Closed Forms for the Apostol-Frobenius-Euler Polynomials

Author(s)	Rashin Dinoy Nuñez, Jenivive Dabon, Stephany Mae Gelido, Winple Menguito, Wilson Castañeda Jr.
Country	Philippines
Abstract	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.
Keywords	closed forms, Apostol-Frobenius-Euler polynomial and numbers, Determinantal expression
Field	Mathematics
Published In	Volume 7, Issue 2, March-April 2025
Published On	2025-04-17
DOI	https://doi.org/10.36948/ijfmr.2025.v07i02.41773
Short DOI	https://doi.org/g9f4v7

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