A New Generalization of m th-Order Laguerre-Based Appell Polynomials Associated with Two-Variable General Polynomials

Waseem Ahmad Khan (), Khidir Shaib Mohamed (), Francesco Aldo Costabile (), Shahid Ahmad Wani and Alawia Adam
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Waseem Ahmad Khan: Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia
Khidir Shaib Mohamed: Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
Francesco Aldo Costabile: Department of Mathematics and Computer Science, University of Calabria, 87036 Rende, CS, Italy
Shahid Ahmad Wani: Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University) (SIU), Pune 412115, India
Alawia Adam: Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia

Mathematics, 2025, vol. 13, issue 13, 1-24

Abstract: This paper presents a novel generalization of the m th-order Laguerre and Laguerre-based Appell polynomials and examines their fundamental properties. By establishing quasi-monomiality, we derive key results, including recurrence relations, multiplicative and derivative operators, and the associated differential equation. Additionally, both series and determinant representations are provided for this new class of polynomials. Within this framework, several subpolynomial families are introduced and analyzed including the generalized m th-order Laguerre–Hermite Appell polynomials. Furthermore, the generalized m th-order Laguerre–Gould–Hopper-based Appell polynomials are defined using fractional operators and we investigate their structural characteristics. New families are also constructed, such as the m th-order Laguerre–Gould–Hopper–based Bernoulli, Laguerre–Gould–Hopper–based Euler, and Laguerre–Gould–Hopper–based Genocchi polynomials, exploring their operational and algebraic properties. The results contribute to the broader theory of special functions and have potential applications in mathematical physics and the theory of differential equations.

Keywords: Laguerre polynomials; mth-order Laguerre-based Appell polynomials; monomiality principle; explicit form; operational connection; recurrence relations; partial differential equations; differential equations; determinant form; fractional derivatives (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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