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Concept
Recurrence relation
Summary
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation. If the values of the first 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 previous terms. A famous example is the recurrence for the Fibonacci numbers,
where the order 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 . For these recurrences, one can express the general term of the sequence as a closed-form expression of . As well, linear recurrences with polynomial coefficients depending on 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 .
The concept of a recurrence relation can be extended to multidimensional arrays, that is, indexed families that are indexed by tuples of natural numbers.
A recurrence relation is an equation that expresses each element of a sequence as a function of the preceding ones. More precisely, in the case where only the immediately preceding element is involved, a recurrence relation has the form
where
is a function, where X is a set to which the elements of a sequence must belong. For any , this defines a unique sequence with as its first element, called the initial value.
It is easy to modify the definition for getting sequences starting from the term of index 1 or higher.
This defines recurrence relation of first order.
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