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# Polynomial sequence

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

Abel polynomials

Bell polynomials

Bernoulli 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.

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