Analytical properties of \(\Delta_h\)-hybrid Laguerre-Appell polynomials and their applications in computer modeling

Volume 39, Issue 4, pp 541--555 https://dx.doi.org/10.22436/jmcs.039.04.08

Publication Date: May 22, 2025 Submission Date: January 25, 2025 Revision Date: February 05, 2025 Accteptance Date: April 02, 2025

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Authors

Sh. A. Wani - Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed) University (SIU), Pune, India. W. A. Khan - Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia. T. Alqurashi - Mathematics Department, Faculty of Science, Al-Baha University, 65779-7738, Albaha city, Kingdom of Saudi Arabia. S. A. Sheikh - Department of Mathematics, University of Kashmir, South campus, Anantnag, Jammu and Kashmir 192101, India. Sh. Tamboli - Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed) University (SIU), Pune, India. D. Salcedo - Computer Science and Electronics Development, Universidad de la Costa, Barranquilla, Colombia.

Abstract

This work thoroughly investigates a new class of polynomials, the \(\Delta_h\) Laguerre-Appell polynomials, which combine certain operational strategies with the principle of monomiality. By using an innovative approach, this study adds to previous research on the subject and presents discoveries. Since these polynomials are essential for simulating entropy in quantum systems, their significance is especially clear in quantum mechanics. In-depth derivations of explicit formulas and an analysis of important characteristics and links to well-known polynomial families are given. Through an examination of the special characteristics and uses of \(\Delta_h\) Laguerre-Appell polynomials, this work greatly expands their theoretical knowledge and increases their prospective applications in a variety of mathematical and scientific contexts.

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Keywords

• \(\Delta_h\) polynomials
• monomiality principle.
• Laguerre-Appell polynomials
• explicit forms
• determinant form

MSC

•  33E20
•  33B10
•  33E30
•  11T23

References

• [1] N. Alam, S. A. Wani, W. A. Khan, F. Gassem, A. Altaleb, Exploring Properties and Applications of Laguerre Special Polynomials Involving the Δh Form, Symmetry, 16 (2024), 16 pages
  • View Article
  • Google Scholar


• [2] N. Alam, S. A. Wani, W. A. Khan, H. N. Zaidi, Investigating the Properties and Dynamic Applications of Δh Legendre– Appell Polynomials, Mathematics, 12 (2024), 14 pages
  • View Article
  • Google Scholar


• [3] M. S. Alatawi, W. A. Khan, New type of degenerate Changhee–Genocchi polynomials, Axioms, 11 (2022), 11 pages
  • View Article
  • Google Scholar


• [4] I. Alazman, B. S. T. Alkahtani, S. A. Wani, Certain properties Δh multi-variate Hermite polynomials, Symmetry, 15 (2023), 10 pages
  • View Article
  • Google Scholar


• [5] L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Co., New York (1985)
  • View Article
  • Google Scholar


• [6] R. Alyusof, S. A. Wani, Certain Properties and Applications of Δh Hybrid Special Polynomials Associated with Appell Sequences, Fractal Fract., 7 (2023), 10 pages
  • View Article
  • Google Scholar


• [7] L. Carlitz, Eulerian numbers and polynomials, Math. Mag., 32 (1959), 247–260
  • View Article
  • Google Scholar


• [8] C. Cesarano, D. Assante, A note on generalized Bessel functions, Int. J. Math. Models Methods Appl. Sci., 7 (2013), 625–629
  • View Article
  • Google Scholar


• [9] C. Cesarano, Y. Quintana, W. Ramírez, Degenerate versions of hypergeometric Bernoulli-Euler polynomials, Lobachevskii J. Math., 45 (2024), 3509–3521
  • View Article
  • Google Scholar
  • MATH


• [10] C. Cesarano, Y. Quintana, W. Ramírez, A Survey on Orthogonal Polynomials from a Monomiality Principle Point of View, Encyclopedia, 4 (2024), 1355–1366
  • View Article
  • Google Scholar


• [11] F. A. Costabile, E. Longo, Δh-Appell sequences and related interpolation problem, Numer. Algorithms, 63 (2013), 165–186
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality principle, In: Advanced special functions and applications (Melfi, 1999), Aracne, Rome, 1 (1999), 147–164
  • View Article
  • Google Scholar
  • MATH


• [13] G. Dattoli, Generalized polynomials operational identities and their applications, J. Comput. Appl. Math., 118 (2000), 111–123
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] G. Dattoli, S. Lorenzutta, A. M. Mancho, A. Torre, Generalized polynomials and associated operational identities, J. Comput. Appl. Math., 108 (1999), 209–218
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] G. Dattoli, P. E. Ricci, A note on Laguerre polynomials, Int. J. Nonlinear Sci. Numer. Simul., 2 (2001), 365–370
  • View Article
  • Google Scholar


