EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6410 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Unified Operational and Algebraic Approach of ∆h-Hybrid Polynomials Associated with Appell Sequences Taghreed Alqurashi1, Waseem ahmad Khan2, Shahid Ahmad Wani3,∗, Semra Kuş4, Shilpa Malge3, Prakash Jadhav5 1 Mathematics Department, Faculty of Science, Al-Baha University, 65779-7738, Albaha City, Kingdom of Saudi, Arabia 2 Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia 3 Symbiosis Institute of Technology PUNE, Symbiosis International (Deemed University), Pune, India 4 Mucur Vocational High School, Kırşehir Ahi Evran University, Kırşehir, Turkey 5 Department of Mechanical Engineering, SRM University AP, Andhra Pradesh 522240, India Abstract. This study introduces a new class of ∆h Legendre-Laguerre-Appell polynomials, con- structed through the synergy of the monomiality framework and operational calculus. A compre- hensive exploration is carried out, beginning with the formulation of their generating function, followed by the derivation of explicit representations and recurrence schemes. Notably, a deter- minantal structure for these polynomials is also established and illustrated through representative examples. The work further investigates how this polynomial family interrelates with well-known ∆h-variants of classical polynomials, including the Bernoulli, Euler, and Genocchi types. Through these connections and properties, the results not only deepen our understanding of the algebraic and analytic behaviour of the ∆h Legendre-Laguerre-Appell polynomials but also highlight their potential applications in broader areas of discrete mathematics and operational theory. 2020 Mathematics Subject Classifications: 33E20, 33B10, 33E30, 11T23 Key Words and Phrases: Monomiality principle, Explicit forms, Determinant form, Operational formalism, Examples ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6410 Email addresses: talqorashi@bu.edu.sa (T. Alqurashi), wkhan1@pmu.edu.sa ( W. A. Khan), shahidwani177@gmail.com (S. A. Wani ), semrakus40@gmail.com (S. Kuş), shilpam@sitpune.edu.in (S. Malge), Prakash.j@srmap.edu.in (P. Jadhav) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 2 of 18 1. Introduction and preliminaries Special polynomials play a vital role in modeling diverse systems across statistical mechanics, quantum mechanics, and several branches of mathematics, including com- binatorics, entropy theory, and algebraic structures. Classical families like “Laguerre, Chebyshev, Legendre, and Jacobi polynomials emerge as solutions to specific second-order differential equations, making them instrumental in approximation theory and physics”. Laguerre polynomials, introduced by Edmond Laguerre in the 19th century, are a prominent class of orthogonal polynomials defined on [0,+∞). These polynomials are inte- gral to various areas such as Fourier analysis, numerical integration (e.g., Gauss–Laguerre quadrature), and solving physical models, notably the radial Schrödinger equation in quan- tum mechanics. These functions are also essential in contexts like heat conduction, wave motion, and diffusion. Recent developments have centered on two-variable extensions of such polynomials, which offer refined tools for analyzing physical phenomena with multiple degrees of free- dom. These include bivariate forms of “Chebyshev, Hermite, and Laguerre polynomials, frequently used in approximation theory, numerical computation, and signal analysis [1– 10]’. Specifically, the bivariate Laguerre polynomials, denotedWϕ(u, v), satisfy a two-variable generalization of the classical Laguerre differential equation. These polynomials