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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 Leonard James Rogers (), and were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913.
The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated explicitly for a broad class of values of its argument.
Leonard James Rogers. Leonard James Rogers FRS [1] (30 March 1862 – 12 September 1933) was a British mathematician who was the first to discover the Rogers–Ramanujan identity and Hölder's inequality, and who introduced Rogers polynomials. [2] The Rogers–Szegő polynomials are named after him.
The functions G and H turn up in the Rogers–Ramanujan identities, and the function Q is the Euler function, which is closely related to the Dedekind eta function. If x = e 2πiτ, then x −1/60 G(x), x 11/60 H(x), x −1/24 P(x), z, κ, ρ, ρ 1, ρ 2, and ρ 3 are modular functions of τ, while x 1/24 Q(x) is a modular form of weight 1/2.
In 2014, a joint paper by Michael J. Griffin, Ono, and S. Ole Warnaar provided a framework for the Rogers–Ramanujan identities and their arithmetic properties, solving a long-standing mystery stemming from the work of Ramanujan. [10] The findings yield new formulas for algebraic numbers.
If instead of counting the number of partitions with distinct parts we count the number of partitions with parts differing by at least 2, a further generalization is possible. It was first discovered by Leonard James Rogers in 1894, and then independently by Ramanujan in 1913 and Schur in 1917, in what are now known as the Rogers-Ramanujan ...
Srinivasa Ramanujan Aiyangar [a] (22 December 1887 – 26 April 1920) was an Indian mathematician.Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then ...
Ramanujan summation; Ramanujan theta function; Ramanujan graph; Ramanujan's tau function; Ramanujan's ternary quadratic form; Ramanujan prime; Ramanujan's constant; Ramanujan's lost notebook; Ramanujan's master theorem; Ramanujan's sum; Rogers–Ramanujan identities; Rogers–Ramanujan continued fraction; Ramanujan–Sato series; Ramanujan ...