A Study on the Apostol Fibonacci-Sigmoid Polynomials of Higher Order \(\alpha\)

Abstract

In this paper, we introduce the Apostol Fibonacci-Sigmoid polynomials of higher order \(\alpha\) by means of a generating function involving the golden exponential function. Using the generalized binomial and golden binomial theorem, we derive explicit representations and several fundamental identities for these polynomials. We also establish recurrence relations, addition formulas and differential properties within the framework of Fibonacci calculus. Furthermore, symmetric identity is obtained through suitably constructed generating functions. These results broaden the theoretical framework of Fibonacci sigmoid polynomials and pave the way for future investigations into higher-order Fibonacci-type special polynomials.