are partic- ularly useful in quantum mechanics, potential theory, and the study of random matrices. Their orthogonality with respect to a bivariate weight function makes them suitable for addressing multivariate problems in mathematical physics and probability theory. As highlighted in [11], the introduction of two-variable Laguerre Wϕ(u, v) and Legendre poly- nomials Sϕ(u, v) provides valuable analytical tools for tackling partial differential equations encountered in various physical models. The Laguerre polynomials (2VLP) with notion Wϕ(u, v) are represented as evξJ0(ξ √ −u) = ∞∑ ϕ=0 Wϕ(u, v) ξϕ ϕ! , (1) where J0(uξ) denotes the Bessel function of the first kind of order zero [12], which is defined by the series: Jϕ(2 √ u) = ∞∑ ν=0 (−1)ν ( √ u) ϕ+ν ν! (ϕ+ ν)! . (2) Additionally, one can use the identity exp(−αD−1 u ) = J0(2 √ αu), D−ϕ u {1} := uϕ ϕ! , (3) where D−1 u stands for the inverse differential operator. An alternative expression for the generating function is evξC0(−uξ) = ∞∑ ϕ=0 Wϕ(u, v) ξϕ ϕ! , (4) T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 3 of 18 with C0(uξ) referring to the Tricomi function of the first kind of order zero [12], described by C0(−uξ) = eD −1 u ξ. (5) Consequently, combining either equation (3) or (5), the generating formulation of the Laguerre polynomials can be equivalently written as: evξ eD −1 u ξ = ∞∑ ϕ=0 Wϕ(u, v) ξϕ ϕ! . (6) Similarly, the Legendre polynomials with notion Sϕ(u, v) are represented as evξJ0(ξ √ −u) = ∞∑ ϕ=0 Sϕ(u, v) ξϕ ϕ! , (7) where J0(uξ) is again the Bessel function of order zero, as given in (2). Alternatively, one may express the generating function using the Tricomi function as: evξC0(−uξ2) = ∞∑ ϕ=0 Sϕ(u, v) ξϕ ϕ! , (8) where C0(uξ) is the same function defined earlier in (5). Therefore, taking into account either (3) or (5), the Legendre polynomial generating function can be restated as: evξ eD −1 u ξ2 = ∞∑ ϕ=0 Sϕ(u, v) ξϕ ϕ! . (9) Recent studies have focused on developing ∆h analogues of special polynomials. In [12], several generalizations were explored. A new class, termed ∆h-special polynomials, was introduced using the classical finite difference operator ∆h in [13–16], due to their broad applications in mathematics, physics, and statistics. The ∆h-Appell polynomials are defined as: A[h] ϕ (u) := Aϕ(u), ϕ ∈ N0 (10) with the recurrence relation: A[h] ϕ (u) = ϕhAϕ−1(u), ϕ ∈ N0, (11) where the finite difference operator is: ∆h H[h](u) = H(u+ h)−H(u). (12) T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 4 of 18 Their generating function is given by [13]: γ(ξ)(1 + hξ) u h = ∞∑ ϕ=0 A[h] ϕ (u) ξϕ ϕ! , (13) with γ(ξ) = ∞∑ ϕ=0 γϕ,h ξϕ ϕ! , γ0,h ̸= 0. (14) Inspired by [13], we define the three-variable ∆h Legendre-Laguerre Appell polynomials (∆h LeLAP ) by: γ(ξ)(1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h = ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! . (15) The paper is formulated as: Section 2 presents their construction and recurrence relations. Section 3 derives explicit formulas. Section 4 discusses the monomiality principle and determinant forms. Section 5 relates these to ∆h-Bernoulli, Euler, and Genocchi polynomials and provides symmetric identities. The conclusion summarizes results and proposes future directions. 