Polynomial sequence explained

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

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.