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

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.