Recurrence Relations

In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms. The term difference equation sometimes (and for the purposes of this article) refers to a specific type of recurrence relation. However, "difference equation" is frequently used to refer to any recurrence relation. An example of a recurrence relation is the logistic map:

$n!!= \begin{cases} 2^{n/2} \cdot \frac{n}{2}!& \text{if }n\mbox{ is even,}\qquad\qquad \\ \dbinom{\frac{n}{2} }{ \frac{n-1}{2}} \cdot 2^{(n-1)/2} \cdot \frac{n-1}{2}! & \text{if }n\text{ is odd} \end{cases}$ with a given constant r; given the initial term x0 each subsequent term is determined by this relation. Some simply defined recurrence relations can have very complex (chaotic) behaviours, and they are a part of the field of mathematics known as nonlinear analysis. Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of n.Watch the video below for further infromation on Recurrence relations.