Determinant approach of the $(p,q)$-Hermite-Appell polynomials and some of their components

In this work, we offer the novel class of $(p,q)$-Hermite-Appell polynomials. Some attributes of this class are constructed, along with the generating function, series definition, $(p,q)$-derivative properties, $(p,q)$-integral representation, summation formulas, and determinate representation. Additionally, we consider a few components for the $(p,q)$ -Hermite-Appell polynomials and infer certain elements of their traits. The generating function and series expansions of some classes of two-dimensional $(p,q)$-Hermite-Appell polynomials are provided. Moreover, we acquire a $(p,q)$-differential operator formula for $(p,q)$-Hermite-Appell polynomials. Finally, the Wolfram Mathematica software is used to plot the graphical diagrams of select components of $(p,q)$-Hermite-Appell, along with two-dimensional $(p,q)$-Hermite-Appell polynomials.