EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6657 ISSN 1307-5543 – ejpam.com Published by New York Business Global Certain Properties and Characterizations of ∆h-Truncated Exponential Based Hermite Polynomials Haitham Qawaqneh 1, Waseem Ahmad Khan 2, Hassen Aydi3,4,∗, Shahid Ahmad Wani5, Prakash Jadhav6 1 Department of Mathematics, 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 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 5 Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed Uni- versity) (SIU), Pune, India 6 Department of Mechanical Engineering, SRM University AP, Andhra Pradesh 522240, India Abstract. This article introduces a novel class of ∆h-truncated exponential-based Hermite poly- nomials and examine their fundamental properties and structural identities. We derive generating functions, recurrence relations, and explicit formulas, along with summation identities. The study further uncovers connections with the monomiality principle, offering insights into their underlying algebraic framework. In addition, an operational formalism is developed, and symmetric identities are established to enhance the theoretical foundation of these polynomials. 2020 Mathematics Subject Classifications: 33C45, 33E20, 33B10, 33E30, 11T23. Key Words and Phrases: Monomiality principle, ∆h-truncated exponential Hermite Appell polynomials, explicit forms, Symmetry identities. 1. Introduction and preliminaries Polynomial families form a foundational pillar in applied mathematics, given their multifaceted characterizations—ranging from orthogonality and generating functions to differential expressions, operational techniques, integral representations, and recurrence relations. Due to their versatile nature and diverse applications, their generalizations ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6657 Email addresses: h.alqawaqneh@zuj.edu.jo (H. Qawaqneh), wkhan1@pmu.edu.sa (W. A. Khan), hassen.aydi@isima.rnu.tn (H. Aydi), shahidwani177@gmail.com (S. A. Wani), Prakash.j@srmap.edu.in (P. Jadhav) 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), 6657 2 of 16 and extensions continue to garner significant attention across mathematical and phys- ical sciences. These formulations are vital not only in providing series expansions for transcendental functions in mathematical physics but also in shaping computational and analytical techniques. Notable studies in this domain include extensive developments in p-adic analysis, q-analysis, and umbral calculus (see, for example, [1–5]). Among these developments, two-variable special polynomials have emerged as highly potent tools, facilitating the derivation of efficient and elegant identities and offering path- ways to novel classes of special polynomials. The inception of two-variable Appell poly- nomials by Bretti et al. [1] via iterated isomorphism marked a pivotal moment, followed by the construction and study of two-variable truncated exponential, Hermite, Legendre, and Laguerre polynomials in works such as [6–13]. Despite their relevance in areas such as quantum mechanics and optics, the TEP (TEP) remains relatively underexplored. Initially defined by Andrews [14], these polynomials are given by: en(ξ1) = n∑ k=0 ξk1 k! , (1) with the limiting behavior lim n→∞ en(ξ1) = eξ1 . A detailed investigation of their properties was later initiated by Dattoli et al. [8]. A key identity for the TEP follows from their integral representation: en(ξ1) = 1 n! ∫ ∞ 0 e−ξ(ξ1 + ξ)n dξ, (2) derived using the classical gamma integral: n! = ∫ ∞ 0 e−ξ ξn dξ. (3) The ordinary generating function of en(µ1) is expressed as [8]: ∞∑ n=0 en(ξ1)t n = eξ1t 