polynomials are obtained, and explore their structural and operational characteristics. The results obtained contribute to the broader study of special polynomials and their applications in mathematical physics and differential equations. 2020 Mathematics Subject Classifications: 33E20, 33C45, 33B10, 33E30, 11T23, 11B83, 11B68 Key Words and Phrases: Laguerre polynomials, Laguerre-based Appell polynomials, Monomi- ality principle, Explicit form, Operational connection, Determinant form 1. Introduction and preliminary results It is well established that special polynomials in two variables provide new analytical tools for solving a broad range of partial differential equations frequently encountered in physical problems. The introduction of the two-variable Laguerre polynomials, denoted as Ln(r1, r2) [1–10], is of significant interest due to their intrinsic mathematical properties and extensive applications in physics. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6658 Email addresses: wkhan1@pmu.edu.sa (W. A. Khan), h.alqawaqneh@zuj.edu.jo (H. Qawaqneh), hassen.aydi@isima.rnu.tn (H. Aydi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 2 of 22 The two-variable Laguerre polynomials (2VLP) Ln(r1, r2) are characterized by the following generating function [11]: er2tJ0(2 √ r1t) = ∞∑ n=0 Ln(r1, r2) tn n! . (1) where J0(r1t) represents the ordinary Bessel function of the first kind of order zero [12], defined as: r n 2 1 Jn(2 √ r1) = ∞∑ k=0 (−1)k ( √ r1) k k!(n+ k)! . (2) Additionally, we note that: exp(−αD−1 r1 ) = J0(2 √ αr1), D−n r1 {1} := rn1 /n! (3) is the inverse differential operator. The class of Appell polynomial sequences [13] appears in numerous applied mathe- matics problems, theoretical physics, approximation theory, and other mathematical dis- ciplines. These sequences are defined through the generating function: R(r1, t) := R(t)er1t = ∞∑ n=0 Rn(r1) tn n! , Rn := Rn(0), (4) where R(t) is an analytic function at t = 0, expressed as: R(t) = ∞∑ n=0 Rn tn n! , R0 ̸= 0, Ri (i = 0, 1, 2, · · · ) being real coefficients. (5) The Appell polynomials Rn(r1) are explicitly given by the series expansion: Rn(r1) = n∑ k=0 ( n k ) Rn−k r k 1 , R′ n(r1) = n Rn−1(r1). (6) By appropriately selecting R(t), various members of the Appell polynomial family can be derived. These are listed in Table 1 below: Table 1. Certain members belonging to the Appell family W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 3 of 22 S.No. Name of polynomials R(t) Generating function Series definition I. Bernoulli t et−1 ( t et−1 ) er1t = ∞∑ n=0 Bn(r1) tn n! Bn(r1) = n∑ k=0 ( n k ) Bkr n−k 1 polynomials ( t et−1 ) = ∞∑ n=0 Bn tn n! and numbers [14] Bn(:= Bn(0) = Bn(1)) II. Euler 2 et+1 ( 2 et+1 ) er1t = ∞∑ n=0 En(r1) tn n! En(r1) = n∑ k=0 ( n k ) Ek 2k ( r1 − 1 2 )n−k polynomials 2et e2t+1 = ∞∑ n=0 En tn n! and numbers [14] En := 2nEn ( 1 2 ) III. Genocchi 2t et+1 ( 2t et+1 ) er1t = ∞∑ n=0 Gn(r1) tn n! Gn(r1) = n∑ k=0 ( n k ) Gkr n−k 1 polynomials 2t et+1 = ∞∑ n=1 Gn tn n! and numbers [? ] Gn := Gn(0) To facilitate further computations, we present the initial values of Bernoulli numbers Bn, Euler numbers En, and Genocchi numbers Gn in Table 2 below: Table 2. Values of five four Bn, En and Gn n 0 1 2 3 4 Bn 1 ± 1 2 1 6 0 − 1 30 En 1 0 -1 0 5 Gn 0 1 -1 0 1 In 2012, Khan and Raza [15] introduced and analyzed a hybrid class of Laguerre-Sheffer polynomials, denoted as LSn(r1, r2), which are defined via the generating function: R(t) er2H(t) J0(2 √ r1H(t)) = ∞∑ n=0 LSn(r1, r2) tn n! . (7) Since Sheffer polynomials Sn(r1) [16] reduce to Appell polynomialsRn(r1) whenH(t) = t, choosing H(t) = t in equation (7) leads to the hybrid Legendre-Appell polynomials (LeAP), defined by: R(t) er2t J0(2 √ r1t) = ∞∑ n=0 LRn(r1, r2) tn n! , (8) or equivalently, R(t) er2t C0(r1t) = ∞∑ n=0 LRn(r1, r2) tn n! , (9) where C0(r1) denotes the 0th order Bessel Tricomi function [14]. The nth-order Tricomi functions Cn(r1) are defined as Cn(r1) = ∞∑ k=0 (−1)krk1 k!(n+ k)! . (10) W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 4 of 22 We also note that exp(−αD̂−1 r1 ) = C0(αx), D̂−n r1 {1} := rn1 n! . (11) Further, it can be expressed as R(t) eyt e−D−1 r1 t = ∞∑ n=0 LRn(r1, r2) tn n! . (12) The hybrid LAP LRn(r1, r2) satisfies the series expansion: LRn(r1, r2) = n! n∑ k=0 Ln−k(r2)r k 1 (n− k)!(k!)2 . (13) By appropriately choosing R(t), various members of the hybrid LAP family can be derived. These are summarized in Table 3: Table 3. Certain members belonging to the LAP family S. Name of hybrid R(t) Generating function Series definition No. polynomials I. Hybrid Laguerre-Bernoulli t et−1 ( t et−1 ) er2t J0(2 √ r1t = ∞∑ n=0 LBn(r1, r2) tn n! LBn(r1, r2) = n! n∑ k=0 (−1)kBn−k(r2)rk1 (n−k)!(k!)2 polynomials II. Hybrid Laguerre-Euler 2 et+1 ( 2 et+1 ) er2t J0(2 √ r1t = ∞∑ n=0 LEn(r1, r2) tn n! LEn(r1, r2) = n! n∑ k=0 (−1)kEn−k(r2)rk1 (n−k)!(k!)2 polynomials III. Hybrid Laguerre-Genocchi 2t et+1 ( 2t et+1 ) er2t J0(2 √ r1t = ∞∑ n=0 LGn(r1, r2) tn n! LGn(r1, r2) = n! n∑ k=0 (−1)kGn−k(r2)rk1 (n−k)!(k!)2 polynomials The 2-variable general polynomials (2VgP) denoted by Pn(r1, r2) are specified by gen- erating relation [17]: exp(r1t)Ψ(r2, t) = ∞∑ n=0 Pn(r1, r2) tn n! , (P0(r1, r2) = 1), (14) where Ψ(r2, t) has (at least the formal) series expansion Ψ(r2, t) = ∞∑ k=0 Ψk(r2) tk k! , (Ψ0(r2) ̸= 0). (15) The foundational idea of the monomiality principle dates back to 1941 when Steffenson [18] first introduced the concept through the notion of poweroid. This approach was later refined and extended by Dattoli [19], paving the way for further advancements in the field. The monomiality principle asserts that the operators M̂ and P̂ act, respectively, as multiplicative and differential operators for a given polynomial sequence {qn(r1)}n∈N. More specifically, these operators satisfy the fundamental recurrence relations: W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 5 of 22 qn+1(r1) = M̂{qn(r1)}, (16) and n qn−1(r1) = P̂{qn(r1)}. (17) A polynomial sequence {qn(r1)}n∈N that adheres to these operator relations is referred to as a quasi-monomial set. Such a set must also satisfy the fundamental commutation relation: [P̂,M̂] = P̂M̂ − M̂P̂ = 1̂, (18) which aligns naturally with the algebraic framework of the Weyl algebra. If a polynomial sequence {qn(r1)}n∈N is quasi-monomial, its defining properties can be derived directly from the characteristics of the operators M̂ and P̂. Specifically, the following key properties hold: (i) The polynomials qn(r1) satisfy a differential equation of the form: M̂P̂{qn(r1)} = n qn(r1), (19) provided that M̂ and P̂ admit suitable differential representations. (ii) An explicit formula for qn(r1) can be expressed as: qn(r1) = M̂n {1}, (20) with the initial condition q0(r1) = 1. (iii) The exponential generating function of qn(r1) is given by: etM̂{1} = ∞∑ n=0 qn(r1) tn n! (|t| <∞) , (21) which follows directly from equation (20). For more details, see [17, 20–27]. The operational framework outlined above has found extensive applications across various fields, including classical optics, quantum mechanics, and different branches of mathematical physics. These techniques offer robust analytical tools for studying diverse polynomial families. Motivated by these developments, we introduce a new generalize Laguerre-based Ap- pell polynomials PLRn(r1, r2, r3). The structure of this paper is as follows: In Section 2, we define the new generalization of Laguerre and Laguerre-based Appell polynomials and explore their key properties, including recurrence relations, associated operators, and differential equations. Section 3 focuses on the series expansions and determinant repre- sentations of these generalized polynomials. In Section 4, we examine specific subfamilies and establish their distinctive properties. Finally, the paper concludes with some remarks summarizing our findings and future research directions. W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 6 of 22 2. The new generalization of Laguerre and Laguerre-based Appell polynomials This section of our research paper introduces the new generalization of three-variable Laguerre-based Appell polynomials, denoted as PLRn(r1, r2, r3). We present their series expansion, quasi-monomial property, operational formulas, and corresponding differential equations. Our study initiates with the construction of a novel generalization of three- variable