Recurrence relation

In mathematics, a recurrence relation is an equation according to which the n {\displaystyle n} th term of a sequence is equal to some combination of the previous terms. Often, only k {\displaystyle k} previous terms of the sequence appear in the equation, for a parameter k {\displaystyle k} that is independent of n {\displaystyle n} ; this number k {\displaystyle k} is called the order of the relation. If the values of the first k {\displaystyle k} numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In linear recurrences, the nth term is equated to a linear function of the k {\displaystyle k} previous terms. A famous example is the recurrence for the Fibonacci numbers, F n = F n − 1 + F n − 2 {\displaystyle F_{n}=F_{n-1}+F_{n-2}} where the order k {\displaystyle k} is two and the linear function merely adds the two previous terms. The sequences that satisfy a recurrence relation are exactly the same as the sequences that satisfy a difference equation. More precisely, a difference equation can be associated to every recurrence relation, and, conversely, a recurrence relation can be associated to every difference equation such that the two processes are inverse one to the other,and the series that satisfy one of the equation staisfies the others. Because of the similarity of difference equations with differential equations, the methods of resolution of differential equations may often be applied to differenceequations aand theusto recurrencerelations. The above example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n . {\displaystyle n.} For these recurrences, one can express the general term of the sequence as a closed-form expression of n {\displaystyle...