recurrence

In mathematics, a recurrence relation is an equation according to which the



n


{\displaystyle n}
th term of a sequence of numbers 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,

where the order



k


{\displaystyle k}
is two and the linear function merely adds the two previous terms. This 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 n}
. As well, linear recurrences with polynomial coefficients depending on



n


{\displaystyle n}
are also important, because many common elementary and special functions have a Taylor series whose coefficients satisfy such a recurrence relation (see holonomic function).
Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of



n


{\displaystyle n}
.
The concept of a recurrence relation can be extended to multidimensional arrays, that is, indexed families that are indexed by tuples of natural numbers.

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