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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
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[1] Berndt, Ramanujan’s Notebooks, Part III (1991) · Zbl 0733.11001
[2] Adiga, Tamsui Oxf. J. Math. Sci. 18 pp 101– (2002)
[3] Adiga, New Zealand J. Math. 31 pp 1– (2002)
[4] Adiga, Ramanujan’s second notebook: Theta-functions and q-series (1985) · Zbl 0565.33002
[5] Ramanujan, The lost notebook and other unpublished papers (1988) · Zbl 0639.01023
[6] Ramanujan, Collected papers (1962)
[7] Berndt, Canad. J. Math. 47 pp 897– (1995) · Zbl 0838.33011
[8] Ramanathan, Indian J. Pure Appl. Math. 16 pp 698– (1985)
[9] Ramanathan, J. Indian. Math. Soc. (N.S.) 52 pp 71– (1987)
[10] DOI: 10.1007/BF02840651 · Zbl 0565.10009
[11] DOI: 10.1023/A:1009767205471 · Zbl 0905.11008
[12] Chan, Acta Arith. 73 pp 343– (1995)
[13] Ramanujan, Notebooks (2 volumes) (1957)
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