2. Results on ∆h LeLAP This section explores the generating function and recurrence formulas associated with a novel class of three-variable ∆h Legendre-Laguerre Appell polynomials. Theorem 1. The generating function for the ∆h LeLAP SLA [h] ϕ (u, v, w) is given by: γ(ξ)(1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h = ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! . (16) Proof. Consider the left-hand side of (16). When expanded as a Newton-type series centered at u = v = w = 0, each term corresponds to a coefficient of ξϕ divided by ϕ!. By identifying these coefficients, the polynomials SLA [h] ϕ (u, v, w) arise naturally as those associated with the generating function expansion. Theorem 2. The following recurrence relations hold for the polynomials SLA [h] ϕ (u, v, w): v∆h h SLA [h] ϕ (u, v, w) = ϕ SLA [h] ϕ−1(u, v, w), u∆h h SLA [h] ϕ (u, v, w) = ϕ(ϕ− 1) SLA [h] ϕ−2(u, v, w), D−1 u → u D−1 w → w. (17) T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 5 of 18 Proof. Differentiating both sides of (16) with respect to v using the finite difference operator v∆h, we obtain: v∆h { γ(ξ)(1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h } = hξ · γ(ξ)(1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h . (18) Substituting the series representation from (16) into the above and shifting indices in the resulting series: v∆h ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! = h ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ+1 ϕ! . (19) Re-indexing and equating coefficients of like powers of ξ on both sides yields the first identity in (17). A similar strategy using differentiation with respect to u leads to the second identity. Theorem 3. The family of polynomials SLA [h] ϕ (u, v, w) admits the following explicit rep- resentation: SLA [h] ϕ (u, v, w) = [ v h ]∑ d=0 ( ϕ d )( v h d ) hd SLA [h] ϕ−d(u,w). (20) Proof. We start from the generating function provided in expression (16): γ(ξ)(1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h = ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! . (21) Observe that when v = 0, this reduces to: γ(ξ)(1 + hξ2) D−1 w h = ∞∑ ϕ=0 SLA [h] ϕ (u, 0, w) ξϕ ϕ! . (22) We now expand the factor (1 + hξ) v−D−1 u h using the binomial theorem, noting that v−D−1 u h = v h− D−1 u h . Since D−1 u is an operator acting on the variable u, we consider its effect separately, and use the binomial expansion: (1 + hξ) v h = v h∑ d=0 ( v h d ) (hξ)d. (23) Multiplying this with the generating function of the form when v = 0, we obtain: T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 6 of 18 γ(ξ) v h∑ d=0 ( v h d ) (hξ)d d! · ∞∑ ϕ=0 SLA [h] ϕ (u, 0, w) ξϕ ϕ! = ∞∑ ϕ=0 min(ϕ, vh)∑ d=0 ( v h d ) hdSLA [h] ϕ−d(u,w)  ξϕ d! · ϕ! . (24) Now applying the identity: 1 d!(ϕ− d)! = 1 ϕ! ( ϕ d ) , (25) we rewrite the series as: ∞∑ ϕ=0 min(ϕ, vh)∑ d=0 ( ϕ d )( v h d ) hdSLA [h] ϕ−d(u,w)  ξϕ ϕ! . (26) Since this matches the original generating function series, by equating the coefficients of ξϕ/ϕ! on both sides, we arrive at the claimed formula: SLA [h] ϕ (u, v, w) = [ v h ]∑ d=0 ( ϕ d )( v h d ) hd SLA [h] ϕ−d(u,w). (27) Theorem 4. The following explicit representation also holds: SLA [h] ϕ (u, v, w) = ϕ∑ k=0 ( ϕ k ) γk,h SLA [h] ϕ−k(u, v, w). (28) Proof. We begin by considering the generating function for the polynomials SLA [h] ϕ (u, v, w), as defined in expression (16): G(ξ) = γ(ξ)(1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h = ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! . (29) Using expansion of γ(ξ) from expression (14) and substituting into the generating function (16) gives: G(ξ) = ( ∞∑ k=0 γk,h ξk k! ) (1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h . (30) Now using the Cauchy product of series, we expand the full generating function: G(ξ) = ∞∑ k=0 γk,h ξk k! · ∞∑ m=0 SLA[h] m (u, v, w) ξm m! = ∞∑ ϕ=0 [ ϕ∑ k=0 ( ϕ k ) γk,h SLA [h] ϕ−k(u, v, w) ] ξϕ ϕ! . (31) T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 7 of 18 By equating the coefficients of ξ ϕ ϕ! on both sides, we deduce: SLA [h] ϕ (u, v, w) = ϕ∑ k=0 ( ϕ k ) γk,h SLA [h] ϕ−k(u, v, w), (32) which completes the proof. 