1− t (t ∈ C, |t| < 1) . (4) A significant extension of the TEP to two variables was established by Dattoli et al. [8], where they proved particularly useful in problems involving integrals of special functions and physical models. The generating function for the two-variable version is: ∞∑ n=0 [2]en(ξ1, µ2)t n = eξ1t 1− ξ2t2 , (5) and the corresponding explicit representation reads: [2]en(ξ1, ξ2) = [n2 ]∑ k=0 ξk2 ξn−2k 1 (n− 2k)! . (6) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 3 of 16 The generalized form involving higher-order polynomials is given by: ∞∑ n=0 [s]en(ξ1, ξ2)t n = eξ1t 1− ξ2ts , (7) with explicit form: [s]en(ξ1, ξ2) = [ns ]∑ k=0 ξk2 ξn−sk 1 (n− sk)! . (8) It is easy to verify from expressions (4), (5), and (7) that: [2]en(ξ1, ξ2) = e(2)n (ξ1, ξ2), en(ξ1) = e(1)n (ξ1, 1). A direct consequence of this formalism links the Chebyshev polynomials of the second kind Un(µ2) to the TEP: Un(ξ2) = [2]en(0, ξ2), (9) whose generating function is well-known [14]: ∞∑ n=0 Un(ξ1)t n = 1 1− 2ξ1t+ t2 , (|t| < 1, ξ1 ≤ 1) . (10) In the operational calculus framework, the multiplicative and derivative operators for the TEP are identified as: M̂e(s) = ξ1 + sξ2Dξ2ξ2D s−1 ξ1 , (11) P̂e(s) = Dξ2 , (12) signifying that [s]en(ξ1, ξ2) form a quasi-monomial sequence [2]. This formalism has been further expanded by composing TEP with Appell-type struc- tures. Khan [15] introduced the truncated exponential-based Appell polynomials through: ∞∑ n=0 [s]en(ξ1, ξ2)t n = A(t) eξ1t 1− ξ2ts , (13) where A(t) denotes the Appell-type generating function. The origin of the monomiality principle dates back to Steffenson’s poweroid method in 1941 [16], later refined by Dattoli [7]. A polynomial sequence {qn(ξ1)} is quasi-monomial if: qn+1(ξ1) = M̂{qn(ξ1)}, (14) n qn−1(ξ1) = P̂{qn(ξ1)}, (15) and the operators satisfy the Weyl algebra: [P̂,M̂] = 1̂. (16) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 4 of 16 The quasi-monomial property yields key operational identities: M̂P̂{qn(ξ1)} = n qn(ξ1), (17) qn(ξ1) = M̂n{1}, (18) etM̂{1} = ∞∑ n=0 qn(ξ1) tn n! , (19) as outlined in [2, 7, 8, 17–19]. In recent years, the introduction of ∆h-type generalizations has expanded the horizon of special polynomial theory. These constructions utilize the forward difference operator: ∆h[g](ξ1) = g(ξ1 + h)− g(ξ1), (20) and its higher-order form: ∆i h[g](ξ1) = i∑ l=0 (−1)i−l ( i l ) g(ξ1 + lh), (21) where ∆0 h = I (identity), ∆1 h = ∆h. The ∆h-Appell polynomials A[h] n (ξ1) are introduced via the generating function [20]: A(h; t)(1 + ht) ξ1 h = ∞∑ n=0 A[h] n (ξ1) tn n! , (22) with the condition: A(h; t) = ∞∑ n=0 An,h tn n! , A0,h ̸= 0. (23) The Stirling numbers of the first kind, S1(n,m), are crucial in expressing rising facto- rials, as given by the relation: (ξ1)n = n∑ m=0 S1(n,m)µm 1 , (24) where (ξ1)0 = 1 and (ξ1)n = ξ(ξ1 − 1) · · · (ξ1 − n+ 1). The Stirling numbers S1(n,m) can be represented by the following generating function [21–23]: 1 m! (log(1 + t))m = ∞∑ n=m S1(n,m) tn n! , (m ≥ 0). (25) For n ≥ 0, the ∆h Stirling numbers of the first kind are defined by: 1 k! (logh(1 + t))k = ∞∑ n=k S1,h(n, k) tn n! , (k ≥ 0). (26) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 5 of 16 It is important to note that limh→0 S1,h(n, k) = S1(n, k). The degenerate Hermite polynomials are defined by [24] (1 + ht) ξ1 h (1 + ht2) ξ2 h = ∞∑ n=0 H[h] n (ξ1, ξ2) tn n! . (27) Note that lim h→0 H[h] n (ξ1, ξ2) = Hn(ξ1, ξ2), where Hn(ξ1, ξ2) is called the 2-variable Hermite polynomials (see [7]). A generalized falling factorial sum σk(n;h) can be defined by the generating function [24–26]: (1 + ht) (n+1) h − 1 (1 + ht) 1 h − 1 = ∞∑ k=0 σk(n;h) tk