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Rogers–Ramanujan Identities
In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and partition (number theory), integer partitions. The identities were first discovered and proved by , and were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof . independently rediscovered and proved the identities. Definition The Rogers–Ramanujan identities are :G(q) = \sum_^\infty \frac = \frac =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots and :H(q) =\sum_^\infty \frac = \frac =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots . Here, (a;q)_n denotes the q-Pochhammer symbol. Combinatorial interpretation Consider the following: * \frac is the generating function for partitions with exactly n parts such that adjacent parts have difference at least 2. * \frac is the generating function for partitions such that each part is congrue ... [...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] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-analogs
In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q''-analogs that arise naturally, rather than in arbitrarily contriving ''q''-analogs of known results. The earliest ''q''-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, , , ''q''-analogues are most frequently studied in the mathematical fields of combinatorics and special functions. In these settings, the limit is often formal, as is often discrete-valued (for example, it may represent a prime power). ''q''-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of chaotic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Identities
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] |
Integer Partitions
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, can be partitioned in five distinct ways: : : : : : The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . A summand in a partition is also called a part. The number of partitions of is given by the partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation, group representation theory in general. Examples The seven partitions of 5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Igor Pak
Igor Pak (russian: link=no, Игорь Пак) (born 1971, Moscow, Soviet Union) is a professor of mathematics at the University of California, Los Angeles, working in combinatorics and discrete probability. He formerly taught at the Massachusetts Institute of Technology and the University of Minnesota, and he is best known for his bijective proof of the Young tableau#Dimension of a representation, hook-length formula for the number of Young tableaux, and his work on random walks. He was a keynote speaker alongside George Andrews (mathematician), George Andrews and Doron Zeilberger at the 2006 Harvey Mudd College Mathematics Conference on Enumerative Combinatorics. Pak is an Associate Editor for the journal Discrete Mathematics (journal), ''Discrete Mathematics''. He gave a László Fejes Tóth, Fejes Tóth Lecture at the University of Calgary in February 2009. In 2018, he was an List of International Congresses of Mathematicians Plenary and Invited Speakers#2018, Rio de Janeiro, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruce C
The English language name Bruce arrived in Scotland with the Normans, from the place name Brix, Manche in Normandy, France, meaning "the willowlands". Initially promulgated via the descendants of king Robert the Bruce (1274−1329), it has been a Scottish surname since medieval times; it is now a common given name. The variant ''Lebrix'' and ''Le Brix'' are French variations of the surname. Actors * Bruce Bennett (1906–2007), American actor and athlete * Bruce Boxleitner (born 1950), American actor * Bruce Campbell (born 1958), American actor, director, writer, producer and author * Bruce Davison (born 1946), American actor and director * Bruce Dern (born 1936), American actor * Bruce Gray (1936–2017), American-Canadian actor * Bruce Greenwood (born 1956), Canadian actor and musician * Bruce Herbelin-Earle (born 1998), English-French actor and model * Bruce Jones (born 1953), English actor * Bruce Kirby (1925–2021), American actor * Bruce Lee (1940–1973), martial ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Q-Hermite Polynomials
In mathematics, the continuous ''q''-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called h ...s by :H_n(x, q)=e^_2\phi_0\left begin q^,0\\ -\end ;q,q^n e^\right\quad x=\cos\,\theta. Recurrence and difference relations : 2x H_n(x\mid q) = H_ (x\mid q) + (1-q^n) H_ (x\mid q) with the initial conditions : H_0 (x\mid q) =1, H_ (x\mid q) = 0 From the above, one can easily calculate: : \begin H_0 (x\mid q) & = 1 \\ H_1 (x\mid q) & = 2x \\ H_2 (x\mid q) & = 4x^2 - (1-q) \\ H_3 (x\mid q) & = 8x^3 - 2x(2-q-q^2) \\ H_4 (x\mid q) & = 16x^4 - 4x^2(3-q-q^2-q^3) + (1- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rogers Polynomials
In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by in the course of his work on the Rogers–Ramanujan identities. They are ''q''-analogs of ultraspherical polynomials, and are the Macdonald polynomials for the special case of the ''A''1 affine root system . and discuss the properties of Rogers polynomials in detail. Definition The Rogers polynomials can be defined in terms of the ''q''-Pochhammer symbol and the basic hypergeometric series In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called ... by : C_n(x;\beta, q) = \frace^ _2\phi_1(q^,\beta;\beta^q^;q,q\beta^e^) where ''x'' = cos(''θ''). References * * * * * *{{Citation , last1=Rogers , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stefano Capparelli
Stefano is the Italian form of the masculine given name Στέφανος (Stefanos, Stephen). The name is of Greek origin, Στέφανος, meaning a person who made a significant achievement and has been crowned. In Orthodox Christianity the achievement is in the realm of virtues, αρετές, therefore the name signifies a person who had triumphed over passions and gained the relevant virtues. In Italian, the stress falls usually on the first syllable, (an exception is the Apulian surname ''Stefano'', ); in English it is often mistakenly placed on the second, . People with the given name Stefano * Stefano (wrestler), ring name of Daniel Garcia Soto, professional wrestler * Stefano Borgia (1731–1804), Italian Cardinal, theologian, antiquarian, and historian * Stefano Bertacco (1962–2020), Italian politician * Stefano Cagol (born 1969), Italian artist * Stefano Casiraghi (1960–1990), Italian socialite * Stefano Cavazzoni (1881–1951), Italian politician * Stefano Erardi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Lie Algebra
In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities. Affine Lie algebras play an important role in string theory and two-dimensional conformal field theory due to the way they are constructed: starting from a simple Lie algebra \mathfrak, one considers the loop algebra, L\mathfrak, formed by the \mathfrak-valued functions on a circle (interpreted as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