Laguerre polynomials, denoted as 3VLP PLn(r1, r2, r3). Utilizing the relations (1) and (14), we introduce the new generalization of three vari- able Laguerre polynomials PLυ(r1, r2, r3) in the following form: er1tΨ(r2, t)C0(r3t) = ∞∑ n=0 PLn(r1, r2, r3) tn n! , (P0(r1, r2) = 1). (22) By simplifying the left-hand side of equation (22) using equations (2) and (15), we derive the subsequent series representations for the generalized three-variable Laguerre polynomials 3V LP PLn(r1, r2, r3) holds: PLn(r1, r2, r3) = n∑ m=0 ( n m ) Ψm(r2)Ln−m(r1, r3). (23) We present the derived quasi-monomial identities for the three-variable Laguerre poly- nomials 3V LeP , denoted as PLn(r1, r2, r3). Theorem 1. The new generalization of three variable Laguerre polynomials 3VLP PLn(r1, r2, r3) demonstrate quasi monomials properties under the following multiplicative and derivative operators: M̂3V gLeP = r1 + Ψ ′ (r2, D̂r1) Ψ(r2, D̂r1) − nD̂−1 r3 , (24) and P̂3V gLeP = D̂r1 , (25) respectively. Proof. By differentiating Equation (22) w.r.t. t on both sides, it follows that ∞∑ n=0 PLn+1(r1, r2, r3) tn n! = xer1tΨ(r2, t)C0(r3t)+ Ψ ′ (r2, t) Ψ(r2, t) er1tΨ(r2, t)C0(r3t)+ ( ∞∑ n=0 (−1)nrn3nt n−1 ([n]!)2 ) er1tΨ(r2, t) ∞∑ n=0 PSn+1(r1, r2, r3) tn n! = ( r1 + Ψ ′ (r2, t) Ψ(r2, t) ) er1tΨ(r2, t)C0(−r3t2)+ ( ∞∑ n=0 (−1)n+1rn+1 3 tn ([n+ 1]!)2 ) er1tΨ(r2, t). Utilizing Equation (22), we have W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 7 of 22 ∞∑ n=0 PLn+1(r1, r2, r3) tn n! = ( r1 + Ψ ′ (r2, t) Ψ(r2, t) ) ∞∑ n=0 PLn(r1, r2, r3) tn n! + ( ∞∑ n=0 (−1)n+1rn+1 3 (n+ 1)tn ([n+ 1]!)2 ) er1tΨ(r2, t). (26) Differentiating the above Equation w.r.t. r3, we have ∞∑ n=0 Dr3 PLn+1(r1, r2, r3) tn n! = ( r1 + Ψ ′ (r2, t) Ψ(r2, t) ) ∞∑ n=0 Dr3 PLn(r1, r2, r3) tn n! − n ∞∑ n=0 PLn(r1, r2, r3) tn n! . (27) Consequently, Dr1 { er1tΨ(r2, t)C0(r3t) } = ter1tΨ(r2, t)C0(r3t), (28) and Ψ ′ (r2,t) Ψ(r2,t) possesses power series expansion in t with Ψ(r2, t) being the invertible series of t. Operating D−1 r3 to Equation (27) on both sides, we have ∞∑ n=0 PLn+1(r1, r2, r3) tn n! = ∞∑ n=0 ( r1 + Ψ ′ (r2, t) Ψ(r2, t) − nD−1 r3 ) PLn(r1, r2, r3) tn n! . (29) In light of (16) and (29), we obtain the assertion (24). Similarly, by applying identity (28) to (22), we get Dr1 { ∞∑ n=0 PLn(r1, r2, r3) tn n! } = ∞∑ n=1 PLn−1(r1, r2, r3) tn (n− 1)! (30) By matching the coefficients of same exponents of t on both sides of (30), it follows that Dr1 {PLn(r1, r2, r3)} = nPLn−1(r1, r2, r3), n ⪰ 1. (31) Thus in view of (17) and (31), we get the assertion (25). Theorem 2. The following differential equations for 3-variable generalized Laguerre poly- nomials 3V GLP PLn(r1, r2, r3) as:( r1D̂r1 + Ψ ′ (r2, D̂r1) Ψ(r2, D̂r1) D̂r1 − nD̂−1 r3 D̂r1 − n ) PLn(r1, r2, r3) = 0, (32) W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 8 of 22 Proof. In view of equations (24) and (25) in (19), we get ( r1Dr1 + Ψ ′ (r2, D̂r1) Ψ(r2, D̂r1) Dr1 − nD̂−1 r3 Dr1 ) PLn(r1, r2, r3) = n PLn(r1, r2, r3). Upon solving the above equation, we get the assertion (32) of Theorem 2.2. Remark 1. Since P0(r1, r2) = 1, therefore in view of monomiality principle equation (15), we have PLn(r1, r2, r3) = ( r1 + Ψ ′ (r2, D̂r1) Ψ(r2, D̂r1) − nD̂−1 r3 )n {1} , (P0(r1, r2) = 1). Also, in view of equations (15), (22) and (24), we have exp ( M̂3V gP ) {1} = er1tΨ(r2, t)C0(r3t) = ∞∑ υ=0 PLn(r1, r2, r3) tn n! . (33) Now, we proceed to introduce the new generalization of 3-variable Laguerre-based Ap- pell polynomials (3VLbAP). To obtain the generating functions for the newly generalized three-variable Laguerre-based Appell polynomials, we utilize the exponential generating function associated with Appell polynomials. Thus, replacing r1 in the left hand side of (5) by the multiplicative operator PLn(r1, r2, r3) given by (24) denoting the new general- ization of 3-variable Laguerre-based Appell polynomials PLRn(r1, r2, r3), we get R(t) exp ( M̂3V gP ) {1} = ∞∑ n=0 PLRn(r1, r2, r3) tn n! , (34) which on using equation (24), we get the following two equivalent forms of PLRn(r1, r2, r3): R(t) exp ( r1 + Ψ ′ (r2, D̂r1 Ψ(r2, D̂r1 − nD̂−1 r3 ) {1} = ∞∑ n=0 PLRn(r1, r2, r3) tn n! . (35) Using the relation (33) in the left hand side of equation (34), the