k! . (28) Note that limh→0 σk(n;h) = Sk(n). This article is organized as follows. In Section 2, we introduce the novel class of ∆h- truncated exponential-based Hermite polynomials and derive their generating functions, recurrence relations, and explicit formulas. Section 3 is devoted to the derivation of sum- mation identities related to these polynomials. In Section 4, we explore their connection with the monomiality principle and develop an operational formalism that highlights their algebraic structure. Section 5 presents symmetric identities to further enrich the theoret- ical framework. Finally, concluding remarks are provided. 2. ∆h-Truncated exponential-based Hermite polynomials This section introduces a new family of three-variable ∆h-truncated exponential-based Hermite polynomials and examines their foundational properties. It significantly enhances the current understanding of polynomial theory and suggests promising directions for further study. The derivation of the generating function for e(r)H [h] n (ξ1, ξ2, ξ3) plays a pivotal role in revealing the structure and analytical behavior of these polynomials. This construction not only aids in uncovering key identities and recurrence relations but also enriches their connection to broader mathematical frameworks. To initiate this, we derive the generating function for e(r)H [h] n (ξ1, ξ2, ξ3) by establishing the following result: Theorem 1. The generating function associated with the three-variable ∆h-truncated exponential-based Hermite polynomials e(r)H [h] n (ξ1, ξ2, ξ3) is expressed as follows: 1 1− ξ3tr (1 + ht) ξ1 h (1 + ht2) ξ2 h = ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! . (29) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 6 of 16 Proof. Consider the expansion of 1 1−ξ3tr (1 + ht) ξ1 h (1 + ht2) ξ2 h around ξ1 = ξ2 = ξ3 = 0 using a Newton series for finite differences. By analyzing the product expansion of the functions (1+ht) ξ1 h and (1+ht2) ξ2 h with respect to powers of t, we identify the coefficients of tn n! as the polynomials e(r)H [h] n (ξ1, ξ2, ξ3), which are defined in equation (29). This confirms the generating function for the three-variable ∆h-truncated exponential-based Hermite polynomials e(r)H [h] n (ξ1, ξ2, ξ3). Theorem 2. For the three-variable ∆h-Truncated exponential-based Hermite polynomials e(r)H [h] n (ξ1, ξ2, ξ3), the following relations hold: ξ1∆h h e(r)H [h] n (ξ1, ξ2, ξ3) = n e(r)H [h] n−1(ξ1, ξ2, ξ3) ξ2∆h h e(r)H [h] n (ξ1, ξ2, ξ3) = n(n− 1) e(r)H [h] n−2(ξ1, ξ2, ξ3). (30) Proof. Differentiating equation (29) with respect to ξ1 and applying expression (20), we get ξ1∆h 1 1− ξ3tr (1+ht) ξ1 h (1+ht2) ξ2 h = 1 1− ξ3tr (1+ht) ξ1+1 h (1+ht2) ξ2 h − 1 1− ξ3tr (1+ht) ξ1 h (1+ht2) ξ2 h = (1 + ht− 1) 1 1− ξ3tr (1 + ht) ξ1 h (1 + ht2) ξ2 h = ht 1 1− ξ3tr (1 + ht) ξ1 h (1 + ht2) ξ2 h . (31) Substituting the right-hand side of equation (29) into (31), we obtain ξ1∆h ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! = h ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn+1 n! . (32) Now, replacing n → n− 1 on the right-hand side of the above expression and comparing the coefficients of like powers of t, we deduce the identities in (30). We now derive an explicit expression satisfied by the three-variable ∆h-Truncated exponential-based Hermite polynomials e(r)H [h] n (ξ1, ξ2, ξ3) as follows: Theorem 3. For the three-variable ∆h-Truncated exponential-based Hermite polynomials e(r)H [h] n (ξ1, ξ2, ξ3), the following expression holds: H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 7 of 16 e(r)H [h] n (ξ1, ξ2, ξ3) = [ ξ1 h ]∑ d=0 ( n d )( ξ1 h d ) hd e(r)H [h] n−d(0, ξ2, ξ3). (33) Proof. Consider