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Difference between revisions of "Recursive sequence"

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''recurrent sequence''
 
''recurrent sequence''
  
A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080330/r0803301.png" /> that satisfies a relation
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A sequence $a_0,a_1,\ldots,$ that satisfies a relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080330/r0803302.png" /></td> </tr></table>
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$$a_{n+p}+c_1a_{n+p-1}+\ldots+c_pa_n=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080330/r0803303.png" /> are constants. The relation permits one to compute the terms of the sequence one by one, in succession, if the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080330/r0803304.png" /> terms are known. A classical example of such a sequence is the Fibonacci sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080330/r0803305.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080330/r0803306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080330/r0803307.png" />). A recursive series is a [[Power series|power series]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080330/r0803308.png" /> whose coefficients form a recursive sequence. Such a series represents an everywhere-defined rational function.
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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=31683
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article