generating function for the new generalization of 3-variable Laguerre-based Appell polynomials PLRn(r1, r2, r3) in the following form: R(t)er1tΨ(r2, t)C0(r3t) = ∞∑ n=0 PLRn(r1, r2, r3) tn n! . (36) where R(t) = ∞∑ k=0 αk tk k! , α0 ̸= 0 Ψ(r2, t) = ∞∑ k=0 ψk(r2) tk k! , ψ0 ̸= 0. (37) W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 9 of 22 Theorem 3. The generalized Laguerre-based Appell polynomials PLRn(r1, r2, r3) satisfy the following recurrence relation: PLRn+1(r1, r2, r3) = n∑ k=0 ( n k ) PLRn−k(r1, r2, r3)γk+r1PLRn(r1, r2, r3)+ n∑ k=0 ( n k ) PLRn−k(r1, r2, r3)ρk(r2) −nD−1 r3 PLRn−1(r1, r2, r3). (38) where R′(t) R(t) = ∞∑ k=0 γk tk k! , Ψt(r2, t) Ψ(r2, t) = ∞∑ k=0 ρk(r2) tk k! , Ψt(r2, t) = ∂ ∂t Ψ(r2, t) (39) and D−1 r3 is the inverse of Dr3. Proof. By differentiating both sides of equation (36) with respect to t, we obtain the following: ∞∑ n=0 PLRn+1(r1, r2, r3) tn n! = R′(t) R(t) R(t)er1tΨ(r2, t)C0(r3t) + r1R(t)er1tΨ(r2, t)C0(r3t) + Ψt(r2, t) Ψ(r2, t) R(t)er1tΨ(r2, t)C0(r3t) + ( ∞∑ n=0 rn3nt n−1 ([n]!)2 ) R(t)er1tΨ(r2, t) (40) Using (39), we have ∞∑ n=0 PLRn+1(r1, r2, r3) tn n! = ( ∞∑ k=0 γk tk k! )( ∞∑ n=0 PLRn(r1, r2, r3) tn n! ) + ( ∞∑ k=0 ρk(r2) tk k! )( ∞∑ n=0 PLRn(r1, r2, r3) tn n! ) +r1 ∞∑ n=0 PLRn(r1, r2, r3) tn n! + ( ∞∑ n=0 rn3nt n−1 ([n]!)2 ) R(t)er1tΨ(r2, t). (41) Hence, when use the Cauchy product, we obtain ∞∑ n=0 PLRn+1(r1, r2, r3) tn n! = ∞∑ n=0 n∑ k=0 ( n k ) PLRn−k(r1, r2, r3)γk tn n! + ∞∑ n=0 n∑ k=0 ( n k ) ρk(r2)PLRn−k(r1, r2, r3) tn n! +r1 ∞∑ n=0 PSRn(r1, r2, r3) tn n! + ( ∞∑ n=0 rn+1 3 (n+ 1)tn ([n+ 1]!)2 ) R(t)er1tΨ(r2, t). (42) Taking the derivative of both sides of the last equation with respect to r3, we get ∞∑ n=0 Dr3 {PLRn+1(r1, r2, r3)] tn n! = ∞∑ n=0 n∑ k=0 ( n k ) Dr3 {PLRn−k(r1, r2, r3)} γk tn n! W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 10 of 22 + ∞∑ n=0 n∑ k=0 ( n k ) ρk(r2)Dr3 {PLRn−k(r1, r2, r3)} tn n! + r1 ∞∑ n=0 Dr3 {PLRn(r1, r2, r3)} tn n! −n ∞∑ n=0 PLRn−1(r1, r2, r3) tn n! . (43) Applying D−1 r3 to both sides of the above equation, we get ∞∑ n=0 PLRn+1(r1, r2, r3) tn n! = ∞∑ n=0 n∑ k=0 ( n k ) PLRn−k(r1, r2, r3)γk tn n! + ∞∑ n=0 n∑ k=0 ( n k ) ρk(r2)PLRn−k(r1, r2, r3) tn n! +r1 ∞∑ n=0 PLRn(r1, r2, r3) tn n! − n ∞∑ n=0 D−1 r3 PLRn−1(r1, r2, r3) tn n! . (44) Thus, equating the coefficients of tn n! on both sides of the above equation (44), we get assertion (38). Theorem 4. The generalized Laguerre-based Appell polynomials PLRn(r1, r2, r3) satisfy the multiplicative and derivative operators as follows: M̂ = r1 + R′(D̂r1) R(D̂r1) + Ψ ′ (r2, D̂r1) Ψ(r2, D̂r1) − nD−1 r3 , (45) and P̂ = Dr1 , (46) respectively. Proof. Taking the derivative with respect to t on both sides of (36), we have ∞∑ n=0 PLRn+1(r1, r2, r3) tn n! = R′(t) R(t) R(t)er1tΨ(r2, t)C0(r3t) + r1R(t)er1tΨ(r2, t)C0(r3t) + Ψ ′ (r2, t) Ψ(r2, t) R(t)er1tΨ(r2, t)C0(r3t) + ( ∞∑ n=0 (−1)nrn3nt n−1 ([n]!)2 ) R(t)er1tΨ(r2, t) (47) ∞∑ n=0 PSRn+1(r1, r2, r3) tn n! = ( r1 + R′(t) R(t) + Ψ ′ (r2, t) Ψ(r2, t) ) R(t)er1tΨ(r2, t)C0(r3t) + ( ∞∑ n=0 (−1)n+1rn+1 3 (n+ 1)tn ([n+ 1]!)2 ) R(t)er1tΨ(r2, t). (48) W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 11 of 22 By using equation (36), we get ∞∑ n=0 PLRn+1(r1, r2, r3) tn n! = ( r1 + R′(t) R(t) + Ψ ′ (r2, t) Ψ(r2, t) ) ∞∑ n=0 PLRn(r1, r2, r3) tn n! + ( ∞∑ n=0 (−1)n+1rn+1 3 (n+ 1)tn ([n+ 1]!)2 ) R(t)er1tΨ(r2, t). (49) Taking the derivative of both sides of the last equation with respect to r3, we get ∞∑ n=0 Dr3PLRn+1(r1, r2, r3) tn n! = ( r1 + R′(t) R(t) + Ψ ′ (r2, t) Ψ(r2, t) ) ∞∑ n=0 Dr3PLRn(r1, r2, r3) tn n! −n ∞∑ n=0 PLRn(r1, r2, r3) tn n! . (50) Applying D−1 r3 to both sides of the above equation, we get ∞∑ n=0 PLRn+1(r1, r2, r3) tn n! = ∞∑ n=0 ( r1 + R′(t) R(t) + Ψ ′ (r2, t) Ψ(r2, t) − nD−1 r3 ) PLRn(r1, r2, r3) tn n! . (51) In view of (16) and (51), we get the assertion (45). Again in view of (17) and (36), we get the assertion (46). Theorem 5. The generalized Laguerre-based Appell polynomials PLRn(r1, r2, r3) satisfies the differential equation as follows:( r1Dr1 + R′(D̂r1) R(D̂r1) Dr1 + Ψ ′ (r2, D̂r1) Ψ(r2, D̂r1) Dr1 − nDr1D −1 r3 − n ) PLRn(r1, r2, r3) = 0. (52) Proof. In view of equations (45), (46) in (19), we get the assertion (52). So, we omit the proof. 