the expansion of the generating function (29): 1 1− ξ3tr (1+ht) ξ1 h (1+ht2) ξ2 h {1} = [ ξ1 h ]∑ d=0 ( ξ1 h d ) (ht)d d! ∞∑ ϕ=0 e(r)H [h] n (0, ξ2, ξ3) tn n! (34) This leads to the equivalent form: ∞∑ ϕ=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! = ∞∑ n=0 [ ξ1 h ]∑ d=0 ( ξ1 h d ) hd e(r)H [h] n (0, ξ2, ξ3) tn+d n! d! . (35) Now, replacing n → n− d on the right-hand side of the above expression, we get: ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! = ∞∑ n=0 [ ξ1 h ]∑ d=0 ( ξ1 h d ) hd e(r)H [h] n−d(0, ξ2, ξ3) tn (n− d)! d! . (36) Multiplying and dividing the right-hand side of equation (36) by n! and then comparing coefficients of like powers of t on both sides yields the required identity in (33). Theorem 4. For the three-variable ∆h-Truncated exponential-based Hermite polynomials e(r)H [h] n (ξ1, ξ2, ξ3), the following series representations hold: e(r)H [h] n (ξ1, ξ2, ξ3) = n! [n 2 ]∑ k=0 (− ξ2 h )k(−h)ke (r) n−2k(ξ1, ξ3;h) k!(n− 2k)! , (37) and e(r)H [h] n (ξ1, ξ2, ξ3) = n! [n r ]∑ k=0 ξk3H [h] n−rk(ξ1, ξ2) (n− rk)! . (38) Proof. By utilizing equations (7), (27), and (29), the results in (37) and (38) follow directly. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 8 of 16 3. Summation formulae In this section, we present concise summation formulae for special two-variable poly- nomials. These expressions offer efficient methods to compute sums and explore structural properties, aiding in analysis across combinatorics, probability, and mathematical physics. We now establish the following results. Theorem 5. Let e(r)H [h] n (ξ1, ξ2, ξ3) denote the ∆h-Truncated exponential-based Hermite polynomials of order r. Then, the following identity holds: e(r)H [h] n (ξ1 + 1, ξ2, ξ3) = n∑ k=0 ( n k ) e(r)H [h] n−k(ξ1, ξ2, ξ3)(− 1 h )k(−h)k. (39) Proof. Utilizing the generating function (29), we proceed as follows: ∞∑ n=0 e(r)H [h] n (ξ1+1, ξ2, ξ3) tn n! − ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! = 1 1− ξ3tr (1+ht) ξ1 h (1+ht2) ξ2 h ( (1 + ht) 1 h − 1 ) = ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! ( ∞∑ k=0 (−1 h )k(−h)k tk k! − 1 ) = ∞∑ n=0 n∑ k=0 ( n k ) e(r)H [h] n−k(ξ1, ξ2, ξ3)(− 1 h )k(−h)k tn n! − ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! (40) By equating the coefficients of powers of t on both sides, the identity (39) is established. Next, we derive the explicit expressions satisfied by the bivariate ∆h-Truncated exponential- based Hermite polynomials e(r)H [h] n (ξ1, ξ2, ξ3) of order r, through the following result: Theorem 6. For the three-variable ∆h-Truncated exponential-based Hermite polynomials e(r)H [h] n (ξ1, ξ2, ξ3) of order r, the following structural identities are valid: e(r)H [h] n (ξ1, ξ2, ξ3) = n∑ k=0 ( n k ) e(r)H [h] n−k(0, ξ2, ξ3) ( −ξ1 h ) k (−h)k, (41) e(r)H [h] n (ξ1, ξ2, ξ3) = n∑ k=0 ( n k ) e(r)H [h] n−k(ξ1 − p, ξ2, ξ3) ( −p h ) k (−h)k, (42) e(r)H [h] n (ξ1 + s, ξ2, ξ3) = n∑ k=0 ( n k ) e(r)H [h] n−k(ξ1, ξ2, ξ3) ( − s h ) k (−h)k, (43) e(r)H [h] n (ξ1, ξ2, ξ3) = n∑ l=0 l∑ k=0 ( n l ) e(r)H [h] n−l(0, ξ2, ξ3) ( ξ1 h )k hlS1(l, k), (44) e(r)H [h] n (ξ1, ξ2, ξ3) = n∑ l=0 l∑ k=0 ( n l ) e(r)H [h] n−l(0, ξ2, ξ3) ( ξ1 h ) k,h hlS1,h(l, k). (45) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 9 of 16 Proof. We expand the generating function (29) in the following way:( ∞∑ n=0 e(r)H [h] n (0, ξ2, ξ3) tn n! )( ∞∑ k=0 ( −ξ1 h ) k (−h)k tk k! ) = ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! , Applying the Cauchy product on the left-hand side and comparing like powers of t, equa- tion (39) is validated. In a similar fashion, identities (40) and (41) follow directly. Furthermore, from equation (29), we have: ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! = 1 1− ξ3tr elog((1+ht) ξ1 h )(1 + ht2) ξ2 h . (46) Employing equation (1), the expression becomes: ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! = ( ∞∑ n=0 e(r)H [h] n (0, ξ2, ξ3) tn n! )( ∞∑ k=0 ( ξ1 h )k log(1 + ht)k k! ) , (47) Substituting equation (25) leads to: ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! = ( ∞∑ n=0 e(r)H [h] n (0, ξ2, ξ3) tn n! )( ∞∑ k=0 ( ξ1 h )k ∞∑ l=k S1(l, k)h l t l l! ) ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! = ( ∞∑ n=0 e(r)H [h] n (0, ξ2, ξ3) tn n! )( ∞∑ l=0 l∑ k=0 ( ξ1 h )k S1(l, k)h l t l l! ) . (48) By replacing n with n− l on the right-hand side and comparing coefficients of tn, identity (44) is obtained. Now, using (26) along with (29), we have: ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! = 1 1− ξ3tr (1 + ht2) ξ2 h e logh((1+ht) ξ1 h ) h . (49) ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! = ( ∞∑ n=0 erH[h] n (0, ξ2, ξ3) tn n! )( ∞∑ k=0 ( ξ1 h ) k,h logh(1 + ht)k k! ) , (50) ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! = ( ∞∑ n=0 e(r)H [h] n (0, ξ2, ξ3) tn n! )( ∞∑ l=0 l∑ k=0 ( ξ1 h ) k,h S1,h(l, k)h l t l l! ) . (51) Substituting n by n− l and comparing corresponding coefficients of t, we confirm identity (45). H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 10 of 16 4. Monomiality Principle The monomiality principle is a core concept in polynomial theory. It states that any polynomial can be written uniquely as a linear combination of monomials (powers of a variable). This form simplifies analysis and helps extract key properties like degree and roots. Monomiality representations are widely used in computations such as interpolation, approximation, and integration. They also appear in physics, control theory, and signal processing, where polynomials model complex systems. The principle was introduced via poweroids by Steffensen in 1941 [16], and later ex- tended by Dattoli [27, 28]. These methods, grounded in mathematical physics, quantum mechanics, and optics, remain vital tools in modern research. In this section, we validate the monomiality principle for the three-variable ∆h-Truncated exponential-based Hermite polynomials, denoted e(r)H [h] n (ξ1, ξ2, ξ3). We confirm this by es- tablishing the following results. Theorem 7. The ∆h-Truncated exponential-based Hermite polynomials e(r)H [h] n (ξ1, ξ2, ξ3) satisfy the following multiplicative and derivative operators: M̂ e(r) H[h] n (ξ1,ξ2,ξ3) = ( ξ1 1 + ξ1∆h + rξ3Dξ3ξ3 ξ1∆h r−1 h + 2nξ2h h+ ξ1∆h 2 ) , (52) and P̂ e(r) H[h] n (ξ1,ξ2,ξ3) = ξ1∆h h . (53) Proof. By differentiating equation (29) with respect to ξ1 and using the identity (12), we get: ξ1∆h { 1 1− ξ3tr (1 + ht) ξ1 h (1 + ht2) ξ2 h } = 1 1− ξ3tr (1 + ht) ξ1+h h (1 + ht2) ξ2 h − 1 1− ξ3tr (1 + ht) ξ1 h (1 + ht2) ξ2 h = (1 + ht− 1) 1 1− ξ3tr (1 + ht) ξ1 h (1 + ht2) ξ2 h = ht 1 1− ξ3tr (1 + ht) ξ1 h (1 + ht2) ξ2 h , (54) which leads to the identity: ξ1∆h h [ ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! ] = t [ ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! ] . (55) Next, differentiating equation (29) with respect to t, we obtain: ∂ ∂t { 1 1− ξ3tr (1 + ht) ξ1 h (1 + ht2) ξ2 h } = ∂ ∂t { ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! } (56) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 11 of 16( ξ1 1 + ht +rξ3Dξ3ξ3t r−1+ 2nξ2 1 + ht2 ){ ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! } = n ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn−1 n! . (57) Applying identity (29) and shifting n → n + 1 on the right-hand side of (57), we derive the operator formula (52). Moreover, from identities (15) and (55), it follows: ξ1∆h h [ ∞∑ n=0 e(r)H [h] n (ξ1, ξ2, ξ3) tn n! ] = n [ ∞∑ n=0 e(r)H [h] n−1(ξ1, ξ2, ξ3) tn n! ] , (58) yielding the derivative operator form in (53). Now, we derive the differential equation satisfied by the ∆h-Truncated exponential- based Hermite polynomials erH [h] n (ξ1, ξ2, ξ3) through the following result: Theorem 8. The ∆h-Truncated exponential-based Hermite polynomials e(r)H [h] n (ξ1, ξ2, ξ3) obey the differential equation:( ξ1 1 + ξ1∆h + rξ3Dξ3ξ3 ξ1∆h r−1 h + 2nξ2h h+ ξ1∆h 2 − nh ξ1∆h ) e(r)H [h] n (ξ1, ξ2, ξ3) = 0. (59) Proof. Substituting the operator forms (52) and (53) into the identity (17), we get:( ξ1 1 + ξ1∆h + rξ3Dξ3ξ3 ξ1∆h r−1 h + 2nξ2h h+ ξ1∆h 2 ) ξ1∆h h e(r)H [h] n (ξ1, ξ2, ξ3) = ne(r)H [h] n (ξ1, ξ2, ξ3). (60) Simplifying the above yields the claimed result (59). We now establish the following operational formula involving e(r)H [h] n (ξ1, ξ2, ξ3): Theorem 9. The following operational relation holds between the ∆h-Truncated exponential- based Hermite polynomials e(r)H [h] n (ξ1, ξ2, ξ3) and the ∆h-Hermite polynomials H[h] n (ξ1, ξ2): e(r)H [h] n (ξ1, ξ2, ξ3) = exp(rξ3Dξ3ξ3 ξ1∆h r h ) { H[h] n (ξ1, ξ2) } . (61) Proof. Using equations (27) and (58), and applying identity (29), the result follows immediately. 5. Symmetric identities In this Section, we investigate symmetric identities inherent to the three-variable ∆h special polynomials. These identities unveil intriguing relationships between the variables and coefficients within the polynomials, shedding light on their underlying symmetrical properties. By exploring how the polynomials behave under transformations that inter- change the variables or coefficients, we uncover profound connections that extend beyond H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 12 of 16 their initial definitions. These symmetric identities not only deepen our understanding of the polynomials themselves but also offer valuable insights into broader mathematical structures and phenomena. Through systematic examination and rigorous derivation, we establish a comprehensive framework for understanding and exploiting the symmetrical properties of these two-variable special polynomials, paving the way for further advance- ments in both theoretical analyses and practical applications. Theorem 10. For a ̸= b, a, b > 0 and xi1, ξ2, ν1, ν2, ϕ1, ϕ2 ∈ C, we have n∑ γ=0 ( n γ ) an−γbγe(r)H [h] n−γ(aξ1, aν1, aϕ1)e(r)H [h] γ (bξ2, bν2, bϕ2) = n∑ γ=0 ( n γ ) aγbn−γ e(r)H [h] n−γ(aξ2, aν2, aϕ2)e(r)H [h] γ (bξ1, bν1, bϕ1). (62) Proof. Let A(t) = 1 1− ϕ1(abt)2 1 1− ϕ2(abt)2 (1 + ht) ab(ξ1+ξ2) h (1 + ht2) ab(ν1+ν2) h (63) = ∞∑ n=0 e(r)H [h] γ (bξ1, bν1, bϕ1) (bt)γ γ! ∞∑ n=0 e(r)H [h] n (aξ2, aν2, aϕ2) (at)n n! = ∞∑ n=0  n∑ γ=0 ( n γ ) an−γbγe(r)H [h] n−γ(aξ1, aν1, aϕ1)e(r)H [h] γ (bξ2, bν2, bϕ2)  tn n! . (64) Similarly, we have A(t) = ∞∑ ϕ=0  n∑ γ=0 ( n γ ) aγbn−γ e(r)H [h] n−γ(aξ2, aν2, aϕ2)e(r)H [h] γ (bξ1, bν1, bϕ1)  tn n! . (65) Comparing the coefficients of t on both sides of last equations, we get (62). Theorem 11. For a ̸= b, a, b > 0 and ξ, ν, ϕ ∈ C, we have n∑ k=0 k∑ γ=0 ( n k )( k γ ) an−γbγ+1βn−k(h)e(r)H [h] k−γ(bξ, bν, bϕ)σγ(a− 1;h) = n∑ k=0 k∑ γ=0 ( n k )( k γ ) bn−γaγ+1βn−k(h)e(r)H [h] k−γ(aξ, aν, aϕ)σγ(b− 1;h). (66) H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 13 of 16 Proof. Consider B(t) = 1 1− ϕ(abt)2 (1 + ht) abξ h (1 + ht2) abν h ((1 + ht) ab h − 1) ((1 + ht) a h − 1)((1 + ht) b h − 1)2 = abt ((1 + ht) a h − 1) 1 1− ϕ(abt)2 (1 + ht) abξ h (1 + ht2) abν h ((1 + ht) ab h − 1) ((1 + ht) b h − 1) = b ∞∑ n=0 βn(h) (at)n n! ∞∑ k=0 e(r)H [h] k (bξ, bν, bϕ) (at)k k! ∞∑ γ=0 σγ(a− 1;h) (bt)γ γ! = b ∞∑ n=0 βn(h) (at)n n! ∞∑ k=0 k∑ γ=0 ( k γ ) ak−γbγe(r)H [h] k−γ(bξ, bν, bϕ)σγ(a− 1;h) tk k! = ∞∑ n=0  n∑ k=0 k∑ γ=0 ( n k )( k γ ) an−γbγ+1βn−k(h)e(r)H [h] k−γ(bξ, bν, bϕ)σγ(a− 1;h)  tn n! . (67) Similarly, we have B(t) = ∞∑ n=0  n∑ k=0 k∑ γ=0 ( n k )( k γ ) bn−γaγ+1βn−k(h)e(r)H [h] k−γ(aξ, aν, aϕ)σγ(b− 1;h)  tn n! . (68) Comparing the coefficients of t on both sides of last equations, we get (66). 