In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a polynomial sequence is a sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

 of polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal

Equal

Equal commonly refers to a state of equality.Equal or equals may also refer to:* Equal , a brand of artificial sweetener* EQUAL Community Initiative, an initiative within the European Social Fund of the European Union...

 to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics

Algebraic combinatorics

Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra....

, as well as applied mathematics

Applied mathematics

Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

.

Examples

Some polynomial sequences arise in physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 and approximation theory

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...

 as the solutions of certain ordinary differential equation

Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

s:

• Laguerre polynomials
  Laguerre polynomials
  In mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:x\,y + \,y' + n\,y = 0\,which is a second-order linear differential equation....
  
  
• Chebyshev polynomials
  Chebyshev polynomials
  In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
  
  
• Legendre polynomials
• Bessel function
  Bessel function
  In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
  
  s
• Jacobi polynomials
  Jacobi polynomials
  In mathematics, Jacobi polynomials are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight ^\alpha ^\beta on the interval [-1, 1]...
  
  

Others come from statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

:

• Hermite polynomials
  Hermite polynomials
  In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where...
  
  

Many are studied in algebra and combinatorics:

• Monomial
  Monomial
  In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
  
  s
• Rising factorials
• Falling factorials
• Abel polynomials
• Bell polynomials
• Bernoulli polynomials
  Bernoulli polynomials
  In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator...
  
  
• Dickson polynomial
  Dickson polynomial
  In mathematics, the Dickson polynomials, denoted Dn, form a polynomial sequence studied by .Over the complex numbers, Dickson polynomials are essentially equivalent to Chebyshev polynomials with a change of variable, and in fact Dickson polynomials are sometimes called Chebyshev polynomials.Dickson...
  
  s
• Fibonacci polynomials
• Lagrange polynomials
• Lucas polynomials
• Spread polynomials
  Spread polynomials
  In the conventional language of trigonometry, the nth-degree spread polynomial Sn, for n = 0, 1, 2, ..., may be characterized by the trigonometric identity\sin^2 = S_n.\,...
  
  
• Touchard polynomials
• Rook polynomials

Classes of polynomial sequences

• Polynomial sequences of binomial type
• Orthogonal polynomials
  Orthogonal polynomials
  In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
  
  
• Secondary polynomials
• Sheffer sequence
• Sturm sequence
• Generalized Appell polynomials