Closed Forms for the Apostol-Frobenius-Euler Polynomials

The Apostol-Frobenius-Euler polynomials are novel extension from the prolific studies on classical polynomials. In recent years, there has been great erudition focused on evaluating special functions. This paper obtained a simple and finite approach to evaluate the Apostol-Frobenius-Euler polynomials through their closed forms. Closed forms are mathematical expressions expressed using a definite number of standard operations. There can be constants, variables, some elementary operations, and functions, but conventionally, limits are excluded. The researchers utilized some properties of the Bell polynomials of the second kind, the Faà di Bruno formula, and the determinantal representation of the quotient rule to obtain the closed forms. The closed forms for the Apostol-Frobenius-Euler number were also presented.