3. Other results This section establishes summation formulae, which serve as essential tools in mathe- matical analysis, revealing intricate relationships, patterns, and symmetries within polyno- mial structures. They aid in combinatorics, probability theory, and mathematical physics while enhancing computational efficiency. Next, we present the summation formulas highlighting key summation properties of the three-variable ∆h Legendre-Laguerre-Appell polynomials SLA [h] ϕ (u, v, w), forming the core of this study. Theorem 5. For ϕ ≥ 0, we have SLA [h] ϕ (u, v, w) = ϕ∑ ψ=0 ( ϕ ψ )( −v h ) ψ (−h)ψSLA [h] ϕ−ψ(u, 0, w). (33) Proof. From (16), we have ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! = γ(ξ)(1+hξ) v h (1+hξ) −D−1 u h (1+hξ2) D−1 w h = ∞∑ ϕ=0 SL [h] ϕ (u, 0, w) ξϕ ϕ! ∞∑ ψ=0 ( −v h ) ψ (−h)ψ ξ ψ ψ! = ∞∑ ϕ=0  ϕ∑ ψ=0 ( ϕ ψ )( −v h ) ψ (−h)ψSLA [h] ϕ−ψ(u, 0, w)  ξϕ ϕ! . (34) When comparing the ϕ coefficients , we obtain (33). Theorem 6. For ϕ ≥ 0, we have SLA [h] ϕ (u, v + 1, w) = ϕ∑ ψ=0 ( ϕ ψ )( −1 h ) ψ (−h)ψSL [h] ϕ−ψ(u, v, w). (35) Proof. From (16), we have ∞∑ ϕ=0 SLA [h] ϕ (u, v+1, w) ξϕ ϕ! − ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! = γ(ξ)(1+hξ) v h (1+hξ) −D−1 u h (1+hξ2) D−1 w h ( (1 + hξ) 1 h − 1 ) T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 8 of 18 = ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ!  ∞∑ ψ=0 ( −1 h ) ψ (−h)ψ ξ ψ ψ! − 1  = ∞∑ ϕ=0  ϕ∑ ψ=0 ( ϕ ψ )( −1 h ) ψ (−h)ψSL [h] ϕ−ψ(u, v, w)  ξϕ ϕ! − ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! . (36) By matching the coefficients of identical powers of ϕ, we obtain (35). We now examine the relationship between the polynomials SLA [h] ϕ (u, v, w) and the Stirling numbers of the first kind. [log(1 + ξ)]k k! = ∞∑ i=k S1(i, k) ξi i! , | ξ |< 1. (37) If we use the definition (37), we get (v)i = i∑ k=0 (−1)i−kS1(i, k)v k. (38) Theorem 7. The polynomials SLA [h] ϕ (u, v, w) have SLA [h] ϕ (u, v, w) = ϕ∑ γ=0 ( ϕ γ ) SLA [h] ϕ−ψ(u, 0, w) ψ∑ j=0 vjS1(ψ, j)h ψ−j , ϕ ≥ 0. (39) Proof. With the help of (16) and (37), we obtain ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! = γ(ξ)e v h log(1+hξ)(1 + hξ) −D−1 u h (1 + hξ2) D−1 w h = γ(ξ)(1 + hξ) −D−1 u h (1 + hξ2) D−1 w h ∞∑ j=0 (v h )j [log(1 + hξ)]j j! = ∞∑ ϕ=0 SLA [h] ϕ (u, 0, w) ξϕ ϕ! ∞∑ ψ=0 ψ∑ j=0 (v h )j S1(ψ, j)h ψ ξ ψ ψ! = ∞∑ ϕ=0  ϕ∑ ψ=0 ( ϕ ψ ) SLA [h] ϕ−ψ(u, 0, w) ψ∑ j=0 (v h )j S1(ψ, j)h ψ  ξϕ ϕ! . (40) If the coefficients of ϕ are equalized in the last equation above and then By matching the coefficients of identical powers of ϕ, we obtain (39). T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 9 of 18 Theorem 8. For ϕ ≥ 0, we have SLA [h] ϕ (u, 0, w) = ϕ∑ ψ=0 ( ϕ ψ ) SL [h] ϕ−ψ(u, v, w) ψ∑ j=0 ( −v h )j S1(ψ, j)h ψ. (41) Proof. From (16), we get γ(ξ)(1 + hξ) −D−1 u h (1 + hξ2) D−1 w h = e− v h log(1+hξ) ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! = ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! ∞∑ j=0 ( −v h )j [log(1 + hξ)]j j! = ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! ∞∑ ψ=0 ψ∑ j=0 ( −v h )j S1(ψ, j)h ψ ξ ψ ψ! = ∞∑ ϕ=0  ϕ∑ ψ=0 ( ϕ ψ ) SL [h] ϕ−ψ(u, v, w) ψ∑ j=0 ( −v h )j S1(ψ, j)h ψ  ξϕ ϕ! . (42) By matching the coefficients of identical powers of ϕ, we obtain (41). Theorem 9. The polynomials SLA [h] ϕ (u, v, w) have the following property for ϕ ≥ 0. SLA [h] ϕ (u, v, w) = ϕ∑ ψ=0 ψ∑ l=0 ( ϕ ψ ) (−h)ψSLA [h] ϕ−ψ(u, 0, w)(1) ψ−lS1(ψ, l) ( −v h )l . (43) Proof. From (16), we have ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! = γ(ξ)(1 + hξ) v h (1 + hξ) −D−1 u h (1 + hξ2) D−1 w h = ∞∑ ϕ=0 SLA [h] ϕ (u, 0, w) ξϕ ϕ! ∞∑ ψ=0 ( −v h ) ψ (−h)ψ ξ ψ ψ! = ∞∑ ϕ=0  ϕ∑ ψ=0 ( ϕ ψ )( −v h ) ψ (−h)ψSLA [h] ϕ−ψ(u, 0, w)  ξϕ ϕ! . (44) Comparing the coefficients of ϕ, we get SLA [h] ϕ (u, v, w) = ϕ∑ ψ=0 ( ϕ ψ )( −v h ) ψ (−h)ψSLA [h] ϕ−ψ(u, 0, w). (45) T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 10 of 18 Using the above equality (38), we get SLA [h] ϕ (u, v, w) = ϕ∑ ψ=0 ψ∑ l=0 ( ϕ ψ ) (−h)ψSLA [h] ϕ−ψ(u, 0, w)(1) ψ−lS1(ψ, l) ( −v h )l . (46) Theorem 10. For ϕ ≥ 0, the polynomials SLA [h] ϕ (u, v, w) have SLA [h] ϕ (u, s, w) = ϕ∑ l=0 l∑ j=0 ( ϕ l ) hlSLA [h] ϕ−l(u, v, w) ( s− v h )j S1(l, j). (47) Proof. From the generating relation (16), we reach γ(ξ)(1 + hξ) −D−1 u h (1 + hξ2) D−1 w h = e− v h log(1+hξ) ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! . (48) Replacing v by s and comparing the resulting equations, we get e s h log(1+hξ)(1 + hξ) −D−1 u h (1 + ht2) D−1 w h = e x−v h log(1+hξ) ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! . By using equations (16) and (37) in the the above equation, we get ∞∑ ϕ=0 SL [h] ϕ (u, s, w) ξϕ ϕ! = ∞∑ ϕ=0 SL [h] ϕ (u, v, w) ξϕ ϕ! ∞∑ j=0 ( s− v h )j [log(1 + hξ)]j j! ∞∑ ϕ=0 SLA [h] ϕ (u, s, w) ξϕ ϕ! = ∞∑ ϕ=0 ϕ∑ l=0 l∑ j=0 ( ϕ l ) hlSLA [h] ϕ−l(u, v, w) ( s− v h )j S1(l, j) ξϕ ϕ! . Finally, assertion (47) is obtained by equating the coefficients corresponding to identical powers of ϕ. 4. Algebraic characteristics Introduced by Steffenson [17] through poweroids and later extended by Dattoli [18, 19], monomiality plays a crucial role in special polynomials. The Ĵ and K̂ operators, as mul- tiplicative and differential operators, further refine polynomial structures, deepening their mathematical significance. The monomiality principle is a key concept in polynomial the- ory, stating that any polynomial can be uniquely expressed as a linear combination of T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 11 of 18 monomials—single-variable terms with non-negative integer exponents. This decomposi- tion simplifies polynomial analysis, aiding in the study of properties like degree, leading coefficient, and roots while enabling advanced mathematical techniques. Beyond theory, the monomiality principle enhances computational methods in inter- polation, approximation, and integration. Its adaptability extends to physics, where poly- nomials model fundamental laws and phenomena. The operators satisfy: Λk+1(λ) = Ĵ {Λk(λ)}, (49) k Λk−1(λ) = K̂{Λk(λ)}. (50) The set {Λk(λ)} forms a quasi-monomial family under these actions. The associated commutator is: [K̂, Ĵ ] = 1̂, (51) indicating Weyl algebra structure. Assuming quasi-monomiality, the following identities hold: (i) Differential equation: Ĵ K̂{Λk(λ)} = k Λk(λ). (52) (ii) Explicit representation: Λk(λ) = Ĵ k{1}, Λ0(λ) = 1. (53) (iii) Generating function: ewĴ {1} = ∞∑ k=0 Λk(λ) wk k! , |w| <∞. (54) These operator-based results support the monomiality framework relevant in physics and applied mathematics. This section affirms the monomiality of the three-variable ∆h Legendre-Laguerre Ap- pell polynomials SLA [h] ϕ (u, v, w), laying the groundwork for further structural analysis and applications. Theorem 11. The ∆h LeLAP SLA [h] ϕ (u, v, w) satisfy the succeeding operators: ˆMSLA = ( v −D−1 u 1 + v∆h + 2 D−1 w v∆h h+ v∆h 2 + γ ′ ( v∆h h ) γ( v∆h h ) ) (55) and ˆDSLR = v∆h h . (56) T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 12 of 18 Proof. To begin, we differentiate equation (16) with respect to v, making use of identity (11). This yields: v∆h { γ(ξ)(1+hξ) v h (1+hξ) D−1 u h (1+hξ2) D−1 w h } = (1+hξ) v+h h (1+hξ) D−1 u h (1+hξ2) D−1 w h − γ(ξ)(1 + hξ) v h (1 + hξ) D−1 u h (1 + hξ2) D−1 w h = hξ γ(ξ)(1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h , (57) which simplifies to the form: v∆h h [ γ(ξ)(1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h ] = ξ [ γ(ξ)(1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h ] , (58) which establishes the identity: v∆h h [ SLA [h] ϕ (u, v, w) ] = ξ [ SLA [h] ϕ (u, v, w) ] . (59) Now, we proceed by differentiating equation (16) with respect to ξ: ∂ ∂ξ { γ(ξ)(1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h } = ∂ ∂ξ { ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! } , (60) resulting in the expression:( v −D−1 u 1 + hξ +2 D−1 w ξ 1 + hξ2 + γ′(ξ) γ(ξ) ){ ∞∑ ϕ=0 SLA [h] ϕ (u, v, w) ξϕ ϕ! } = ∞∑ ϕ=0 ϕ SLA [h] ϕ (u, v, w) ξϕ ϕ! . (61) By employing identity (49) and making the substitution n → n+ 1 in the right-hand side of equation (61), the conclusion in (55) is validated. Moreover, utilizing expression (50), we derive the following: v∆h h [ SLA [h] ϕ (u, v, w) ] = ϕ SLA [h] ϕ−1(u, v, w), (62) which corresponds precisely to expression (56). Next, the differential equation for the polynomials SLA [h] ϕ (u, v, w) is derived. T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 13 of 18 Theorem 12. The ∆h LeLAP SLA [h] ϕ (u, v, w) satisfy the differential equation:( v −D−1 u 1 + v∆h + 2 D−1 w v∆h h+ v∆h 2 + γ ′ ( v∆h h ) γ( v∆h h ) − ϕh v∆h ) SLR[h] n (u, v, w) = 0. (63) Proof. Inserting expression (55) and (56) in the expression (52), the assertion (63) is proved. We now derive the determinant representation of the ∆h LeLAP SLA [h] ϕ (u, v, w) by establishing the following result: Theorem 13. The ∆h LeLAP SLA [h] ϕ (u, v, w) admit the determinant form given by: SLA [h] ϕ (u, v, w) = (−1)ϕ (γ0,h) ϕ+1 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 SL [h] 1 (u, v, w) SL [h] 2 (u, v, w) · · · SL [h] ϕ−1(u, v, w) SL [h] ϕ (u, v, w) γ0,h γ1,h γ2,h · · · γϕ−1,h γϕ,h 0 γ0,h ( 2 1 ) γ1,h · · · ( ϕ−1 1 ) γϕ−2,h ( ϕ 1 ) γϕ−1,h 0 0 γ0,h · · · ( ϕ−1 2 ) γϕ−3,h ( ϕ 2 ) γϕ−2,h . . . · · · . . . . . · · · . . 0 0 0 · · · γ0,h ( ϕ ϕ−1 ) γ1,h ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ , (64) where γϕ,h, ϕ = 0, 1, · · · being the coefficients of the Maclaurin series of 1 γ(ξ) . Proof. By multiplying expression (16) with 1 γ(ξ) = ∑∞ ϕ=0 γϕ,h ξϕ ϕ! on both sides, it follows ∞∑ ϕ=0 SL [h] ϕ (u, v, w) ξϕ ϕ! = ∞∑ ϕ=0 ∞∑ m=0 γm,h ξm m! SLA [h] ϕ (u, v, w) ξϕ ϕ! , (65) which in consideration of the well-know C.P. formulae gives SL [h] ϕ (u, v, w) = ϕ∑ m=0 ( ϕ m ) γm,h SLA [h] ϕ−m(u, v, w). (66) T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 14 of 18 5. Examples The Appell polynomial family, presented with the equality (13) at the beginning of our study, offers the opportunity to obtain a wide range of members by choosing an appropriate function γ(ξ). These members take the names of different polynomials and associated numbers as the appropriate function γ(ξ) changes. Thus, each member has a new generating function. Now, we give information on the generating function for these polynomials. For the “∆h Bernoulli polynomials β [h] ϕ (v), Euler polynomials E [h] ϕ (v) and Genocchi polynomials G [h] ϕ (v)” the generating relations are given by log(1 + hξ) 1 h (1 + hξ) 1 h − 1 (1 + hξ) v h = ∞∑ ϕ=0 β [h] ϕ (v) ξϕ ϕ! , | t |< 2π, (67) 2 (1 + hξ) 1 h + 1 (1 + hξ) v h = ∞∑ ϕ=0 E [h] ϕ (v) ξϕ ϕ! , | t |< π, (68) and 2 log(1 + hξ) 1 h (1 + hξ) 1 h + 1 (1 + hξ) v h = ∞∑ ϕ=0 G [h] ϕ (v) ξϕ ϕ! , | t |< π, (69) respectively. As h→ 0, the polynomials reduce to the Bernoulli Bϕ(v), Euler Eϕ(v), and Genocchi Λϕ(v) polynomials [20]. These polynomials, related to ∆h, are vital in number theory, combinatorics, and numerical analysis, aiding in problem-solving and formula derivation. Bernoulli numbers are key in Taylor expansions and number theory, Euler numbers in secant function