# Difference between revisions of "Recursive sequence"

From Encyclopedia of Mathematics

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''recurrent sequence'' | ''recurrent sequence'' | ||

− | A sequence | + | A sequence $a_0,a_1,\ldots,$ that satisfies a relation |

− | + | $$a_{n+p}+c_1a_{n+p-1}+\ldots+c_pa_n=0,$$ | |

− | where | + | where $c_1,\ldots,c_p$ are constants. The relation permits one to compute the terms of the sequence one by one, in succession, if the first $p$ terms are known. A classical example of such a sequence is the Fibonacci sequence $1,1,2,3,5,8$ ($a_{n+2}=a_{n+1}+a_n$, $a_0=a_1=1$). A recursive series is a [[Power series|power series]] $a_0+a_1x+a_2x^2+\ldots$ whose coefficients form a recursive sequence. Such a series represents an everywhere-defined rational function. |

## Revision as of 20:46, 14 April 2014

*recurrent sequence*

A sequence $a_0,a_1,\ldots,$ that satisfies a relation

$$a_{n+p}+c_1a_{n+p-1}+\ldots+c_pa_n=0,$$

where $c_1,\ldots,c_p$ are constants. The relation permits one to compute the terms of the sequence one by one, in succession, if the first $p$ terms are known. A classical example of such a sequence is the Fibonacci sequence $1,1,2,3,5,8$ ($a_{n+2}=a_{n+1}+a_n$, $a_0=a_1=1$). A recursive series is a power series $a_0+a_1x+a_2x^2+\ldots$ whose coefficients form a recursive sequence. Such a series represents an everywhere-defined rational function.

#### Comments

A good reference treating many aspects of such sequences is [a1].

#### References

[a1] | A.J. van der Poorten, "Some facts that should be better known, especially about rational functions" R.A. Molin (ed.) , Number theory and applications (Proc. First Conf. Canadian Number Theory Assoc., Banff, April 1988) , Kluwer (1989) pp. 497–528 |

**How to Cite This Entry:**

Recursive sequence.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Recursive_sequence&oldid=12807

This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article