Insights into New Generalization of q -Legendre-Based Appell Polynomials: Properties and Quasi Monomiality

Naeem Ahmad () and Waseem Ahmad Khan ()
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Naeem Ahmad: Mathematics Department, College of Science, Jouf University, Sakaka 72388, Saudi Arabia
Waseem Ahmad Khan: Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia

Mathematics, 2025, vol. 13, issue 6, 1-17

Abstract: In this paper, by using the zeroth-order q -Tricomi functions, the theory of three-variable q -Legendre-based Appell polynomials is introduced. These polynomials are studied by means of generating functions, series expansions, and determinant representation. Further, by utilizing the concepts of q -quasi-monomiality, these polynomials are examined as several q -quasi-monomial and operational representations; the q -differential equations for the three-variable q -Legendre-based Appell polynomials were obtained. In addition, we established a new generalization of three-variable q -Legendre-Hermite-Appell polynomials, and we derive series expansion, determinant representation, and q -quasi-monomial and q -differential equations. Some examples are framed to better illustrate the theory of three-variable q -Legendre-based Appell polynomials, and this is characterized by the above properties.

Keywords: q-calculus; q-general polynomials; q-Legendre polynomials; three-variable q-Legendre-Appell polynomials; monomiality principle; explicit form; operational connection; determinant form (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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