|
Stirling Polynomial
In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, which are closely related to the Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials. There are multiple variants of the ''Stirling polynomial'' sequence considered below most notably including the Sheffer sequence form of the sequence, S_k(x), defined characteristically through the special form of its exponential generating function, and the ''Stirling (convolution) polynomials'', \sigma_n(x), which also satisfy a characteristic ''ordinary'' generating function and that are of use in generalizing the Stirling numbers (of both kinds) to arbitrary complex-valued inputs. We consider the "''convolution polynomial''" variant of this sequence and its properties second in the last subsection of the article. Still other variants of the Stirling polynomials are studied in the supplementary links to the ar ... [...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 t ... [...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 Be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (excep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Polynomials
In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and the Stirling interpolation polynomials as special cases. Definition The general difference polynomial sequence is given by :p_n(z)=\frac where is the binomial coefficient. For \beta=0, the generated polynomials p_n(z) are the Newton polynomials :p_n(z)= = \frac. The case of \beta=1 generates Selberg's polynomials, and the case of \beta=-1/2 generates Stirling's interpolation polynomials. Moving differences Given an analytic function f(z), define the moving difference of ''f'' as :\mathcal_n(f) = \Delta^n f (\beta n) where \Delta is the forward difference operator. Then, provided that ''f'' obeys certain summability conditions, then it may be represented in terms of these polynomials as :f(z)=\sum_^\infty p_n(z) \mathcal_n(f). The conditions for sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Appell Sequence
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence \_ satisfying the identity :\frac p_n(x) = np_(x), and in which p_0(x) is a non-zero constant. Among the most notable Appell sequences besides the trivial example \ are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments. Equivalent characterizations of Appell sequences The following conditions on polynomial sequences can easily be seen to be equivalent: * For n = 1, 2, 3,\ldots, ::\frac p_n(x) = n p_(x) :and p_0(x) is a non-zero constant; * For some sequence \_^ of scalars with c_0 \neq 0, ::p_n(x) = \sum_^n \binom c_k x^; * For the same sequence of scalars, ::p_n(x) = \left(\sum_^\infty \frac D^k\right) x^n, :where ::D = \frac; * For n=0,1,2,\ldots, ::p_n(x+y) = \sum_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Polynomials Of The Second Kind
The Bernoulli polynomials of the second kind , also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating function: : \frac= \sum_^\infty z^n \psi_n(x) ,\qquad , z, -1 and :\gamma=\sum_^\infty\frac\Big\, \quad a>-1 where is Euler's constant. Furthermore, we also have : \Psi(v)= \frac\left\,\qquad \Re(v)>-a, where is the gamma function. The Hurwitz and Riemann zeta functions may be expanded into these polynomials as follows : \zeta(s,v)= \frac + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom (k+v)^ and : \zeta(s)= \frac + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom (k+1)^ and also : \zeta(s) =1 + \frac + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom (k+2)^ The Bernoulli polynomials of the second kind are also involved in the following relationship : \big(v+a-\tfrac\big)\zeta(s,v) = -\frac + \zeta(s-1,v) + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom (k+v)^ between the zeta functions, as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Functions
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Equation
In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a ''functional equation'' is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the ''logarithmic functional equation'' \log(xy)=\log(x) + \log(y). If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term ''functional equation'' is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bell Numbers
In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, they are named after Eric Temple Bell, who wrote about them in the 1930s. The Bell numbers are denoted B_n, where n is an integer greater than or equal to zero. Starting with B_0 = B_1 = 1, the first few Bell numbers are :1, 1, 2, 5, 15, 52, 203, 877, 4140, ... . The Bell number B_n counts the number of different ways to partition a set that has exactly n elements, or equivalently, the number of equivalence relations on it. B_n also counts the number of different rhyme schemes for n -line poems. As well as appearing in counting problems, these numbers have a different interpretation, as moments of probability distributions. In particular, B_n is the n -th moment of a Poisson distribution with mean 1. Counting Set partitions In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre Polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only if is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of xy'' + (\alpha + 1 - x)y' + ny = 0~. where is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin). More generally, a Laguerre function is a solution when is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form \int_0^\infty f(x) e^ \, dx. These polynomials, usually denoted , , …, are a polynomial sequence which may be defined by the Rodrigues for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Interpolation
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 deg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers \left\ in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities :p_n(x+y)=\sum_^n\, p_k(x)\, p_(y). Many such sequences exist. The set of all such sequences forms a Lie group under the operation of umbral composition, explained below. Every sequence of binomial type may be expressed in terms of the Bell polynomials. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type). Polynomial sequences put on firm footing the vague 19th century notions of umbral calculus. Examples * In consequence of this definition the binomial theorem can be stated by saying that the sequence is of binomial type. * The sequence of " lower factorials" is defined by(x)_n=x(x-1)(x-2)\cdot\cdots\cdot(x-n+1).(In the theory of special functions, this same notation denotes upper f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
