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Ramanujan’s formulas for the explicit evaluation of the Rogers-Ramanujan continued fraction and theta-functions. (English) Zbl 0933.11003

For \( |q|<1,\) let \( (a;q)_\infty :=\prod_{n=0}^\infty (1-aq^n) \),\( \chi(-q) =(q;q^2)_\infty,\) and \(\varphi(q) =\sum_{n=-\infty}^\infty q^{n^2}.\) In this paper, the author proves various identities found in Ramanujan’s Lost Notebook using modular equations of degrees 5 and 25, one of which is the following: \[ s =\dfrac{1-2t-2t^2+t^3+\sqrt{1-4t-10t^3-4t^5+t^6}}{2},\tag{1} \] where \(s= {\varphi(-q^{1/5})/}{\varphi(q^5)}\) and \(t=q^{1/5}{\chi(-q^{1/5})/}{\chi(-q^5)}.\) This identity is equivalent to an identity recently discovered by B. C. Berndt, H. H. Chan and L. C. Zhang [see J. Reine Angew. Math. 480, 141-159 (1996; Zbl 0862.33017)]. Using (1), the author proves the following “formidable” identity recorded by Ramanujan: \[ \begin{split} R(q) &= \dfrac{1}{4t}\left(\left(1+t\dfrac{\sqrt{5}+1}{2}\right)\sqrt{1-t}-\sqrt{(1-t)\left(1+t\dfrac{\sqrt{5}+1}{2}\right)^2-2t(\sqrt{5}+1)}\right)\\ &\quad\times \left(-\left(1-t\dfrac{\sqrt{5}-1}{2}\right)\sqrt{1-t}+\sqrt{(1-t)\left(1-t\dfrac{\sqrt{5}-1}{2}\right)^2+2t(\sqrt{5}-1)}\right)\end{split} \] where \(R(q)\) is the Rogers-Ramanujan continued fraction.

MSC:

11A55 Continued fractions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
11F27 Theta series; Weil representation; theta correspondences

Citations:

Zbl 0862.33017
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