EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5886 ISSN 1307-5543 – ejpam.com Published by New York Business Global Certain Properties of ∆h Legendre Polynomials and Applications in Computer Modelling Shahid Ahmad Wani1,∗, Waseem Ahmad Khan2, Taghreed Alqurashi3, Javid Gani Dar1, Dxion Salcedo4 1 Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed) University, Pune, India 2 Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia 3 Mathematics Department, Faculty of Science, Al-Baha University, 65779-7738, Albaha City, Kingdom of Saudi Arabia 4 Computer Science and Electronics Development, Universidad de la Costa, Barranquilla, Colombia Abstract. This study investigates the development of polynomials, with a particular focus on the unique ∆h Legendre polynomials. Explicit formulas for these polynomials are derived, along with summation formulae that provide further structural insights. Additionally, the monomiality principle is established, reinforcing the algebraic framework of these polynomials. Symmetric identities are also formulated, highlighting their fundamental properties. The study concludes with remarks summarizing the key findings and potential directions for future research. 2020 Mathematics Subject Classifications: 33E20, 33B10, 33E30, 11T23 Key Words and Phrases: ∆h sequences, Monomiality principle, explicit forms, Symmetric identities 1. Introduction and preliminaries Many fields, such as statistical mechanics and quantum mechanics, have used specific polynomials to represent and characterize the behavior of complex systems. There are several other fields, including statistics and quantum mechanics, where complex systems have been described and analyzed using these special polynomials. In a number of mathe- matical fields, including combinatorics, entropy, and algebraic combinatorics, polynomial sequences are essential. In approximation theory and physics, Legendre polynomials, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5886 Email addresses: shahidwani177@gmail.com (S. A. Wani), wkhan1@pmu.edu.sa ( W. A. Khan), talqorashi@bu.edu.sa (T. Alqurashi), javid.dar@sitpune.edu.in (J. G. Dar), dsalcedo2@cuc.edu.co (D. Salcedo) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 2 of 16 named after the French mathematician Edmond Legendre, were introduced in the 19th century. These polynomials arise as solutions to the second-order Legendre differential equation and are defined on the interval [0,+∞), commonly denoted by Sn(u), where n represents the degree. Legendre polynomials exhibit orthogonality with respect to the weight function e−u on [0,+∞), ensuring that the weighted integral of two polynomials with different degrees is zero. This orthogonality and their recurrence relation enable efficient computation of higher-degree polynomials from lower-degree ones. Additionally, they allow functions to be expressed as series expansions using their generating function, simplifying the derivation of closed-form solutions for certain differential equations. Widely applied in mathematics, physics, and engineering, Legendre polynomials are integral to solving the Schrödinger equation for spherically symmetric quantum systems, including the hydrogen atom. They also play a crucial role in problems involving diffusion, wave propagation, and heat con- duction. Much recent work has focused on two-variable special polynomials in mathematical physics. There are characteristics of a class of polynomials called two-variable special polynomials, such as [1–10]. They are widely studied in the subject of algebraic geometry and have many applications