expansions, and Genocchi numbers in graph theory and orthogonal polynomials. By choosing an appropriate γ(ξ) in equation 16, we derive “generating functions for the ∆h Legendre-Laguerre-based Bernoulli, Euler, and Genocchi polynomials”. log(1 + hξ) 1 h (1 + hξ) 1 h − 1 (1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h = ∞∑ ϕ=0 SLB [h] ϕ (u, v, w) ξϕ ϕ! , (70) 2 (1 + hξ) 1 h + 1 (1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h = ∞∑ ϕ=0 SLE [h] ϕ (u, v, w) ξϕ ϕ! , (71) and 2 log(1 + hξ) (1 + hξ) 1 h + 1 (1 + hξ) v−D−1 u h (1 + hξ2) D−1 w h = ∞∑ ϕ=0 SLG [h] ϕ (u, v, w) ξϕ ϕ! . (72) Furthermore, the polynomials SLB [h] ϕ (u, v, w), SLE [h] ϕ (u, v, w) and SLG [h] ϕ (u, v, w) satisfy the following explicit form in light of expression (28): T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 15 of 18 SLB [h] ϕ (u, v, w) = n∑ k=0 ( n k ) Bk,h SL [h] n−k(u, v, w), (73) SLE [h] ϕ (u, v, w) = n∑ k=0 ( n k ) Ek,h SL [h] n−k(u, v, w) (74) and SLG [h] ϕ (u, v, w) = n∑ k=0 ( n k ) Gk,h SL [h] n−k(u, v, w). (75) In view of expressions (64), the polynomials SLB [h] ϕ (u, v, w), SLE [h] ϕ (u, v, w) and SLG [h] ϕ (u, v, w) satisfy the following determinant representations: SLB [h] ϕ (u, v, w) = (−1)n ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 SL [h] 1 (u, v, w) SL [h] 2 (u, v, w) · · · SL [h] n−1(u, v, w) SL [h] n (u, v, w) γ0,h γ1,h γ2,h · · · γn−1,h γn,h 0 γ0,h ( 2 1 ) γ1,h · · · ( n−1 1 ) γn−2,h ( n 1 ) γn−1,h 0 0 γ0,h · · · ( n−1 2 ) γn−3,h ( n 2 ) γn−2,h . . . · · · . . . . . · · · . . 0 0 0 · · · γ0,h ( n n−1 ) γ1,h ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ , (76) SLE [h] ϕ (u, v, w) = (−1)n ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 SL [h] 1 (u, v, w) SL [h] 2 (u, v, w) · · · SL [h] n−1(u, v, w) SL [h] n (u, v, w) γ0,h γ1,h γ2,h · · · γn−1,h γn,h 0 γ0,h ( 2 1 ) γ1,h · · · ( n−1 1 ) γn−2,h ( n 1 ) γn−1,h 0 0 γ0,h · · · ( n−1 2 ) γn−3,h ( n 2 ) γn−2,h . . . · · · . . . . . · · · . . 0 0 0 · · · γ0,h ( n n−1 ) γ1,h ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ , (77) T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 16 of 18 and SLG [h] ϕ (u, v, w) = (−1)n ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 SL [h] 1 (u, v, w) SL [h] 2 (u, v, w) · · · SL [h] n−1(u, v, w) SL [h] n (u, v, w) γ0,h γ1,h γ2,h · · · γn−1,h γn,h 0 γ0,h ( 2 1 ) γ1,h · · · ( n−1 1 ) γn−2,h ( n 1 ) γn−1,h 0 0 γ0,h · · · ( n−1 2 ) γn−3,h ( n 2 ) γn−2,h . . . · · · . . . . . · · · . . 0 0 0 · · · γ0,h ( n n−1 ) γ1,h ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ . (78) 6. Conclusion In this study, the ∆h LeLAP were systematically introduced and explored through various mathematical frameworks, including the monomiality principle and operational rules. By deriving their generating function, explicit formulas, recurrence relations, and determinant form, we established a comprehensive foundation for their theoretical devel- opment. Furthermore, the connections between these polynomials and renowned families such as ∆h-Bernoulli, ∆h-Euler, and ∆h-Genocchi polynomials underscore their relevance in the broader mathematical landscape. These results contribute to a deeper understand- ing of polynomial structures and their interrelations, making them valuable tools for future mathematical investigations. Looking ahead, further research can focus on exploring the orthogonal properties of ∆h Legendre-Laguerre-Appell polynomials and their potential uses in numerical analy- sis, approximation theory, and differential equations. Additionally, investigating their q-analogues and extensions in multivariable settings could provide new insights into their algebraic and analytical properties. Another promising direction