Some properties of Apostol–Hermite–Kampé de Fériet–Bell–Bernoulli-type polynomials and their fractional extension

This study presents a new, generalized family of special polynomials. These polynomials are designated as the Apostol–Hermite–Kampé de Fériet–Bell–Bernoulli-type polynomials. Based on their generating function, the corresponding series expansions and summation formulas are obtained. Furthermore, determinant representation and several differential and integral representations of these polynomials are derived. The fractional extension of this polynomial family is explored, resulting in the formulation of several associated identities. Finally, the study utilizes Mathematica to deliver zero distributions and graphical representations.