in mathematics and other domains. Notable instances of two-variable special polynomials include bivariate Chebyshev, Hermite, Laguerre, and Legendre polynomials, etcetra. Approximation theory, numerical analysis, and signal processing all make extensive use of them. Bivariate Legendre polynomials are Legendre polynomials extended to two variables. They are useful in quantum mechanics, potential theory, and random matrix theory, and they satisfy a bivariate counterpart of the Legendre differential equation. The behavior of two-degree-of-freedom systems can be studied with the help of bivariate Legendre polynomials. These polynomials are widely studied in mathematical physics, probability theory, and approximation theory because they meet a specific orthogonality condition concerning a weight function. These two-variable special polynomials are important because they can be used to solve problems in a variety of mathematical and scientific fields, offer a rich framework for expressing and analyzing multivariate functions, and have particular characteristics that make them appropriate for particular applications. Two-variable Legendre polynomials, denoted as Sn(u, v) [11], hold significant math- ematical value and serve as essential tools in physics due to their broad applications. These polynomials provide a powerful framework for analyzing solutions to various partial differential equations commonly arising in physical contexts. The 2-variable Legendre polynomials (2VLeP) Sn(u, v) are defined through the follow- ing generating function: evtJ0(2t √ −u) = ∞∑ n=0 Sn(u, v) tn n! , (1) S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 3 of 16 where J0(ut) is the 0th order ordinary Bessel function of first kind [12] defined by Jn(2 √ u) = ∞∑ k=0 (−1)k ( √ u) n+2k k! (n+ k)! . (2) We also note that exp(−αD−1 u ) = J0(2 √ αu), D−n u {1} := un n! (3) is the inverse derivative operator. Or, Alternatively by evtC0(−ut2) = ∞∑ n=0 Sn(u, v) tn n! , (4) where C0(ut) is the 0th order Tricommi function of first kind [12] with C0(−ut2) = eD −1 u t2 . (5) Thus, in view of equation (3) or (5), the generating expression for Legendre polynomials can be casted as: evt eD −1 u t2 = ∞∑ n=0 Sn(u, v) tn n! . (6) In recent years, mathematicians have shown a growing interest in the study of ∆h forms of special polynomials, motivated by their analytical significance and computational utility. Several generalizations of these polynomials have been explored in works such as [13–18]. Expanding upon these studies, the classical finite difference operator ∆h has been employed to introduce a novel class known as ∆h special polynomials, as discussed in [19–23]. These polynomials are not only significant in pure mathematical theory but also hold substantial applications in computational modeling. In numerical analysis, ∆h special polynomials serve as essential tools for approximating solutions to differential and inte- gral equations, particularly in discretized frameworks. In physics and engineering, they provide efficient computational schemes for modeling wave propagation, heat conduction, and fluid dynamics. Their structured recurrence relations and explicit forms allow for the development of stable numerical algorithms, making them valuable in computer simula- tions. Additionally, in statistical computing, these polynomials facilitate the design of dis- crete probability distributions and contribute to data interpolation techniques, enhancing predictive modeling. Their role in symbolic computation further extends to software im- plementations for solving partial difference equations in