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.
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
[edit]Some polynomial sequences arise in physics and approximation theory as the solutions of certain ordinary differential equations:
Others come from statistics:
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
[edit]- Polynomial sequences of binomial type
- Orthogonal polynomials
- Secondary polynomials
- Sheffer sequence
- Sturm sequence
- Generalized Appell polynomials
See also
[edit]References
[edit]- 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.