# Recurrence relation

*recurrence formula*

A relation of the form

permitting one to compute all members of the sequence if its first members are given. Examples of recurrence relations are: 1) , a geometric progression; 2) , an arithmetic progression; 3) , the sequence of Fibonacci numbers.

In the case where the recurrence relation is linear (see Recursive sequence) the problem of describing the set of all sequences that satisfy a given recurrence relation has an analogy with solving an ordinary homogeneous linear differential equation with constant coefficients.

#### References

[1] | A.I. Markushevich, "Rekursive Folgen" , Deutsch. Verlag Wissenschaft. (1973) (Translated from Russian) |

#### Comments

A sequence of elements of a commutative ring with a unit element satisfies a linear recurrence relation , , if and only if the formal power series is a rational function of the form , with and a polynomial of degree .

**How to Cite This Entry:**

Recurrence relation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Recurrence_relation&oldid=15946