image processing and digital signal analysis. Given their computational efficiency and adaptability, ∆h special polynomials are emerging as powerful mathematical tools with extensive applications in computer- aided modeling and simulations. In symbolic computation and computer algebra sys- tems (CAS), these polynomials aid in the efficient manipulation of large-scale algebraic S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 4 of 16 expressions. Their structured operational rules allow for automated simplifications and symbolic differentiation, making them valuable in software implementations for solving dis- crete versions of differential equations, such as in image processing, digital signal analysis, and network modelling. Furthermore, in graph theory and combinatorial optimization, ∆h polynomials provide analytical tools for studying discrete structures and optimizing network algorithms. Their application extends to shortest path problems, spanning tree calculations, and network flow optimizations, which are crucial in computer science and logistics. “These ∆h-Appell polynomial are represented as: A[h] n (u) := An(u), n ∈ N0 (7) and defined by A[h] n (u) = nhAn−1(u), n ∈ N, (8) where ∆h is the finite difference operator: ∆h H[h](u) = H(u+ h)−H(u)”. (9) The ∆h-Appell polynomials An(u) are specified by the following generating function [19]: γ(t)(1 + ht) u h = ∞∑ n=0 A[h] n (u) tn n! , (10) where γ(t) = ∞∑ n=0 γn,h tn n! , γ0,h ̸= 0. (11) Motivated by Costabile [19], here we introduced the two variable ∆h Legendre poly- nomials: (1 + ht) v h (1 + ht2) D−1 u h = S[h]n (u, v) tn n! (12) through the generating function concept. The article is organized to give readers a thorough grasp of the ∆h Legendre polyno- mials, with Section 2 outlining how they are generated, how they recur, and how they are evaluated. Effective calculation is made possible by the formulas provided in Section 3 for evaluating or adding up these polynomials under particular circumstances. By analyzing the behavior of ∆h Legendre polynomials under different operations and determining their determinant form, Section 4 explores the Momomiality principle. In Section 5, symmet- ric identities are obtained for these polynomials. By summarizing the results and going over applications, consequences, and possible future research areas related to ∆h Legendre polynomials, the conclusion enhances understanding of their behavior and usefulness in a variety of mathematical contexts. S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 5 of 16 2. Two variable ∆h Legendre polynomials This section explores a new class of two-variable ∆h Legendre polynomials, establishing their fundamental properties and expanding the scope of polynomial theory. The devel- opment of their generating function, S[h]ω (u, v), provides deeper insight into their structure and behavior, which is crucial for applications in combinatorics, analysis, and mathe- matical physics. By linking these polynomials to their generating function, this study enhances the understanding of polynomial families and their applications. The findings offer valuable perspectives on their special properties, paving the way for further research in mathematical and scientific domains. First, we derive the generating function for these ∆h Legendre polynomials S[h]n (u, v) by proving the following result: Theorem 1. For the two variable ∆h Legendre polynomials S[h]n (u, v), the succeeding gen- erating relation holds true: (1 + ht) v h (1 + ht2) D−1 u h = ∞∑ n=0 S[h]n (u, v) tn n! , (13) or equivalently (1 + ht) v hC0 ( −u h log(1 + ht2) ) = ∞∑ n=0 