Polynomial sequence: Difference between revisions

Revision as of 14:57, 4 December 2007

Latest revision as of 15:36, 14 August 2021

(44 intermediate revisions by 26 users not shown)

Line 1:

{{Use American English|date = March 2019}}

In [[mathematics]], a '''polynomial sequence''' is a [[sequence]] of [[polynomial]]s indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Various special polynomial sequences are known by [[eponym]]s; among these are:

{{Short description|Sequence valued in polynomials}}

In [[mathematics]], a '''polynomial sequence''' is a [[sequence]] of [[polynomial]]s indexed by the nonnegative integers[[integer]]s 0, 1, 2, 3, ..., in which each [[indexed family|index]] is equal to the [[degree of a polynomial|degree]] of the corresponding polynomial. Various special polynomialPolynomial sequences are knowna bytopic of interest in [[eponymenumerative combinatorics]]s; amongand these[[algebraic are:combinatorics]], as well as [[applied mathematics]].

== Examples ==

<div style="-moz-column-count:3; column-count:3;">

Some polynomial sequences arise in [[physics]] and [[approximation theory]] as the solutions of certain [[ordinary differential equation]]s:

* [[Laguerre polynomials]]

* [[Chebyshev polynomials]]

* [[Legendre polynomials]]

* [[Jacobi polynomials]]

Others come from [[statistics]]:

* [[Hermite polynomials]]

Many are studied in [[algebra]] and combinatorics:

* [[Monomial]]s

* [[Rising factorial]]s

* [[Falling factorial]]s

* [[All-one polynomial]]s

* [[Abel polynomials]]

* [[Bell polynomials]]

* [[Bernoulli polynomials]]

* [[BoubakerCyclotomic polynomialspolynomial]]s

* [[ChebyshevDickson polynomialspolynomial]]s

* [[Fibonacci polynomials]]

* [[HermiteLagrange polynomials]]

* [[LegendreLucas polynomials]]

* [[Laguerre polynomials]]

* [[Spread polynomials]]

* [[Touchard polynomials]]

* [[Rook polynomials]]

</div>

==Classes of polynomial sequences==

* Polynomial sequences of [[binomial type]]

* [[Orthogonal polynomials]]

* [[Secondary polynomials].]

* [[Sheffer sequence]]

* [[Sturm sequence]]

* [[Generalized Appell polynomials]]

==See also==

*[[Umbral calculus]]

==References==

* Aigner, Martin. "A course in enumeration", GTM Springer, 2007, {{isbn|3-540-39032-4}} p21.

* Roman, Steven "The Umbral Calculus", Dover Publications, 2005, {{isbn|978-0-486-44139-9}}.

* Williamson, S. Gill "Combinatorics for Computer Science", Dover Publications, (2002) p177.

{{DEFAULTSORT:Polynomial Sequence}}

[[Category:Polynomials]]

[[Category:Sequences and series]]

[[fr:Suite de polynômes]]

[[it:Sequenza polinomiale]]