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Evaluations of the Roger-Ramanujan continued fraction \(R(q)\) by modular equations. (English) Zbl 0982.33010
In his first letter to Hardy, Ramanujan stated two specializations of his now famous continued fraction \[ R(q):={q^{1/5}} \big/{1}+q\big/1+q^2\big/1+q^3\big/1+\cdots, \] namely explicit values of \(R(q)\) for \(q=e^{-2\pi}\) and \(q=-e^{-\pi}\). Several other special evaluations are recorded in his notebooks. Since then, mathematicians have been fascinated by these specializations. A table of known values of \(R(q)\) by 1999 has been set up by S.-Y. Kang [Acta Arith. 90, No. 1, 49-68 (1999; Zbl 0933.11003)]. In the paper under review, the author uses modular equations related to degrees that are multiples of 5 to rederive some known special evaluations of \(R(q)\), and to derive more than 20 new ones. As an example let me cite \[ R(e^{-\pi})=\tfrac {1} {8}(3+\sqrt 5)(\root 4\of 5-1)\big( \sqrt{10+2\sqrt 5}-(3+\root 4\of 5)(\root 4\of 5-1)\big). \]

33D99 Basic hypergeometric functions
11A55 Continued fractions
11F25 Hecke-Petersson operators, differential operators (one variable)
Zbl 0933.11003
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