Advancements in q-Hermite-Appell polynomials: a three-dimensional exploration

Abstract

In this research, we leverage various $ q $-calculus identities to introduce the notion of $ q $-Hermite-Appell polynomials involving three variables, elucidating their formalism. We delve into numerous properties and unveil novel findings regarding these $ q $-Hermite-Appell polynomials, encompassing their generating function, series representation, summation equations, recurrence relations, $ q $-differential formula, and operational principles. Our investigation sheds light on the intricate nature of these polynomials, elucidating their behavior and facilitating deeper understanding within the realm of $ q $-calculus.

Keywords

• special polynomials
• $ q $-calculus
• monomiality principle
• explicit form
• operational connection
• determinant form
• symmetric identities
• summation formulae