A statement describing some quantity such as f(n) (where f is some function and n is a positive integer) in terms of values of f(m), where m is a nonnegative integer smaller than n; initial values such as f(0) or f(1) can be assumed to be defined. The concept can be extended to include functions of several variables. A recurrence will then involve defining f(m,n), say, in terms of f(m′,n′) where in some sense (m′,n′) is smaller than (m,n); again initial values can be assumed. The numbers in the Fibonacci series can be defined by a recurrence.
In general, a recurrence can be considered as an equation connecting the values of the function at a number of related points. It has the form g(n, f(n), f(n−1),…, f(n−k) = 0n = k, k + 1,…, N Assuming initial values for f(0), f(1),…, f(k−1), values for other points n can be calculated.
g(n, f(n), f(n−1),…, f(n−k) = 0
n = k, k + 1,…, N
Equations of this type arise naturally in the discretization of continuous problems, and in a slightly different form, known as a difference equation, appear repeatedly in combinatorics.