S[h]n (u, v) tn n! . (14) Proof. Expanding (1 + ht) v h (1 + ht2) D−1 u h around u = v = 0 using a Newton series for finite differences and analyzing the product’s development with respect to t’s powers, we identify the polynomials S[h]n (u, v) as coefficients of tn n! . These coefficients, given in equa- tion (13), represent the generating function of the two-variable ∆h-Legendre polynomials S[h]n (u, v). Theorem 2. For the two variable ∆h Legendre polynomials S[h]n (u, v), the succeeding re- lations hold true: v∆h h S[h]n (u, v) = n S[h]n−1(u, v) u∆h h S[h]n (u, v) = n(n− 1) S[h]n−2(u, v), D−1 u → u. (15) Proof. By differentiating (13) w.r.t. v by taking into consideration of expression (5), we have v∆h(1 + ht) v h (1 + ht2) D−1 u h = (1 + ht) v+h h (1 + ht2) D−1 u h − (1 + ht) v h (1 + ht2) D−1 u h = (1 + ht− 1)(1 + ht) v h (1 + ht2) D−1 u h S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 6 of 16 = ht (1 + ht) v h (1 + ht2) D−1 u h . (16) By substituting r.h.s. of expression (13) in (16), we find v∆h ∞∑ n=0 S[h]n (u, v) tn n! = h ∞∑ n=0 S[h]n (u, v) tn+1 n! . (17) Substituting n → n − 1 into the right-hand side of (16) and equating the coefficients of identical powers of t in the resulting expression leads to the derivation of (15). We now derive the explicit form of the two-variable ∆h Legendre polynomials, S[h]n (u, v), as stated in the following theorem: Theorem 3. The two-variable ∆h Legendre polynomials S[h]n (u, v) satisfy the following relations: S[h]n (u, v) = v h∑ d=0 ( n d )( v h d ) hd S[h]n−d(u). (18) Proof. Expanding generating relation (13) in the given manner: (1 + ht) v h (1 + ht2) D−1 u h = v h∑ d=0 ( v h d ) (ht)d d! ∞∑ n=0 S[h]n (u, 0) tn n! (19) which can further be written as S[h]n (u, v) tn n! = ∞∑ n=0 v h∑ d=0 ( v h d ) hd S[h]n (u, 0) tn+d n! d! . (20) By replacing n → n− d in the r.h.s. of previous expression, it follows that S[h]n (u, v) tn n! = ∞∑ n=0 v h∑ d=0 ( v h d ) hd S[h]n (u, 0) tn (n− d)! d! . (21) On multiplying and dividing by n! in the r.h.s. of previous expression (21) and com- paring the coeffecients of same exponents of t on both sides, assertion (18) is deduced. S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 7 of 16 3. Summation formulae This section presents summation formulae, essential tools in mathematical analysis for understanding polynomial structures in two variables. These formulas systematically compute sums of specific polynomials, revealing hidden symmetries and interrelationships between variables. Their applications extend to mathematical physics, probability the- ory, and combinatorics, aiding in the development of efficient computational techniques. Summation equations serve as fundamental building blocks for advancing mathematical theory and its real-world applications. Below, we establish these summation formulae by proving the following key results: Theorem 4. For n ≥ 0, we have S[h]n (v + 1, u) = n∑ m=0 ( n m )( −1 h ) m (−h)mS[h]n−m(u, v). (22) Proof. From (13), we have ∞∑ n=0 S[h]n (v + 1, u) tn n! − ∞∑ n=0 S[h]n (u, v) tn n! = (1 + ht) v h (1 + ht2) D−1 u h ( (1 + ht) 1 h − 1 ) = ∞∑ n=0 S[h]n (u, v) tn n! ( ∞∑ m=0 ( −1 h ) m (−h)m tm m! − 1 ) = ∞∑ n=0 ( n∑ m=0 ( n m )( −1 h ) m (−h)mS[h]n−m(u, v) ) tn n! − ∞∑ n=0 S[h]n (u, v) tn n! . (23) Comparing the coefficients of t, we obtain (22). Theorem 5. For n ≥ 0, we have S[h]n (u, v) = [n 2 ]∑ j=0 ( −v h ) n−2j (−h)n−j ( −u h ) j (−1)j n! (n− 2j)!(j!)2 . (24) Proof. From (13), we have ∞∑ n=0 S[h]n (u, v) tn n! = (1 + ht) v h (1 + ht2) D−1 u h = ∞∑ n=0 ( −v h ) n (−h)n tn n! ∞∑ j=0 ( −u h ) j (−1)j(−h)j t2j j!j! = ∞∑ n=0 [n 2 ]∑ j=0 ( −v h ) n−2j (−h)n−j ( −u h ) j (−1)j tn (n− 2j)!