Non-Linear Recursive Sequences

Linear Recursive Sequences

A sequence is an infinite list of numbers, like The numbers in the sequence are called its terms. The general form of a sequence is a 1 , a 2 , a 3 ,. .. where a n is the n-th term of the sequence. In the example (1) above, a 1 = 1, a 2 = 2, a 3 = 4, and so on. The notations {a n } or {a n } ∞ n=1 are abbreviations for a 1 , a Occasionally the indexing of the terms will start with something oth...

Intersections of Second - Order Linear Recursive Sequences

Equivalence-motivated non-linear recursive estimation

Recursive non-linear Bayesian estimation is addressed using equivalence approach as motivating framework. Its specific form – tailored to a model class covering non-normal ARX (auto-regression with exogenous variables) models, models with discrete outputs and continuous-valued regression vectors and their dynamic mixtures – is presented. The resulting algorithms provide efficient solutions of d...

On the Appearance of Primes in Linear Recursive Sequences

We present an application of difference equations to number theory by considering the set of linear second-order recursive relations,Un+2( √ R,Q) =RUn+1 −QUn,U0 = 0,U1 = 1, and Vn+2( √ R,Q) =RVn+1 −QVn, V0 = 2,V1 = √ R, where R and Q are relatively prime integers and n∈ {0,1, . . .}. These equations describe the set of extended Lucas sequences, or rather, the Lehmer sequences. We add that the r...

An Introduction to Linear Recursive Sequences in Spread Spectrum Systems

Spread Spectrum technology is being applied to many areas of modern communications such as Cellular Telephones, Global Positioning Satellite (GPS), and Very Small Aperture Satellite Terminals (VSATS) just to name a few. The core concept behind this technology involves spreading the data to be transmitted over a much broader spectrum than what is conventionally done. The spreading is usually ach...