• [16] G. Dattoli, P. E. Ricci, C. Cesarano, L. Vázquez, Special polynomials and fractional calculus, Math. Comput. Model., 37 (2003), 729–733
  • View Article
  • MathSciNet
  • Google Scholar


• [17] G. Dattoli, P. E. Ricci, C. Cesarano, The Lagrange polynomials, the associated generalizations and the umbral calculus, Integral Transforms Spec. Funct., 14 (2003), 181–186
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] G. Dattoli, A. Torre, Operational methods and two variable Laguerre polynomials, Atti Acad. Sci. Torino CL. Sci. Fis. Mat. Natur., 132 (1998), 3–9
  • View Article
  • Google Scholar
  • MATH


• [19] G. Dattoli, A. Torre, Exponential operators, quasi-monomials and generalized polynomials, Radiat. Phys. Chem., 57 (2000), 21–26
  • View Article
  • Google Scholar


• [20] G. Dattoli, A. Torre, The Laguerre and Legendre polynomials from an operational point of view, Appl. Math. Comput., 124 (2001), 117–127
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] W. A. Khan, A note on degenerate Hermite poly-Bernoulli polynomials, J. Classical Anal., 1 (2016), 65–76
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] W. A. Khan, M. S. Alatawi, Analytical properties of degenerate Genocchi polynomials the second kind and some of their applications, Symmetry, 14 (2022), 15 pages
  • View Article
  • Google Scholar


• [23] Y. Quintana, W. Ramírez, A degenerate version of hypergeometric Bernoulli polynomials: announcement of results, Commun. Appl. Ind. Math., 15 (2024), 36–43
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] W. Ramírez, C. Cesarano, Some new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol- Genocchi polynomials, Carpathian Math. Publ., 14 (2022), 354–363
  • View Article
  • Google Scholar
  • MATH


• [25] W. Ramírez, C. Cesarano, Applying the monomiality principle to the new family of Apostol Hermite Bernoulli-type polynomials, Commun. Appl. Ind. Math., 15 (2024), 28–35
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] M. Riyasat, S. A. Wani, S. Khan, On some classes of differential equations and associated integral equations for the Laguerre-Appell polynomials, Adv. Pure Appl. Math., 9 (2018), 185–194
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] S. Roshan, H. Jafari, D. Baleanu, Solving FDEs with Caputo-Fabrizio derivative by operational matrix based on Genocchi polynomials, Math. Methods Appl. Sci., 41 (2018), 9134–9141
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] J. F. Steffensen, The poweriod, an extension of the mathematical notion of power, Acta. Math., 73 (1941), 333–366
  • View Article
  • Google Scholar


• [29] S. A. Wani, S. Khan, Properties and applications of the Gould-Hopper-Frobenius-Euler polynomials, Tbilisi Math. J., 12 (2019), 93–104
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] S. A. Wani, Two-iterated degenerate Appell polynomials: properties and applications, Arab J. Basic Appl. Sci., 31 (2024), 83–92
  • View Article
  • Google Scholar


• [31] S. A. Wani, K. Abuasbeh, G. I. Oros, S. Trabelsi, Studies on special polynomials involving degenerate Appell polynomials and fractional derivative, Symmetry, 15 (2023), 12 pages
  • View Article
  • Google Scholar


• [32] S. A. Wani, I. Alazman, B. Alkahtani, Certain properties and applications of convoluted Δh multi-variate Hermite and Appell sequences, Symmetry, 15 (2023), 10 pages
  • View Article
  • Google Scholar


• [33] S. A. Wani, A. Warke, J. G. Dar, Degenerate 2D bivariate Appell polynomials: properties and applications, Appl. Math. Sci. Eng., 31 (2023), 14 pages
  • View Article
  • Google Scholar


• [34] A. Wrülich, Beam Life-Time in Storage Rings, CERN Accelerator School, (1994), 409–435
  • View Article


• [35] M. Zayed, S. A. Wani, A Study on Generalized Degenerate Form of 2D Appell Polynomials via Fractional Operators, Fractal Fract., 7 (2023), 14 pages
  • View Article
  • Google Scholar


• [36] M. Zayed, S. A. Wani, Y. Quintana, Properties of Multivariate Hermite Polynomials in Correlation with Frobenius–Euler Polynomials, Mathematics, 11 (2023), 17 pages
  • View Article
  • Google Scholar