(j!)2 . (25) S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 8 of 16 Comparing the coefficients of t, we obtain (24). Now, we investigate the connection between the Stirling numbers of the first kind and 2-variable ∆h Legendre polynomials. [log(1 + t)]k k! = ∞∑ i=k S1(i, k) ti i! , | t |< 1. (26) From the above definition, we have (v)i = i∑ k=0 (−1)i−kS1(i, k)v k. (27) Theorem 6. For n ≥ 0, we have S[h]n (u, v) = n∑ m=0 ( n m ) S[h]n (u, 0) m∑ j=0 xjS1(m, j)hm−j . (28) Proof. From (13), we have ∞∑ n=0 S[h]n (u, v) tn n! = e v h log(1+ht)(1 + ht2) D−1 u h = (1 + ht2) D−1 u h ∞∑ j=0 (v h )j [log(1 + ht)]j j! = ∞∑ n=0 S[h]n (u, 0) tn n! ∞∑ m=0 m∑ j=0 (v h )j S1(m, j)hm tm m! = ∞∑ n=0  n∑ m=0 ( n m ) S[h]n−m(u, 0) m∑ j=0 (v h )j S1(m, j)hm  tn n! . (29) Comparing the coefficients of t, we obtain (28). Theorem 7. For n ≥ 0, we have S[h]n (u, v) = n∑ l=0 n−l∑ m=0 n! (n−m− l)!(m+ l)! hmS[h]n−m−1(0, u)S1(m+ l, l)vl. (30) Proof. From (13), we have ∞∑ n=0 S[h]n (u, v) tn n! = (1 + ht) v h (1 + ht2) D−1 u h S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 9 of 16 = ∞∑ n=0 S[h]n (0, u) tn n! ∞∑ m=0 ( −v h ) m (−h)m tm m! = ∞∑ n=0 ( n∑ m=0 ( n m )( −v h ) m (−h)mS[h]n−m(0, u) ) tn n! . (31) Comparing the coefficients of t, we get S[h]n (u, v) = n∑ m=0 ( n m )( −u h ) m (−h)mS[h]n−m(0, v). (32) Using the above equality (3.5), we get S[h]n (u, v) = ( n∑ m=0 ( n m ) (−h)mS[h]n−m(0, u) )( m∑ l=0 (−1)m−lS1(m, l)(−h)−lvl ) = n∑ l=0 n∑ m=l n! (n−m)!m! (−h)m−lS[h]n−m(0, u)(−1)m−lS1(m, l)vl = n∑ l=0 n−l∑ m=0 n! (n−m− l)!(m+ l)! (−h)mS[h]n−m−1(0, u)(−1)mS1(m+ l, l)vl. (33) The complete proof of the theorem. Theorem 8. For n ≥ 0, we have S[h]n (v + w, u) = n∑ l=0 n−l∑ m=0 n! (n−m− l)!(m+ l)! hmS[h]n−m−l(u, v)S1(m+ l, l)wl. (34) Proof. Taking v + w instead of u in (13), we have ∞∑ n=0 S[h]n (v + w, v) tn n! = (1 + ht) v+w h (1 + ht2) D−1 u h = ( ∞∑ n=0 S[h]n (u, v) tn n! )( ∞∑ m=0 ( −w h ) m (−h)m tm m! ) . (35) Using the Cauchy rule and after comparing the coefficients of t on both sides of the resulting equation, we have S[h]n (v + w, u) = n∑ m=0 ( n m )( −w h ) m (−h)mS[h]n−m(u, v). (36) Then using (25) for ( −w h ) m , we have obtain (34). S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 10 of 16 Theorem 9. For n ≥ 0, we have S[h]n (u, v) = n∑ m=0 m∑ j=0 ( n m ) (z − v)jS1(m, j)hm−jS[h]n−m(0, u). (37) Proof. From (13), we have (1 + ht2) D−1 u h = e− v h log(1+ht) ∞∑ n=0 S[h]n (u, v) tn n! . (38) Replacing v by z and comparing the resulting equations, we get e z−v h log(1+ht)(1 + ht2) D−1 u h = ∞∑ n=0 S[h]n (u, v) tn n! . (39) Finally, expanding the exponential function and then comparing the coefficients of equal powers of t, we come to assertion (39) of theorem 3.8. Remark 1. Taking v = 0 in Theorem 3.8, we immediately deduce the following conse- quence of Theorem 3.8: S[h]n (0, u) = n∑ m=0 m∑ j=0 ( n m ) zjS1(m, j)hm−jS[h]n−m(0, u). 4. Monomiality Principle The monomiality principle serves as a fundamental framework for understanding and manipulating polynomial expressions. It states that any polynomial can be uniquely ex- pressed as a linear combination of monomials, simplifying their structure and facilitating mathematical analysis. This principle aids in extracting key properties such as degree, leading coefficient, and roots, enabling the development of advanced mathematical tech- niques and algorithms. Beyond its theoretical significance, the monomiality principle plays a crucial role in var- ious scientific and engineering applications. In computational mathematics, it ensures the efficiency of numerical