Polynomial sequenceIn 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. Some polynomial sequences arise in physics and approximation theory as the solutions of certain ordinary differential equations: Laguerre polynomials Chebyshev polynomials Legendre
Horner's methodIn mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. After the introduction of computers, this algorithm became fundamental for computing efficiently with polynomials.
Classical orthogonal polynomialsIn mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials). They have many important applications in such areas as mathematical physics (in particular, the theory of random matrices), approximation theory, numerical analysis, and many others.
Generalized hypergeometric functionIn mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series.
Monomial basisIn mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an immediate consequence of the definition of a polynomial). The polynomial ring K[x] of univariate polynomials over a field K is a K-vector space, which has as an (infinite) basis.
Heaviside step functionThe Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one. The function was originally developed in operational calculus for the solution of differential equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely.