3. Series representation and determinant form Hybrid special polynomials play a crucial role in mathematical analysis due to their rich structural properties. Their series representation provides explicit forms and recur- rence relations, aiding in solving differential and functional equations. The determinant form of these polynomials offers a compact and elegant way to analyze their algebraic and combinatorial properties. It facilitates the study of orthogonality, symmetry, and trans- formation identities. Hybrid polynomials also bridge classical and modern polynomial families, extending their applicability in mathematical physics and engineering. Their determinant representation helps in computing higher-order coefficients efficiently. These polynomials are widely used in approximation theory and numerical analysis. Overall, they contribute significantly to both theoretical and applied mathematical research. W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 12 of 22 Theorem 6. The 3-variable generalized Laguerre-based Appell polynomials PLRn(r1, r2, r3) are defined by the series: PLRn(r1, r2, r3) = n∑ k=0 ( n k ) Rk PLn−k(r1, r2, r3), (53) with Rk is given by equation (5). Proof. In view of equation (36), we can write ∞∑ υ=0 PLRn(r1, r2, r3) tn n! = R(t) ∞∑ n=0 PLn(r1, r2, r3) tn n! . (54) Using the expansion (5) of R(t) from the left-hand side of equation (54), we simplify and then equate the coefficients of like powers of δ on both sides of the resulting equation, leading us to assertion (53). Theorem 7. The generalized Laguerre-based Appell polynomials PLRn(r1, r2, r3) has the following determinant representation PLRn,q(r1, r2, r3) = (−1)n (β0)n+1 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 PL1(r1, r2, r3) PL2(r1, r2, r3) ... PLn−1(r1, r2, r3) PL(m) n (r1, r2, r3) β0 β1 β2 ... βn−1 βn 0 β0 ( 2 1 ) β1 ... ( n−1 1 ) βn−2 ( n 1 ) βn−1 0 0 β0 ... ( n−1 1 ) βn−3 ( n 2 ) βn−2 ... ... ... ... ... 0 0 0 ... β0 ( n n−1 ) β1 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ , (55) where ∑∞ n=0 PLn(r1, r2, r3) tn n! = er1tΨ(r2, t)C0(r3t) , 1 R(t) = ∑∞ k=0 βk tk k! . Proof. Using the series representation of 1 R(t) as follows: [R(t)]−1 = ∞∑ k=0 βk tk k! , using the generation function (22), we get er1tΨ(r2, t)C0(r3t) = ( ∞∑ k=0 βk tk k! )( ∞∑ n=0 PLRn(r1, r2, r3) tn n! ) . Hence ∞∑ n=0 PLRn(r1, r2, r3) tn n! = ( ∞∑ k=0 βk tk k! )( ∞∑ n=0 PLRn(r1, r2, r3) tn n! ) . W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 13 of 22 Applying the Cauchy product, we have ∞∑ n=0 PLRn(r1, r2, r3) tn n! = ∞∑ n=0 n∑ k=0 ( n k ) βk PLRn−k(r1, r2, r3) tn n! . By comparing the coefficients of tn n! from the polynomial equation, we get PLRn(r1, r2, r3) = n∑ k=0 ( n k ) βk PLRn−k(r1, r2, r3), n ∈ N0. So, we obtain the system of equations as follows: PL0(r1, r2, r3) = β0 PLR0(r1, r2, r3), PL1(r1, r2, r3) = β0 PLR1(r1, r2, r3) + β1 PLR0(r1, r2, r3), PL2(r1, r2, r3) = β0 PLR2(r1, r2, r3) + ( 2 1 ) β1 PLR1(r1, r2, r3) + β2 PLR0(r1, r2, r3), ... PLn−1(r1, r2, r3) = β0 PLRn−1(r1, r2, r3)+ ( n− 1 1 ) β1 PLRn−2(r1, r2, r3)+· · ·+βn−1 PLR0(r1, r2, r3), PLRn(r1, r2, r3) = β0 PLRn(r1, r2, r3)+ ( n 1 ) β1 PLRn−1(r1, r2, r3)+· · ·+βn PLR0(r1, r2, r3). Applying Cramers’ rule, we get PLRn(r1, r2, r3) = ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ β0 0 ... 0 PL0(r1, r2, r3) β1 β0 ... 0 PL1(r1, r2, r3) β2 ( 2 1 ) β1 ... 0 PL2(r1, r2, r3) β3 ( 3 2 ) β2 ... 0 PL3(r1, r2, r3) ... ... ... ... ... βn−1 ( n−1 1 ) βn−2 ... β0 PLn−1(r1, r2, r3) βn ( n 1 ) βn−1 ... ( n n−1 ) β1 PLm(r1, r2, r3) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ β0 0 ... 0 0 β1 β0 ... 0 0 β2 ( 2 1 ) β1 ... 0 0 β3 ( 3 2 ) β2 ... 0 0 ... ... ... ... ... βn−1 ( n−1 1 ) βn−2 ... β0 0 βn ( n 1 ) βn−1 ... ( n n−1 ) β1 β0 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ . W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 14 of 22 By taking the transpose in the last equation, we have PLRn(r1, r2, r3) = 1 (β0)n+1 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ β0 β1 ... βn−1 βn 0 β0 ... ( n−1 1 ) βn−2 ( n 1 ) βn−1 0 0 ... ( n−1 1 ) βn−3 ( n 2 ) βn−2 ... ... ... ... ... 0 0 ... β0 ( n n−1 ) β1 PL0(r1, r2, r3) PL1(r1, r2, r3) ... PLn−1(r1, r2, r3) PL(m) n (r1, r2, r3) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ . Thus, simple row operations are used to finish the proof. 