Polynomial sequence: Difference between revisions

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

A '''polynomial sequence''' is a sequence of polynomials 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}}

AIn [[mathematics]], a '''polynomial sequence''' is a [[sequence]] of polynomials[[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==

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

* [[HermiteLaguerre polynomials]]

* [[Chebyshev polynomials]]

* [[Hermite polynomials]]

* [[Legendre polynomials]]

* [[LaguerreJacobi 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]]

* [[Cyclotomic polynomial]]s

* [[Dickson polynomial]]s

* [[Fibonacci polynomials]]

* [[Lagrange polynomials]]

* [[Lucas polynomials]]

* [[Spread polynomials]]

* [[Touchard polynomials]]

* [[Rook polynomials]]

==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.

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[This article is a stub; it needs a lot of work.]

[[Category:Polynomials]]

[[Category:Sequences and series]]