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A note on a continued fraction of Ramanujan. (English) Zbl 1062.33016
This paper gives a number of results on the structure of Ramanujan’s continued fraction $c(q)=\frac 11\,{\quad\atop +}\,\frac{2q}{1-q^2}\,{\quad\atop +}\,\frac{q^2(1+q^2)^2}{1-q^6}\,{\quad\atop +}\,\frac{q^4(1+q^4)^2}{1-q^{10}}\,{\quad\atop +\dots}; \quad | q| <1.$ Not surprisingly, the identities obtained involve Ramanujan’s theta functions, since it is proved that $c(q)=\varphi(q^4)/\varphi(q);\quad \varphi(q):=(-q;-q)_\infty /(q;-q)_\infty.$ The authors also prove that if $q=\exp\Big(-\pi^2\, _2F_1(\tfrac 12,\tfrac 12;1;1-\alpha)/_2F_1(\tfrac 12,\tfrac 12,1;\alpha)\Big),$ then $\alpha=1-(2c(q)-1)^4.$ This allows them to derive relations between $$c(q)$$ and $$c(q^n)$$ for positive integers $$n$$. It also allows them to evaluate $$c(q)$$ at $$q=e^{-\pi\sqrt{n}}$$.

##### MSC:
 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$ 40A15 Convergence and divergence of continued fractions
##### Keywords:
Ramanujan; continued fraction; modular equations
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##### References:
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