4. Applications This study extends the exploration of recently introduced polynomials, focusing on the examination of the generalization of three-variable Laguerre-based Appell polynomials. Specifically, when considering the case where Ψ(r2, t) = er2t 2 in the generating function (36), which results in the reduction of 3V LAP PLRn(r1, r2, r3) to the Laguerre-Hermite- Appell polynomials (LHAP ) PHRn(r1, r2, r3) can be characterized by a specific generating function. R(t)er1t+r2t2C0(r3t) = ∞∑ n=0 PHRn(r1, r2, r3) tn n! . (56) In other words, we note that PHRn(r1, r2, r3) = exp ( −D̂−1 r3 ∂ ∂r1 ) {HRn(r1, r2)} = exp ( r2 ∂2 ∂r21 ) {LRn(r1, r3)} . (57) Theorem 8. The three variable Laguerre-Hermite-based Appell polynomials are defined by the series: PHRn(r1, r2, r3) = n∑ k=0 ( n k ) Rk PHn−k(r1, r2, r3). (58) Proof. In view of (56), we have ∞∑ n=0 PHRn(r1, r2, r3) tn n! = R(t) ∞∑ n=0 PHn(r1, r2, r3) tn n! . (59) Now, by using the expansion of R(t) from the left-hand side of equation (59), we can simplify and equate the coefficients of like powers of t on both sides of the resulting equation to obtain assertion (58). Next, we will demonstrate the determinant form for PHRn(r1, r2, r3) using an approach similar to that presented in [28, 29], taking into account equation (56). W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 15 of 22 Theorem 9. The determinant representation of 3-variable Laguerre-Hermite-Appell poly- nomials PHRn(r1, r2, r3) of degree n is PHR0(r1, r2, r3) = 1 β0 , PHRn(r1, r2, r3) = (−1)n (β0)n+1 ∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 PH1(r1, r2, r3) PH2(r1, r2, r3) ... PHn−1(r1, r2, r3) PHn(r1, r2, r3) β0 β1 β2 ... βn−1 βn 0 β0 ( 2 1 ) β1 ... ( n−1 1 ) βn−2 ( n 1 ) βn−1 0 0 β0 ... ( n−1 1 ) βn−3 ( n 2 ) βn−2 ... ... ... ... ... 0 0 0 ... β0,q ( n n−1 ) β1 ∣∣∣∣∣∣∣∣∣∣∣∣∣ , (60) βn = − 1 R0 ( n∑ k=1 ( n k ) Rkβn−k ) , n = 0, 1, 2, . . . , where β0 ̸= 0, β0 = 1 R0,q and PHn(r1, r2, r3), n = 0, 1, 2, . . . , are the three variable q- Legendre-Hermite polynomials. Proof. By inserting the series forms of the new generalization of the three-variable Laguerre-Hermite polynomials into the generating function of the three-variable Laguerre- Hermite-Appell polynomials, we obtain: R(t) ∞∑ n=0 PHn(r1, r2, r3) tn n! = ∞∑ n=0 PHRn(r1, r2, r3) tn n! . (61) By multiplying 1 R(t) = ∞∑ k=0 βk tk k! , (62) on both sides, it follows that ∞∑ n=0 PHn(r1, r2, r3) tn n! = ∞∑ k=0 βk tk k! ∞∑ n=0 PHRn(r1, r2, r3) tn n! . (63) Applying Cauchy product in (63) gives PHn(r1, r2, r3) = n∑ k=0 ( n k ) βk PHRn−k(r1, r2, r3). (64) This equality leads to a system of n equations with the unknowns Rn(r1, r2, r3), where n = 0, 1, 2, . . . To solve this system using Cramer’s rule, we note that the denominator is the de- terminant of a lower triangular matrix, which has a determinant of (β0) n+1. By taking W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 16 of 22 the transpose of the numerator and replacing the ith row with the (i + 1)th position for i = 1, 2, . . . , n− 1, we obtain the desired result. We will now demonstrate the multiplicative and derivative operators of PHRn(r1, r2, r3). The following theorem is presented: Theorem 10. The generalization of Laguerre-Hermite-based Appell polynomials satisfies the multiplicative and derivative operators as follows: M̂ = r1 + R′(D̂r1) R(D̂r1) − n ( r2 +D−1 r3 ) , (65) and P̂ = Dr1 , (66) respectively. Proof. Utilizing the derivative with respect to t on both sides of equation (56), we find ∞∑ n=0 PHRn+1(r1, r2, r3) tn n! = R′(t) R(t) R(t)er1t+r2t2C0(r3t)+r1R(t)er1t+r2t2C0(r3t)+2r2tR(t)er1t+r2t2C0(r3t) + ( ∞∑ n=0 (−1)nrn3nt n−1 ([n]!)2 ) R(t)er1t+r2t2 (67) ∞∑ n=0 PHRn+1(r1, r2, r3) tn n! = ( r1 + R′(t) R(t) − nr2 ) R(t)er1t+r2t2C0(r3t) + ( ∞∑ n=0 (−1)n+1rn+1 3 (n+ 1)tn ([n+ 1]!)2 ) R(t)er1t+r2t2 . (68) By using equation (56), we get ∞∑ n=0 PHRn+1(r1, r2, r3) tn n! = ( r1 + R′(t) R(t) − nr2 ) ∞∑ n=0 PHRn(r1, r2, r3) tn n! + ( ∞∑ n=0 (−1)n+1rn+1 3 (n+ 1)tn ([n+ 1]!)2 ) R(t)er1t+r2t2 . (69) On differentiating both sides of the last equation with respect to r3, we obtain: ∞∑ n=0 Dr3PHRn+1(r1, r2, r3) tn