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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.
The connection between the continued fraction and the Rogers–Ramanujan functions was already found by Rogers in 1894 (and later independently by Ramanujan). The continued fraction can also be expressed by the Dedekind eta function : [ 6 ]
The diagonal of a half square forms the basis for the geometrical construction of a golden rectangle.. The golden ratio φ is the arithmetic mean of 1 and . [4] The algebraic relationship between , the golden ratio and the conjugate of the golden ratio (Φ = − 1 / φ = 1 − φ) is expressed in the following formulae:
In mathematics, regular continued fractions play an important role in representing real numbers, and have a rich general theory touching on a variety of topics in number theory. Moreover, generalized continued fractions have important and interesting applications in complex analysis .
Another meaning for generalized continued fraction is a generalization to higher dimensions. For example, there is a close relationship between the simple continued fraction in canonical form for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx. Generalizing this idea, one might ...
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The values of the Rogers-Ramanujan continued fraction where is algebraic and < | | <. [44] The lemniscatic values of theta function ∑ n = − ∞ ∞ q n 2 {\displaystyle \sum _{n=-\infty }^{\infty }q^{n^{2}}} (under the same conditions for q {\displaystyle {q}} ) are also transcendental.
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