In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 called ...
of
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s indexed by the nonnegative
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s 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
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
.
Examples
Some polynomial sequences arise in
physics
Physics is the natural science that studies matter, its 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 which ...
and
approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' wil ...
as the solutions of certain
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s:
*
Laguerre polynomials
*
Chebyshev polynomials
The Chebyshev polynomials are two sequences of polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric ...
*
Legendre polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applica ...
*
Jacobi polynomials
Others come from
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
:
*
Hermite polynomials
Many are studied in
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
and combinatorics:
*
Monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
s
*
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
In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They a ...
*
Sturm sequence
In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of loc ...
*
Generalized Appell polynomials
See also
*
Umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blis ...
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