Polynomial sequence: Difference between revisions

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

== Definition ==

{{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 [[enumerative combinatorics]] and [[algebraic 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:

* [[Laguerre polynomials]]

* [[TouchardChebyshev polynomials]]

* [[Legendre polynomials]]

* [[Jacobi polynomials]]

Others come from [[statistics]]:

==See also==

* [[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.

{{DEFAULTSORT:Polynomial Sequence}}

[[Category:Polynomials]]

[[Category:Sequences and series]]