6. Conclusion In this study, we have introduced a novel class of ∆h-truncated exponential-based Hermite polynomials and established their fundamental properties, including generating functions, recurrence relations, explicit formulas, and summation identities. The connec- tion with the monomiality principle has been explored to reveal their underlying algebraic structure, and an operational formalism has been developed. Additionally, symmetric identities have been presented to further deepen the theoretical understanding of these polynomials. These results lay a solid foundation for future investigations and potential applications in both pure and applied mathematics. Future research can explore several directions, including the development of q-analogues and degenerate forms of the proposed polynomials to study associated q-difference equa- tions and limiting behaviors. Investigating orthogonality conditions and suitable weight functions will help identify inner product spaces where these polynomials are orthogo- nal. Their application in interpolation, approximation theory, and spectral methods also warrants attention, particularly in solving differential or integral equations. Potential uses in mathematical physics and engineering—such as quantum systems and signal anal- ysis—highlight their applied significance. Additionally, a detailed study of asymptotic properties and zero distributions using analytic and numerical tools could offer deeper insights into their structural behavior. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 14 of 16 Availability of data and materials Not applicable. Competing interests The authors declare no competing interests. Acknowledgments The authors acknowledge the financial support from Al-Zaytoonah University of Jor- dan, Amman 11733, Jordan. References [1] Gabriella Bretti, C. Cesarano, and Paolo Emilio Ricci. Laguerre-type exponentials and generalized appell polynomials. Computers & Mathematics with Applications, 48(5-6):833–839, 2004. [2] H. M. Srivastava, Serkan Araci, Waseem A. Khan, and Mehmet Acikgoz. A note on the truncated-exponential based apostol-type polynomials. Symmetry, 11(4):538, 2019. [3] Noor Alam, Shahid Ahmad Wani, Waseem Ahmad Khan, and Hasan Nihal Zaidi. Investigating the properties and dynamic applications of δh legendre–appell polyno- mials. Mathematics, 12(13):1973, 2024. [4] Noor Alam, Shahid Ahmad Wani, Waseem Ahmad Khan, Fakhredine Gassem, and Anas Altaleb. Exploring properties and applications of laguerre special polynomials involving the δh form. Symmetry, 16(9):1154, 2024. [5] Waseem Ahmad Khan and Maryam Salem Alatawi. A note on modified degenerate changhee–genocchi polynomials of the second kind. Symmetry, 15(1):136, 2023. [6] Paul Appell and Joseph Kampé de Fériet. Fonctions Hypergéométriques et Hyper- sphériques: Polynômes d’Hermite. Gauthier-Villars, 1926. [7] G. Dattoli. Hermite-bessel and laguerre-bessel functions: A by-product of the mono- miality principle. Proceedings of the Melfi School on Advanced Topics in Mathematics and Physics, pages 147–164. [8] Giuseppe Dattoli, Clemente Cesarano, and Dario Sacchetti. A note on truncated polynomials. Applied Mathematics and Computation, 134(2-3):595–605, 2003. [9] G. Dattoli, S. Lorenzutta, A. M. Mancho, and A. Torre. Generalized polynomials and associated operational identities. Journal of Computational and Applied Mathematics, 108(1-2):209–218, 1999. [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(2), 2023. