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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Poisson Distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson (; ). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume. For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution with mean 3: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very small probability it could be 10. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Constant Term
In mathematics, a constant term is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial :x^2 + 2x + 3,\ the 3 is a constant term. After like terms are combined, an algebraic expression will have at most one constant term. Thus, it is common to speak of the quadratic polynomial :ax^2+bx+c,\ where x is the variable, as having a constant term of c. If the constant term is 0, then it will conventionally be omitted when the quadratic is written out. Any polynomial written in standard form has a unique constant term, which can be considered a coefficient of x^0. In particular, the constant term will always be the lowest degree term of the polynomial. This also applies to multivariate polynomials. For example, the polynomial :x^2+2xy+y^2-2x+2y-4\ has a constant term of −4, which can be considered to be the coefficient of x^0y^0, where the variables are eliminated by being exponentiated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 (exc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). The integral is evaluated for all values of shift, producing the convolution function. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution (f*g) differs from cross-correlation (f \star g) only in that either or is reflected about the y-axis in convolution; thus it is a cross-correlation of and , or and . For complex-valued fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Andrew Odlyzko
Andrew Michael Odlyzko (Andrzej Odłyżko) (born 23 July 1949) is a Polish- American mathematician and a former head of the University of Minnesota's Digital Technology Center and of the Minnesota Supercomputing Institute. He began his career in 1975 at Bell Telephone Laboratories, where he stayed for 26 years before joining the University of Minnesota in 2001. Work in mathematics Odlyzko received his B.S. and M.S. in mathematics from the California Institute of Technology and his Ph.D. from the Massachusetts Institute of Technology in 1975. In the field of mathematics he has published extensively on analytic number theory, computational number theory, cryptography, algorithms and computational complexity, combinatorics, probability, and error-correcting codes. In the early 1970s, he was a co-author (with D. Kahaner and Gian-Carlo Rota) of one of the founding papers of the modern umbral calculus. In 1985 he and Herman te Riele disproved the Mertens conjecture. In mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Gian-Carlo Rota
Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, probability theory, and phenomenology. Early life and education Rota was born in Vigevano, Italy. His father, Giovanni, an architect and prominent antifascist, was the brother of the mathematician Rosetta, who was the wife of the writer Ennio Flaiano. Gian-Carlo's family left Italy when he was 13 years old, initially going to Switzerland. Rota attended the Colegio Americano de Quito in Ecuador, and graduated with an A.B. in mathematics from Princeton University in 1953 after completing a senior thesis, titled "On the solubility of linear equations in topological vector spaces", under the supervision of William Feller. He then pursued graduate studies at Yale University, where he received a Ph.D. in mathematics in 1956 after completing a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, ''c'' (the ''center'' of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Difference Operator
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Delta Operator
In mathematics, a delta operator is a shift-equivariant linear operator Q\colon\mathbb \longrightarrow \mathbb /math> on the vector space of polynomials in a variable x over a field \mathbb that reduces degrees by one. To say that Q is shift-equivariant means that if g(x) = f(x + a), then :.\, In other words, if f is a "shift" of g, then Qf is also a shift of Qg, and has the same "shifting vector" a. To say that an operator ''reduces degree by one'' means that if f is a polynomial of degree n, then Qf is either a polynomial of degree n-1, or, in case n = 0, Qf is 0. Sometimes a ''delta operator'' is defined to be a shift-equivariant linear transformation on polynomials in x that maps x to a nonzero constant. Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition when \mathbb has characteristic zero, since shift-equivariance is a fairly strong condition. Examples * The forward difference operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Shift-equivariant
In mathematics, a delta operator is a shift-equivariant linear operator Q\colon\mathbb \longrightarrow \mathbb /math> on the vector space of polynomials in a variable x over a field \mathbb that reduces degrees by one. To say that Q is shift-equivariant means that if g(x) = f(x + a), then :.\, In other words, if f is a "shift" of g, then Qf is also a shift of Qg, and has the same "shifting vector" a. To say that an operator ''reduces degree by one'' means that if f is a polynomial of degree n, then Qf is either a polynomial of degree n-1, or, in case n = 0, Qf is 0. Sometimes a ''delta operator'' is defined to be a shift-equivariant linear transformation on polynomials in x that maps x to a nonzero constant. Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition when \mathbb has characteristic zero, since shift-equivariance is a fairly strong condition. Examples * The forward difference operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |