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Slater's Identities
In mathematics, a Bailey pair is a pair of sequences satisfying certain relations, and a Bailey chain is a sequence of Bailey pairs. Bailey pairs were introduced by while studying the second proof Rogers 1917 of the Rogers–Ramanujan identities, and Bailey chains were introduced by . Definition The q-Pochhammer symbol In the mathematical field of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer ...s (a;q)_n are defined as: :(a;q)_n = \prod_(1-aq^j) = (1-a)(1-aq)\cdots(1-aq^). A pair of sequences (α''n'',β''n'') is called a Bailey pair if they are related by :\beta_n=\sum_^n\frac or equivalently :\alpha_n = (1-aq^)\sum_^n\frac. Bailey's lemma Bailey's lemma states that if (α''n'',β''n'') is a Bailey pair, then so is (α'''n'',β'''n'') where :\alpha^\prime_n= \frac :\beta^\prime_n = \sum_\frac. In other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rogers–Ramanujan Identities
In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and 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 congruent to either 1 or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-Pochhammer Symbol
In the mathematical field of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symbol (x)_n = x(x+1)\dots(x+n-1), in the sense that \lim_ \frac = (x)_n. The ''q''-Pochhammer symbol is a major building block in the construction of ''q''-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike the ordinary Pochhammer symbol, the ''q''-Pochhammer symbol can be extended to an infinite product: (a;q)_\infty = \prod_^ (1-aq^k). This is an analytic function of ''q'' in the interior of the unit disk, and can also be considered as a formal power series in ''q''. The special case \phi(q) = (q;q)_\infty=\prod_^\infty (1-q^k) is known as Euler's function, and is important in combinatorics, number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pacific Journal Of Mathematics
The Pacific Journal of Mathematics is a mathematics research journal supported by several universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisation, and the University of California, Berkeley. It was founded in 1951 by František Wolf and Edwin F. Beckenbach and has been published continuously since, with five two-issue volumes per year and 12 issues per year. Full-text PDF versions of all journal articles are available on-line via the journal's website with a subscription. The journal is incorporated as a 501(c)(3) organization A 501(c)(3) organization is a United States corporation, Trust (business), trust, unincorporated association or other type of organization exempt from federal income tax under section 501(c)(3) of Title 26 of the United States Code. It is one of .... The 255-page proof of the odd order theorem, by Walter Feit and John Griggs Thompson, was publi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the list of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
