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Wilf–Zeilberger Pair
In mathematics, specifically combinatorics, a Wilf–Zeilberger pair, or WZ pair, is a pair of functions that can be used to certify certain combinatorial identities. WZ pairs are named after Herbert S. Wilf and Doron Zeilberger, and are instrumental in the evaluation of many sums involving binomial coefficients, factorials, and in general any hypergeometric series. A function's WZ counterpart may be used to find an equivalent and much simpler sum. Although finding WZ pairs by hand is impractical in most cases, Gosper's algorithm provides a sure method to find a function's WZ counterpart, and can be implemented in a symbolic manipulation program. Definition Two functions ''F'' and ' ... [...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] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity (mathematics)
In mathematics, an identity is an equality relating one mathematical expression ''A'' to another mathematical expression ''B'', such that ''A'' and ''B'' (which might contain some variables) produce the same value for all values of the variables within a certain range of validity. In other words, ''A'' = ''B'' is an identity if ''A'' and ''B'' define the same functions, and an identity is an equality between functions that are differently defined. For example, (a+b)^2 = a^2 + 2ab + b^2 and \cos^2\theta + \sin^2\theta =1 are identities. Identities are sometimes indicated by the triple bar symbol instead of , the equals sign. Common identities Algebraic identities Certain identities, such as a+0=a and a+(-a)=0, form the basis of algebra, while other identities, such as (a+b)^2 = a^2 + 2ab +b^2 and a^2 - b^2 = (a+b)(a-b), can be useful in simplifying algebraic expressions and expanding them. Trigonometric identities Geometrically, trigonometric ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert S
Herbert may refer to: People Individuals * Herbert (musician), a pseudonym of Matthew Herbert Name * Herbert (given name) * Herbert (surname) Places Antarctica * Herbert Mountains, Coats Land * Herbert Sound, Graham Land Australia * Herbert, Northern Territory, a rural locality * Herbert, South Australia. former government town * Division of Herbert, an electoral district in Queensland * Herbert River, a river in Queensland * County of Herbert, a cadastral unit in South Australia Canada * Herbert, Saskatchewan, Canada, a town * Herbert Road, St. Albert, Canada New Zealand * Herbert, New Zealand, a town * Mount Herbert (New Zealand) United States * Herbert, Illinois, an unincorporated community * Herbert, Michigan, a former settlement * Herbert Creek, a stream in South Dakota * Herbert Island, Alaska Arts, entertainment, and media Fictional entities * Herbert (Disney character) * Herbert Pocket (''Great Expectations'' character), Pip's close friend and roommate in the Cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doron Zeilberger
Doron Zeilberger (דורון ציילברגר, born 2 July 1950 in Haifa, Israel) is an Israeli mathematician, known for his work in combinatorics. Education and career He received his doctorate from the Weizmann Institute of Science in 1976, under the direction of Harry Dym, with the thesis "New Approaches and Results in the Theory of Discrete Analytic Functions." He is a Board of Governors Professor of Mathematics at Rutgers University. Contributions Zeilberger has made contributions to combinatorics, hypergeometric identities, and q-series. Zeilberger gave the first proof of the alternating sign matrix conjecture, noteworthy not only for its mathematical content, but also for the fact that Zeilberger recruited nearly a hundred volunteer checkers to "pre-referee" the paper. In 2011, together with Manuel Kauers and Christoph Koutschan, Zeilberger proved the ''q''-TSPP conjecture, which was independently stated in 1983 by George Andrews and David P. Robbins. Zeilberger is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Summation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined. Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article. The summation of an explicit sequence is denoted as a succession of additions. For example, summation of is denoted , and results in 9, that is, . Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0. Very often, the elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book '' Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential function an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosper's Algorithm
In mathematics, Gosper's algorithm, due to Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has ''a''(1) + ... + ''a''(''n'') = ''S''(''n'') − ''S''(0), where ''S''(''n'') is a hypergeometric term (i.e., ''S''(''n'' + 1)/''S''(''n'') is a rational function of ''n''); then necessarily ''a''(''n'') is itself a hypergeometric term, and given the formula for ''a''(''n'') Gosper's algorithm finds that for ''S''(''n''). Outline of the algorithm Step 1: Find a polynomial ''p'' such that, writing ''b''(''n'') = ''a''(''n'')/''p''(''n''), the ratio ''b''(''n'')/''b''(''n'' − 1) has the form ''q''(''n'')/''r''(''n'') where ''q'' and ''r'' are polynomials and no ''q''(''n'') has a nontrivial factor with ''r''(''n'' + ''j'') for ''j'' = 0, 1, 2, ... . (This is always possible, whether or not the series is summable in cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Algebra System
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "computer algebra" or "symbolic computation", which has spurred work in algorithms over mathematical objects such as polynomials. Computer algebra systems may be divided into two classes: specialized and general-purpose. The specialized ones are devoted to a specific part of mathematics, such as number theory, group theory, or teaching of elementary mathematics. General-purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions. To be useful, a general-purpose computer algebra system must include various features such as: *a user interface allo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
