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Polynomial Sequence
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as in connection with Brownian motion; * combinatorics, as an example of an Appell sequence, obeying the umbral calculus; * numerical analysis as Gaussian quadrature; * physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term \beginxu_\end is present); * systems theory in connection with nonlinear operations on Gaussian noise. * random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Polynomials
In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials. Definition These Fibonacci polynomials are defined by a recurrence relation:Benjamin & Quinn p. 141 :F_n(x)= \begin 0, & \mbox n = 0\\ 1, & \mbox n = 1\\ x F_(x) + F_(x),& \mbox n \geq 2 \end The Lucas polynomials use the same recurrence with different starting values: :L_n(x) = \begin 2, & \mbox n = 0 \\ x, & \mbox n = 1 \\ x L_(x) + L_(x), & \mbox n \geq 2. \end They can be defined for negative indices bySpringer :F_(x)=(-1)^F_(x), :L_(x)=(-1)^nL_(x). The Fibonacci polynomials form a sequence of orthogonal polynomials with A_n=C_n=1 and B_n=0. Examples The first few Fibonacci polynomials are: :F_0(x)=0 \, :F_1(x)=1 \, :F_2(x)=x \, :F_3(x)=x^2+1 \, :F_4(x)=x^3+2x \, :F_5(x)=x^4+3x^2+1 \, :F_6(x)=x^5+4x^3+3x \, The first few Lucas polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Polynomials
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' and the y_j are called ''values''. The Lagrange polynomial L(x) has degree \leq k and assumes each value at the corresponding node, L(x_j) = y_j. Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler. Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration and Shamir's secret sharing scheme in cryptography. For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. Definition Given a set of k + 1 nodes \, which must all be distinct, x_j \neq x_m for indices j \neq m, the Lagrange basis for polynomials of degre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Polynomials
In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials. Definition These Fibonacci polynomials are defined by a recurrence relation:Benjamin & Quinn p. 141 :F_n(x)= \begin 0, & \mbox n = 0\\ 1, & \mbox n = 1\\ x F_(x) + F_(x),& \mbox n \geq 2 \end The Lucas polynomials use the same recurrence with different starting values: :L_n(x) = \begin 2, & \mbox n = 0 \\ x, & \mbox n = 1 \\ x L_(x) + L_(x), & \mbox n \geq 2. \end They can be defined for negative indices bySpringer :F_(x)=(-1)^F_(x), :L_(x)=(-1)^nL_(x). The Fibonacci polynomials form a sequence of orthogonal polynomials with A_n=C_n=1 and B_n=0. Examples The first few Fibonacci polynomials are: :F_0(x)=0 \, :F_1(x)=1 \, :F_2(x)=x \, :F_3(x)=x^2+1 \, :F_4(x)=x^3+2x \, :F_5(x)=x^4+3x^2+1 \, :F_6(x)=x^5+4x^3+3x \, The first few Lucas polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dickson Polynomial
In mathematics, the Dickson polynomials, denoted , form a polynomial sequence introduced by . They were rediscovered by in his study of Brewer sums and have at times, although rarely, been referred to as Brewer polynomials. 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 polynomials are generally studied over finite fields, where they sometimes may not be equivalent to Chebyshev polynomials. One of the main reasons for interest in them is that for fixed , they give many examples of ''permutation polynomials''; polynomials acting as permutations of finite fields. Definition First kind For integer and in a commutative ring with identity (often chosen to be the finite field ) the Dickson polynomials (of the first kind) over are given by :D_n(x,\alpha)=\sum_^\frac \binom (-\alpha)^i x^ \,. The first few Dickson poly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 primitive roots of unity e^ , where ''k'' runs over the positive integers not greater than ''n'' and coprime to ''n'' (and ''i'' is the imaginary unit). In other words, the ''n''th cyclotomic polynomial is equal to : \Phi_n(x) = \prod_\stackrel \left(x-e^\right). It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive ''n''th-root of unity ( e^ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is :\prod_\Phi_d(x) = x^n - 1, showing that is a root of x^n - 1 if and only if it is a ''d''th primitive root of unity for some ''d'' that divides ''n''. Examples If ''n'' is a prim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Polynomials