is their utilization in solv- ing integral transforms, fractional calculus problems, and mathematical physics equations, thereby broadening their scope in applied mathematics. Funding Not applicable. References [1] SAWani and S Khan. Properties and applications of the gould-hopper-frobenius-euler polynomials, tbilisi math. J, 12(1):93–104, 2019. T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 17 of 18 [2] W Ramı́rez and C Cesarano. Some new classes of degenerated generalized apostol- bernoulli, apostol-euler and apostol-genocchi polynomials. Carpathian Mathematical Publications, 14(2):354–363, 2022. [3] Mohra Zayed, Shahid AhmadWani, and Yamilet Quintana. Properties of multivariate hermite polynomials in correlation with frobenius–euler polynomials. Mathematics, 11(16):3439, 2023. [4] Shahid Ahmad Wani, Kinda Abuasbeh, Georgia Irina Oros, and Salma Trabelsi. Studies on special polynomials involving degenerate appell polynomials and fractional derivative. Symmetry, 15(4):840, 2023. [5] Shahid Ahmad Wani. Two-iterated degenerate appell polynomials: properties and applications. Arab Journal of Basic and Applied Sciences, 31(1):83–92, 2024. [6] G Dattoli, PE Ricci, C Cesarano, and L Vázquez. Special polynomials and fractional calculus. Mathematical and computer modelling, 37(7-8):729–733, 2003. [7] G Dattoli, S Lorenzutta, AM Mancho, and A Torre. Generalized polynomials and associated operational identities. Journal of computational and applied mathematics, 108(1-2):209–218, 1999. [8] Rashad A. Al-Jawfi, Abdulghani Muhyi, and Wadia Faid Hassan Al-shameri. On generalized class of bell polynomials associated with geometric applications. Axioms, 13(2), 2024. [9] Rashad A. Al-Jawfi, Abdulghani Muhyi, and Wadia Faid Hassan Al-shameri. A new family of appell-type changhee polynomials with geometric applications. Axioms, 13(2), 2024. [10] Subuhi Khan, Mumtaz Riyasat, and Shahid Ahmad Wani. On some classes of differ- ential equations and associated integral equations for the laguerre–appell polynomials. Advances in Pure and Applied Mathematics, 9(3):185–194, 2018. [11] Giuseppe Dattoli, Paolo E Ricci, and Clemente Cesarano. A note on legendre poly- nomials. International Journal of Nonlinear Sciences and Numerical Simulation, 2(4):365–370, 2001. [12] Richard A Silverman et al. Special functions and their applications. Courier Corpo- ration, 1972. [13] Francesco A Costabile and Elisabetta Longo. δ h-appell sequences and related inter- polation problem. Numerical Algorithms, 63:165–186, 2013. [14] Mumtaz Riyasat, Amal S Alali, and Subuhi Khan. Certain properties of 3d degenerate generalized fubini polynomials and applications. Afrika Matematika, 35(2):47, 2024. [15] Ibtehal Alazman, Badr Saad T Alkahtani, and Shahid Ahmad Wani. Certain prop- erties of δ h multi-variate hermite polynomials. Symmetry, 15(4):839, 2023. [16] R Alyusof and SA Wani. Certain properties and applications of deltah hybrid special polynomials associated with appell sequences, fractal fract., 7 (2023), 233. [17] JF5953 Steffensen. The poweroid, an extension of the mathematical notion of power. 1941. [18] G Dattoli. Generalized polynomials, operational identities and their applications. Journal of Computational and Applied mathematics, 118(1-2):111–123, 2000. [19] G Dattoli. Hermite-bessel and laguerre-bessel functions: A by-product ot the mono- T. Alqurashi et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6410 18 of 18 miality principle, advanced special functions and applications. Proceedings of the Melfi School on Advanced Topics in Mathematics and Physics, pages 147–164. [20] L Carlitz. Eulerian numbers and polynomials. Math. Mag., 32(4):247–260, 1959.