Polynomial sequence

{{Use American English|date = March 2019}} {{Short description|Sequence valued in polynomials}} In mathematics, 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. Polynomial sequences are a topic 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 equations: * Laguerre polynomials * Chebyshev polynomials * Legendre polynomials * Zernike polynomials * Jacobi polynomials

Others come from statistics: * Hermite polynomials

Many are studied in algebra and combinatorics: * Monomials * Rising factorials * Falling factorials * All-one polynomials * Abel polynomials * Bell polynomials * Bernoulli polynomials * Cyclotomic polynomials * Dickson polynomials * 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