integration, approximation, and interpolation methods. Similarly, in fields like image analysis, control theory, and signal processing, it provides a struc- tured approach for modeling complex systems. Its relevance extends to physics, where polynomials describe fundamental laws and phenomena. The study of hybrid special polynomials has further explored monomiality and its operational principles. Originally introduced by Steffenson in 1941 with the concept of poweroids, the principle was later expanded by Dattoli. In this framework, multiplicative and derivative operators Ĵ and K̂ play a crucial role in defining polynomial sequences {gk(u1)}k∈N, reinforcing its mathematical and practical significance. S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 11 of 16 The following expressions are satisfied by these operators: gk+1(u1) = Ĵ {gk(u1)} (40) and k gk−1(u1) = K̂{gk(u1)}. (41) Therefore, a quasi-monomial domain is produced when the polynomial set gk(u1)m∈N is subjected to multiplicative and derivative operations. For this quasi-monomial, it is essential to adhere to the following formula: [K̂, Ĵ ] = K̂Ĵ − Ĵ K̂ = 1̂. (42) It consequently displays a Weyl group structure. It is possible to determine the significance of the underlined set by using the significance and application of the operators Ĵ and K̂, assuming that the set {gk(u1)}k∈N is quasi- monomial. As a result, the following axioms hold: (i) gk(u1) gives differential equation Ĵ K̂{gk(u1)} = k gk(u1), (43) provided Ĵ and K̂ exhibits differential traits. (ii) The expression gk(u1) = Ĵ k {1}, (44) gives the explicit form, with g0(u1) = 1. (iii) Further, the expression ewĴ {1} = ∞∑ k=0 gk(u1) wk k! , |w| < ∞ , (45) demonstrates generating expression behavior and is obtained by applying identity (44). These methods, rooted in quantum mechanics, mathematical physics, and classical optics, remain highly relevant in modern research. They serve as reliable tools for ana- lyzing complex phenomena across various fields, reinforcing our understanding of intricate systems. Recognizing their significance, we validate the concept of monomiality for the ∆h Leg- endre polynomials, denoted as S[h]n (u, v). These polynomials form a crucial mathematical framework for modeling diverse phenomena. By establishing their monomiality, we aim to highlight their fundamental properties and applications. This section presents our validation results, confirming the integrity and utility of ∆h Legendre polynomials. Through rigorous analysis, we affirm their relevance for both theoretical and applied research, thereby substantiating the monomiality principle for S[h]n (u, v). S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 12 of 16 Theorem 10. The ∆h Legendre polynomials S[h]n (u, v) satisfy the succeeding multiplicative and derivative operators: M̂SA = ( v 1 + v∆h + 2 D−1 u v∆h h+ v∆h 2 ) (46) and D̂S = v∆h h . (47) Proof. In consideration of expression (5), taking derivatives w.r.t. v of expression (13), we have v∆h { (1 + ht) v h (1 + ht2) D−1 u h } = (1 + ht) v+h h (1 + ht2) D−1 u h − (1 + ht) v h (1 + ht2) D−1 u h = (1 + ht− 1)(1 + ht) v h (1 + ht2) D−1 u h = ht (1 + ht) v h (1 + ht2) D−1 u h , (48) thus, we have v∆h h [ (1 + ht) v h (1 + ht2) D−1 u h ] = t [ (1 + ht) v h (1 + ht2) D−1 u h ] , (49) which gives the identity v∆h h [ S[h]n (u, v) ] = t [ S[h]n (u, v) ] . (50) Now, differentiating expression (13) w.r.t. t, we have ∂ ∂t { (1 + ht) v h (1 + ht2) D−1 u h } = ∂ ∂t { ∞∑ n=0 S[h]n (u, v) tn n! } , (51) ( v 1 + ht + 2 D−1 u t 1 + ht2 ){ ∞∑ n=0 S[h]n (u, v) tn n! } = ∞∑ n=0 n S[h]n (u, v) tn−1 n! . (52) Which