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 15 of 16 [11] Duha Abu Judeh and M. Abu Hammad. Applications of conformable fractional pareto probability distribution. International Journal of Advances in Soft Computing and Applications, 14(2):115–124, 2022. [12] Haitham Qawaqneh, Mohd Salmi Md Noorani, Hassen Aydi, Amjed Zraiqat, and Arslan Hojat Ansari. On fixed point results in partial b-metric spaces. Journal of Function Spaces, 2021(1):8769190, 2021. [13] Haitham Qawaqneh, Mohd Salmi Md Noorani, and Hassen Aydi. Some new character- izations and results for fuzzy contractions in fuzzy b-metric spaces and applications. AIMS Mathematics, 8(3):6682–6696, 2023. [14] Larry C. Andrews. Special Functions of Mathematics for Engineers, volume 49. SPIE Press, 1998. [15] Subuhi Khan, Ghazala Yasmin, and Naeem Ahmad. A note on truncated exponential- based appell polynomials. Bulletin of the Malaysian Mathematical Sciences Society, 40:373–388, 2017. [16] J. Steffensen. The poweroid, an extension of the mathematical notion of power. 1941. [17] Ibtehal Alazman, Badr Saad T. Alkahtani, and Shahid Ahmad Wani. Certain prop- erties of δh multi-variate hermite polynomials. Symmetry, 15(4):839, 2023. [18] Shahid Ahmad Wani, Waseem Ahmad Khan, Taghreed Alqurashi, Javid Gani Dar, and Dxion Salcedo. Certain properties of δh legendre polynomials and applications in computer modelling. European Journal of Pure and Applied Mathematics, 18(2):5886, 2025. [19] R. Alyusof and S. A. Wani. Certain properties and applications of δh hybrid special polynomials associated with appell sequences, 2023. [20] Francesco A. Costabile and Elisabetta Longo. δh-appell sequences and related inter- polation problem. Numerical Algorithms, 63:165–186, 2013. [21] H. Qawaqneh, H. A. Hammad, and H. Aydi. Exploring new geometric contraction mappings and their applications in fractional metric spaces, 2024. [22] Muhammad Nazam, Hassen Aydi, Mohd Salmi Noorani, and Haitham Qawaqneh. Existence of fixed points of four maps for a new generalized f-contraction and an application. Journal of Function Spaces, 2019(1):5980312, 2019. [23] Haitham Qawaqneh, Mohd Salmi Noorani, Hassen Aydi, and Wasfi Shatanawi. On common fixed point results for new contractions with applications to graph and in- tegral equations. Mathematics, 7(11):1082, 2019. [24] Waseem A. Khan. A note on degenerate hermite poly-bernoulli numbers and poly- nomials. Journal of Classical Analysis, 8(1):65–76, 2016. [25] Waseem Ahmad Khan, Ugur Duran, Jihad Younis, and Cheon Seoung Ryoo. On some extensions for degenerate frobenius-euler-genocchi polynomials with applications in computer modeling. Applied Mathematics in Science and Engineering, 32(1):2297072, 2024. [26] Waseem A. Khan, Abdulghani Muhyi, Rifaqat Ali, Khaled Ahmad Hassan Alzobydi, Manoj Singh, and Praveen Agarwal. A new family of degenerate poly-bernoulli poly- nomials of the second kind with its certain related properties. AIMS Mathematics, 6(11):12680–12697, 2021. H. Qawaqneh et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6657 16 of 16 [27] G. Dattoli. Generalized polynomials, operational identities and their applications. Journal of Computational and Applied Mathematics, 118(1-2):111–123, 2000. [28] G. Dattoli, P. E. Ricci, C. Cesarano, and L. Vázquez. Special polynomials and fractional calculus. Mathematical and Computer Modelling, 37(7-8):729–733, 2003.