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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 $R(e^{-\pi})=\tfrac {1} {8}(3+\sqrt 5)(\root 4\of 5-1)\big( \sqrt{10+2\sqrt 5}-(3+\root 4\of 5)(\root 4\of 5-1)\big).$

##### MSC:
 33D99 Basic hypergeometric functions 11A55 Continued fractions 11F25 Hecke-Petersson operators, differential operators (one variable)
##### Keywords:
Ramanujan continued fraction; modular equations
Zbl 0933.11003
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