Gould–Hopper-based Fibonacci–Frobenius–Sigmoid polynomials: Properties and applications in computer modeling

Abstract Fibonacci extensions of several special polynomials, including Fibonacci–Bernoulli, Fibonacci-harmonic, Fibonacci–Euler, and Fibonacci–Hermite polynomials, have recently been studied, and numerous properties and relations of these polynomials have been thoroughly examined utilizing the content of the golden calculus. This paper aims to consider the generating functions of the Fibonacci–Gould–Hopper polynomials and the Gould–Hopper-based Fibonacci–Frobenius-sigmoid polynomials, from which we derive several beneficial relations and properties. These include explicit formulas, summation formulas, correlation formulas with the new and old Fibonacci-type polynomials, symmetric properties, recurrence relation, addition formulas, golden derivative properties, and golden integral representation for these polynomials. Moreover, graphical illustrations of the Fibonacci–Gould–Hopper polynomials and the Gould–Hopper-based Fibonacci–Frobenius-sigmoid polynomials are presented. Their numerical analyses are used to validate theoretical results and reveal distinctive scattering patterns in the distribution of their zeros across the complex plane, offering insights into their underlying analytic structure. Furthermore, interesting patterns in the zeros (real and complex zeros) distributions of these two new families of polynomials are examined and drawn, forming 2D and 3D structures. In addition, the approximate real and complex zeros of the mentioned polynomials for some special cases are presented in four tables. Lastly, four conjectures about the zeros of these polynomials are given.