A recurrence relation is an equation that recursively defines a sequence, once one more initial terms are given. Each further term of the sequence is defined as a function of the preceding terms. The term difference equation is sometimes referred to as 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:
Xn+1 = rxn(1-xn)
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 behaviours and they are a part of the field of mathematics known as nonlinear analysis. Solving the recurrence relation means obtaining a closed-form solution: a non-linear recursive function of n.
Recurrence relation has many different applications. These can include biological applications, digital signal processing and economics. In biology, some of the best-known difference equations have their origins in the attempt to model population dynamics. The logistical map is used either directly to model population growth, or as a starting point for more detailed models. In digital signal processing, recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in infinite impulse response digital filters. Recurrence relations, especially linear recurrence relations, are used extensively in both theoretical and empirical economics.