A new class of generalized Apostol–type Frobenius–Euler polynomials

The paper presents a new type of generalized Apostol-type Frobenius–Euler polynomials and numbers with specific order $\kappa$ and level $m$. We establish fundamental identities and properties using generating function techniques, such as summation formulas, differential and integral relations, and addition theorems. Additionally, we explore the connections between these polynomials and the Stirling numbers of the second kind, as well as other polynomial families. Lastly, we derive a differential equation and a recurrence relation for these new classes of polynomials. Finally, we show applications that can be obtained using these polynomials where the graphs of the zero functions and the meshes are displayed.