Horadam sequences: A survey update and extension.

We give an update on work relating to Horadam sequences that are generated by a general linear recurrence formula of order two. This article extends a first ever survey published in early 2013 in this Bulletin, and includes coverage of a new research area opened up in recent times.N/

Deformed Differential Calculus on Generalized Fibonacci Polynomials

In this paper a differential calculus defined on generalized Fibonacci polynomials is given. The main objective is to generalize the $q$-calculus and the Golden calculus or Fibonacci calculus and thus obtain the Pell calculus, Jacobsthal calculus, Chebysheff calculus, Mersenne calculus, among others. This calculus will serve as a framework for the solutions of equations in differences with proportional delay. For this reason, the deformed $(s,t)$-analogs of the binomial formula and of the ordinary exponential and trigonometric functions are defined, together with their operational and analytical properties

Spreadsheets and the discovery of new knowledge

The paper introduces a new class of polynomials discovered as an extended inquiry into a two-parametric difference equation using a spreadsheet. These polynomials possess a number of interesting properties connected to the notion of a generalized golden ratio and can be used as a background for a spreadsheet-enhanced teaching of mathematics. The paper reflects on activities designed for a technology-rich mathematics education course for prospective teachers of secondary mathematics

Fibonacci and Lucas Differential Equations

The second-order linear hypergeometric differential equation and the hypergeometric function play a central role in many areas of mathematics and physics. The purpose of this paper is to obtain differential equations and the hypergeometric forms of the Fibonacci and the Lucas polynomials. We also write again these polynomials by means of Olver’s hypergeometric functions. In addition, we present some relations between these polynomials and the other well-known functions

An introduction of the theory of nonlinear error-correcting codes

Nonlinear error-correcting codes are the topic of this thesis. As a class of codes, it has been investigated far less than the class of linear error-correcting codes. While the latter have many practical advantages, it the former that contain the optimal error-correcting codes. In this project the theory (with illustrative examples) of currently known nonlinear codes is presented. Many definitions and theorems (often with their proofs) are presented thus providing the reader with the opportunity to experience the necessary level of mathematical rigor for good understanding of the subject. Also, the examples will give the reader the additional benefit of seeing how the theory can be put to use. An introduction to a technique for finding new codes via computer search is presented