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, p21.
- Roman, Steven "The Umbral Calculus", Dover Publications, 2005, .
- Williamson, S. Gill "Combinatorics for Computer Science", Dover Publications, (2002) p177.