In this paper, we introduce the Fibonacci polynomials (FPs) and approximate functions using them. Furthermore, several lemmas and corollaries present the properties of FPs. Also, we derive the Fibonacci polynomials operational matrix for the fractional derivative in the Caputo sense, which has not been undertaken before.  As applications of the Fibonacci polynomials operational matrix, we solve the system of Volterra Fredholm integro-fractional differential equations. In this scheme, we approximate one and two variable functions based on Fibonacci basis. Then by applying Fibonacci polynomials operational matrix, the system of Volterra Fredholm integro-fractional differential equations is reduced to a system of algebraic equations that is easily solvable with the help of a software (version 13 of the Mathematica software). The obtained results are in good agreement with the exact solutions and with those in literature. As anticipated, the solutions converge to classical solutions as the fractional derivative order approaches integer values.

In this paper, a class of new polynomials based on Fibonacci sequence using Newton interpolation is introduced. This target is performed once using Newton forward-divided-difference formula and another more using Newton backward-divided-difference formula. Some interesting results are obtained for forward and backward differences. The relationship between forward (and backward) differences and the Khayyam-Pascal’, s triangle are also examined.

Here, based on the Fibonacci polynomials, a new collocation method is presented in order to solve the system of linear fuzzy Volterra integral equations of the second kind. By using this method, these systems are reduced to a linear system of algebraic equations which are simply solvable. Also, the error analysis and existence of the solution of the suggested method are discussed. Finally, to show the importance and application of this method, we have used some rational examples. The method is computationally very attractive and gives very accurate results. Easy implementation and simple operations are the essential features of the Fibonacci polynomials.

In this paper, a new interconnection network structure called hierarchical extended Fibonacci cubes interconnection networks HEFC1(n) is proposed, and its properties are evaluated. A set of the HEFC1(n)s constructed by the proposed method is a two level conventional hierarchical network. For each n (dimension of extended Fibonacci cube-EFC1(n)), a HEFC1(n)   interconnection network can be constructed. All EFC1(n) for n>4 are recursively constructible, hence HEFC1(n)s for n>4 are also recursively constructible with some extra EFC1(n-1)s and EFC1(n-2)s. Furthermore, since a HEFC1(n) has a hierarchically structured character and the feature of uniformity, a wide variety of inter-cluster connections are possible. The comparisons of HEFC1(n)s with some of the traditional cubic and hierarchical cubic networks are presented in this study. The routing in HEFC1(n+2)s is as easy as routing in HCN(n,n) and the scalability of HEFC1(n+2) is better than the scalability of HCN(n,n) and H(2n). H(2n) and HCN(n,n) are symmetric interconnection networks, however, HEFC1(n+2) is not a symmetric interconnection network and HEFC1(n+2)s have a recurrent structure as H(2n) and HCN(n,n). The cost of HEFC1(n+2) is better than the costs of HCN(n,n) and H(2n).The edge connectivity of HEFC1(n+2) is worse than the edge connectivity of H(2n) and HCN(n,n). This makes the VLSI design of HEFC1(n+2) simpler than the VLSI designs of H(2n) and HCN(n,n).

This article deals with two new subclasses of analytic and bi-univalent functions in the open unit disk, which is defined  by applying subordination principle between analytic functions and the generalized Bivariate Fibonacci polynomials. Bounds for coefficients $\left|a_{2}\right|$ and $\left|a_{3}\right|$ of functions in these subclasses are estimated in terms of generalized Bivariate Fibonacci polynomials. In addition, the Fekete-Szeg\"{o} problem is handled for the members of these subclasses and several consequences and examples of the main results are presented. The results of article generalize some of the previously published papers in the literature.