Properties and applications of generalized 1-parameter 3-variable Hermite-based Appell polynomials

We present a novel framework for introducing generalized 3-variable 1-parameter Hermite-based Appell polynomials. These polynomials are characterized by generating function, series definition, and determinant definition, elucidating their fundamental properties. Moreover, utilizing a factorization method, we established recurrence relations, shift operators, and various differential equations, including differential, integrodifferential, and partial differential equations. Special attention is given to exploring the specific cases of 3-variable 1-parameter generalized Hermite-based Bernoulli, Euler, and Genocchi polynomials, offering insights into their unique features and applications.