In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a polynomial sequence is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s indexed by the nonnegative
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s 0, 1, 2, 3, ..., in which each
index
Index (: indexes or indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on the Halo Array in the ...
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
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
.
Examples
Some polynomial sequences arise in
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
and
approximation theory as the solutions of certain
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
s:
*
Laguerre polynomials
*
Chebyshev polynomials
*
Legendre polynomials
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
*
Jacobi polynomials
Others come from
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
:
*
Hermite polynomials
Many are studied in
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
and combinatorics:
*
Monomials
*
Rising factorials
*
Falling factorials
*
All-one polynomials
*
Abel polynomials
*
Bell polynomials
*
Bernoulli polynomials
*
Cyclotomic polynomial
In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th prim ...
s
*
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
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geom ...
*
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.
{{DEFAULTSORT:Polynomial Sequence
Polynomials
Sequences and series