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Polynomial sequence

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
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

References

Aigner, Martin. "A course in enumeration", GTM Springer, 2007, p21.
Roman, Steven "The Umbral Calculus", Dover Publications, 2005, .
Williamson, S. Gill "Combinatorics for Computer Science", Dover Publications, (2002) p177.

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