n! = ( r1 + R′(t) R(t) − nr2 ) ∞∑ n=0 Dr3PHRn(r1, r2, r3) tn n! −n ∞∑ n=0 PHRn(r1, r2, r3) tn n! . (70) W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 17 of 22 Applying D−1 r3 to both sides of the above equation, we get ∞∑ n=0 PHRn+1(r1, r2, r3) tn n! = ∞∑ n=0 ( r1 + R′(t) R(t) − n ( r2 +D−1 r3 )) PHRn(r1, r2, r3) tn n! . (71) In view of (16) and (71), we get the assertion (65). Again in view of (17) and (56), we get the assertion (66). Theorem 11. The following differential equation for PHRn(r1, r2, r3) holds true:( r1Dr1 + R′(Dr1) R(Dr1) − n ( r2 +D−1 r3 ) Dr1 − n ) PHRn(r1, r2, r3) = 0. (72) Proof. Using (65) and (66) in (19), we get( r1Dr1 + R′(D̂r1) R(D̂r1) Dr1 − n ( r2 +D−1 r3 ) Dr1 ) PHRn(r1, r2, r3) = n PHRn(r1, r2, r3). (73) Upon the simplification, we get the assertion (72). 5. Examples The Appell polynomial family, defined by the parameter function R(t), provides solu- tions to specific differential equations. Different choices of R(t) generate various polynomi- als, allowing adaptability in mathematical modeling. This flexibility makes them valuable in physics, engineering, and other scientific fields. Table 1 systematically presents their generating functions, series definitions, and numerical values. Generating functions of- fer concise power series representations, aiding analytical manipulation. Series definitions provide formal expressions crucial for problem-solving and computation. Numerical values enhance practical understanding and facilitate real-world applications. These polynomials are widely used in probability theory, quantum mechanics, and signal processing. Their versatility allows specialized solutions for complex mathematical problems. Overall, the Appell polynomial family serves as a powerful tool in scientific research. The Bernoulli, Euler, and Genocchi numbers are foundational in mathematics, with applications in number theory, combinatorics, algebraic geometry, and more. Bernoulli numbers appear in polynomials and the Euler-Maclaurin formula, while Euler numbers contribute to modular forms and elliptic curve theory. Genocchi numbers play a key role in combinatorial problems, graph theory, and automata theory. These numbers connect to hyperbolic secant functions and have implications in quantum field theory and signal processing. By treating them as members of the Appell family, new polyno- mials like the two-iterated degenerate Hermite-Appell polynomials are derived, offering rich research opportunities in their generating expressions and characteristics. As a re- sult, different members of PHRn(r1, r2, r3) appear as Laguerre-Hermite-based Bernoulli W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 18 of 22 polynomials PHBn(r1, r2, r3), Laguerre-Hermite-Euler polynomials PHEn(r1, r2, r3), and Laguerre-Hermite-Genocchi polynomials PHGn(r1, r2, r3). The following expressions can be used to cast these polynomials: t et − 1 er1t+r2t2C0(r3t) = ∞∑ n=0 PHBn(r1, r2, r3) tn n! , (74) 2 et + 1 er1t+r2t2C0(r3t) = ∞∑ n=0 PHEn(r1, r2, r3) tn n! , (75) and 2t et + 1 er1t+r2t2C0(r3t) = ∞∑ n=0 PHGn(r1, r2, r3) tn n! . (76) For instance the Laguerre-Hermite-based Bernoulli polynomials PHBn(r1, r2, r3), Laguerre- Hermite-Euler polynomials PHEn(r1, r2, r3), and Laguerre-Hermite-Genocchi polynomials PHGn(r1, r2, r3) are defined by the following operational identities: PHBn(r1, r2, r3) = exp ( −D̂−1 r3 ∂ ∂r1 ) {HBn(r1, r2)} = exp ( r2 ∂2 ∂r21 ) {LBn(r1, r3)} , (77) PHEn(r1, r2, r3) = exp ( −D̂−1 r3 ∂ ∂r1 ) {HEn(r1, r2)} = exp ( r2 ∂2 ∂r21 ) {LEn(r1, r3)} , (78) and PHGn(r1, r2, r3) = exp ( −D̂−1 r3 ∂ ∂r1 ) {HGn(r1, r2)} = exp ( r2 ∂2 ∂r21 ) {LGn(r1, r3)} . (79) Similarly, a similar method can be used to draw corresponding results for these poly- nomials. The monomiality principle, which looks at how polynomials behave in respect to their monomial coefficients, is one such result. Extensive study of the monomiality principle provides important information about the structure and characteristics of these polynomials. Moreover, the polynomials’ explicit expressions can be found, offering a distinct de- piction of their coefficients and terms. This makes it easier to comprehend and analyze the