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

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