on usage of identity expression (50) and replacing n → n+1 in the r.h.s. of previous expression (52), assertion (46) is established. S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 13 of 16 Further, in view of identity expression (50), we have v∆h h [ S[h]n (u, v) ] = [ n S[h]n−1(u, v) ] , (53) which gives expression for the derivative operator (47). Next, we deduce the differential equation for the ∆h Legendre polynomials S[h]n (u, v) by demonstrating the succeeding result: Theorem 11. The ∆h Legendre polynomials S[h]n (u, v) satisfy the differential equation:( v 1 + v∆h + 2 D−1 u v∆h h+ v∆h 2 − nh v∆h ) S[h]n (u, v) = 0. (54) Proof. Inserting expression (46) and (47) in the expression (43), the assertion (54) is proved. 5. Symmetric identities The two-variable ∆h special polynomials have symmetric identities that we examine in this section. These identities reveal fascinating connections between the polynomials’ variables and coefficients, illuminating their fundamental symmetrical characteristics. By examining the behavior of the polynomials when the variables or coefficients are changed, we can find deep relationships that go beyond their original definitions. In addition to help- ing us comprehend the polynomials better, these symmetric identities provide important new information on more general mathematical occurrences and structures. We provide a thorough framework for comprehending and taking use of the symmetrical character- istics of these two-variable special polynomials by methodical investigation and exacting derivation, opening the door for future developments in theoretical analysis and real-world applications. Consequently, we discover a few of the Legenedre polynomials formulae and characteristics. Theorem 12. For a ̸= b, a, b > 0 and u1, u2, v1, v2 ∈ C, we have n∑ m=0 ( n m ) an−mbmS[h]n−m(au1, av1)S[h]m (bu2, bv2) = n∑ m=0 ( n m ) anbn−mS[h]n−m(au2, av2)S[h]m (bu1, bv1). (55) Proof. Let A(t) = (1 + ht) ab(v1+v2) h C0 ( −abu1 h log(1 + ht2) ) C0 ( −abu2 h log(1 + ht2) ) S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 14 of 16 = ∞∑ n=0 S[h]n (bu1, bv1) (at)n n! ∞∑ m=0 S[h]n (au2, av2) (bt)m m! = ∞∑ n=0 ( n∑ m=0 ( n m ) an−mbmS[h]n−m(au1, av1)S[h]m (bu2, bv2) ) tn n! . (56) Similarly, we have A(t) = ∞∑ n=0 ( n∑ m=0 ( n m ) anbn−mS[h]n−m(au2, av2)S[h]m (bu1, bv1) ) tn n! . (57) Comapring the coefficients of t on both sides of last equations, we get (55). Theorem 13. For a ̸= b, a, b > 0 and u1, u2, v ∈ C, we have n∑ k=0 k∑ m=0 ( n k )( k m ) an−mbm+1βn−k(h)S [h] n−m(bu, bv)σm(a− 1;h) = n∑ k=0 k∑ m=0 ( n k )( k m ) bn−mam+1βn−k(h)S [h] n−m(au, av)σm(b− 1;h). (58) Proof. Consider B(t) = abt(1 + ht) abv h C0 (−abu h log(1 + ht2) ) ((1 + ht) ab h − 1) ((1 + ht) a h − 1)((1 + ht) b h − 1) = abt ((1 + ht) a h − 1) (1 + ht) abv h C0 ( −abu h log(1 + ht2) ) ((1 + ht) ab h − 1) ((1 + ht) b h − 1) = b ∞∑ n=0 βn(h) (at)n n! ∞∑ k=0 S[h]k (bu, bv) (at)k k! ∞∑ m=0 σm(a− 1;h) (bt)m m! = b ∞∑ n=0 βn(h) (at)n n! ∞∑ k=0 k∑ m=0 ( k m ) ak−mbmS[h]k−m(bu, bv)σm(a− 1;h) tk k! = ∞∑ n=0 ( n∑ k=0 k∑ m=0 ( n k )( k m ) an−mbm+1βn−k(h)S [h] n−m(bu, bv)σm(a− 1;h) ) tn n! . (59) Similarly, we have B(t) = ∞∑ n=0 ( n∑ k=0 k∑ m=0 ( n k )( k m ) bn−mam+1βn−k(h)S [h] n−m(au, av)σm(b− 1;h) ) tn n! . (60) Comparing the both sides of the above equations, we get (58). S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5886 15 of 16 6. Conclusion The introduction and study of ∆h Legendre polynomials mark a significant advance- ment in polynomial theory, particularly in quantum mechanics and entropy modeling. 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