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Recurrence relation — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Recurrence_relation
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-06T13:16:45.718Z
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Recurrence relation — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Recurrence_relation (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_7ea5d4ddd162 (Metraipolis archive dc_7ea5d4ddd162, sha256 33171a6fb36ea850aa3e555c585986925701d6c63dab865765c26afb8ae586a4)

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@misc{dc7ea5d4ddd162,
  title = {Recurrence relation — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Recurrence_relation},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_7ea5d4ddd162; sha256 33171a6fb36ea850aa3e555c585986925701d6c63dab865765c26afb8ae586a4},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Recurrence_relation. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In mathematics, a recurrence relation is an equation according to which the n {\displaystyle n} th term of a sequence 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, F n = F n − 1 + F n − 2 {\displaystyle F_{n}=F_{n-1}+F_{n-2}} where the order k {\displaystyle k} is two and the linear function merely adds the two previous terms. The sequences that satisfy a recurrence relation are exactly the same as the sequences that satisfy a difference equation. More precisely, a difference equation can be associated to every recurrence relation, and, conversely, a recurrence relation can be associated to every difference equation such that the two processes are inverse one to the other, and the sequences that satisfy one of the equations satisfies the other. Because of the similarity of difference equations with differential equations, the methods of resolution of differential equations may often be applied to difference equations and thus to recurrence relations. The above example is a linear recurrence with constant coefficients. 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 are also important, since many common elementary functions 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.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «differential equation»).
Fonte: Wikipedia in inglese, voce «Recurrence relation» — https://en.wikipedia.org/wiki/Recurrence_relation
Wikidata Q740970: «sequence or array in which each further term is defined as a function of the preceding terms» (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q740970
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Recurrence_relation?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1377276563, consultata il 2026-10-06T11:32:15.969Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 2699724064ef06f59c028dea3a612490286b9c4105bec8ee8afdaf5e1b08eb02