polynomials. Examining the differential equations that these polynomials satisfy is another line of inquiry. There are relationships between these polynomials and other mathematical ideas that can be found by investigating the related differential equations. It also enables in-depth comprehension by enabling researchers to examine their behavior in a variety of scenarios. Furthermore, determinant forms investigation for these polynomials offers an impor- tant and interesting research avenue. You can use determinants, which are mathematical entities that describe particular aspects of matrices, to make links between these polynomi- als and linear algebra. This endeavor creates new opportunities for further understanding and applications in a larger mathematical environment. W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 19 of 22 Furthermore, in view of expressions (60), the polynomials PHBn(r1, r2, r3), PHEn(r1, r2, r3) and PHGn(r1, r2, r3) satisfy the following determinant representations: PHBn(r1, r2, r3) = (−1)n ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 PH1(r1, r2, r3) PH2(r1, r2, r3) · · · PHn−1(r1, r2, r3) PHn(r1, r2, r3) 1 1 2 1 3 · · · 1 n 1 n+1 0 1 ( 2 1 ) 1 2 · · · ( n−1 1 ) 1 n−1 ( n 1 ) 1 n 0 0 1 · · · ( n−1 2 ) 1 n−2 ( n 2 ) 1 n−1 . . . · · · . . . . . · · · . . 0 0 0 · · · 1 ( n n−1 ) 1 2 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ , (80) PHEn(r1, r2, r3) = (−1)n ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 PH1(r1, r2, r3) PH2(r1, r2, r3) · · · PHn−1(r1, r2, r3) PHn(r1, r2, r3) 1 1 2 1 2(1)2 · · · 1 2(1)n−1 1 2(1)n 0 1 ( 2 1 ) 1 2 · · · ( n−1 1 ) 1 2(1)n−2 ( n 1 ) 1 2(1)n−1 0 0 1 · · · ( n−1 2 ) 1 2(1)n−3 ( n 2 ) 1 2(1)n−2 . . . · · · . . . . . · · · . . 0 0 0 · · · 1 ( n n−1 ) 1 2 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ , (81) and PHGn(r1, r2, r3) = (−1)n ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 PH1(r1, r2, r3) PH2(r1, r2, r3) · · · PHn−1(r1, r2, r3) PHn(r1, r2, r3) 0 1 2 ( 2 1 ) (1)2 4 · · · ( n−1 1 ) (1)n−1 2(n−1) ( n 1 ) (1)n 2n 0 0 1 2 · · · ( n−1 2 ) (1)n−2 2(n−2) ( n 2 ) (1)n−1 2(n−1) ... ... ... . . . ... ... 0 0 0 · · · 1 2 ( n n−1 ) (1)2 4 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ . (82) W. A. Khan, H. Qawaqneh, H. Aydi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6658 20 of 22 6. Concluding remarks In this study, we introduced and systematically analyzed a new generalization of La- guerre and Laguerre-based Appell polynomials. Through a detailed exploration of their fundamental properties, we established recurrence relations, multiplicative and derivative operators, and a governing differential equation via quasi-monomiality. The derivation of both series and determinant representations further highlights the structural depth of this novel polynomial family. Additionally, the introduction of generalized Laguerre-Hermite Appell polynomials, along with their specific cases involving Bernoulli, Euler, and Genoc- chi polynomials, enriches the framework of special functions. These findings contribute to the broader understanding of polynomial sequences and their applications in mathematical physics and differential equations. For future research, the investigation of these polynomials in the context of orthogonal- ity and integral transforms could provide further insights into their analytical properties. Exploring their connections with fractional calculus and special function theory may yield new results with applications in approximation theory and signal processing. Further- more, the study of their extensions in the framework of q-calculus and number theory could reveal deeper algebraic and combinatorial properties. Finally, the numerical aspects and computational implementations of these polynomials can be explored for potential applications in scientific computing and engineering. Availability of data and materials Not applicable. Competing interests The authors declare no competing interests. Authors’ contributions All authors contributed equally to the article. Acknowledgements The authors acknowledge the financial support from Al-Zaytoonah University of Jor- dan, Amman 11733, Jordan. 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