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in the study of many special functions and, in particular, the Riemann zeta function and the Hurwitz zeta function. They are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator). For the Bernoulli polynomials, the number of crossings of the ''x''-axis in the unit interval does not go up with the degree. In the limit of large degree, they approach, when appropriately scaled, the sine and cosine functions. A similar set of polynomials, based on a generating function, is the family of Euler polynomials. Representations The Bernoulli polynomials ''B''''n'' can be defined by a generating function. They also admit a variety of derived representations. Generating functions The generating function for the Bernoulli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bell Polynomials
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno's formula. Definitions Exponential Bell polynomials The ''partial'' or ''incomplete'' exponential Bell polynomials are a triangular array of polynomials given by :B_(x_1,x_2,\dots,x_) = \sum \left(\right)^\left(\right)^\cdots\left(\right)^, where the sum is taken over all sequences ''j''1, ''j''2, ''j''3, ..., ''j''''n''−''k''+1 of non-negative integers such that these two conditions are satisfied: :j_1 + j_2 + \cdots + j_ = k, :j_1 + 2 j_2 + 3 j_3 + \cdots + (n-k+1)j_ = n. The sum :B_n(x_1,\dots,x_n)=\sum_^n B_(x_1,x_2,\dots,x_) is called the ''n''th ''complete exponential Bell polynomial''. Ordinary Bell polynomials Likewise, the partial ''ordinary'' Bell polynomial is defined by :\hat_(x_1,x_2,\ldots,x_) = \sum \frac x_1^ x_2^ \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel Polynomials
The Abel polynomials in mathematics form a polynomial sequence, the ''n''th term of which is of the form :p_n(x)=x(x-an)^. The sequence is named after Niels Henrik Abel (1802-1829), the Norwegian mathematician. This polynomial sequence is of binomial type: conversely, every polynomial sequence of binomial type may be obtained from the Abel sequence in the 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 .... Examples For , the polynomials are :p_0(x)=1; :p_1(x)=x; :p_2(x)=-2x+x^2; :p_3(x)=9x-6x^2+x^3; :p_4(x)=-64x +48x^2-12x^3+x^4; For , the polynomials are :p_0(x)=1; :p_1(x)=x; :p_2(x)=-4x+x^2; :p_3(x)=36x-12x^2+x^3; :p_4(x)=-512x +192x^2-24x^3+x^4; :p_5(x)=10000x-4000x^2+600x^3-40x^4+x^5; :p_6(x)=-248832x+103680x^2-17280x^3+1440x^4-60x^5+x^6; References * Ext ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
All-one Polynomial
In mathematics, an all one polynomial (AOP) is a polynomial in which all coefficients are one. Over the finite field of order two, conditions for the AOP to be irreducible are known, which allow this polynomial to be used to define efficient algorithms and circuits for multiplication in finite fields of characteristic two.. The AOP is a 1- equally spaced polynomial. Definition An AOP of degree ''m'' has all terms from ''x''''m'' to ''x''0 with coefficients of 1, and can be written as :AOP_m(x) = \sum_^ x^i or :AOP_m(x) = x^m + x^ + \cdots + x + 1 or :AOP_m(x) = . Thus the roots of the all one polynomial of degree ''m'' are all (''m''+1)th roots of unity other than unity itself. Properties Over GF(2) the AOP has many interesting properties, including: *The Hamming weight of the AOP is ''m'' + 1, the maximum possible for its degree *The AOP is irreducible if and only if ''m'' + 1 is prime and 2 is a primitive root modulo ''m'' + 1 (over GF(''p'') with prime ''p'', it is irre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Falling Factorial
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \end The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as :\begin x^ = x^\overline &= \overbrace^ \\ &= \prod_^n(x+k-1) = \prod_^(x+k) \,. \end The value of each is taken to be 1 (an empty product) when . These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation , where is a non-negative integer. It may represent ''either'' the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used with yet another meaning, namely to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
