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Explicit evaluations of Ramanujan-Göllnitz-Gordon continued fraction. (English) Zbl 1170.11002
The continued fraction in the title is \[ K(q)= {q^{1/2}\over 1+ q+}\, {q^2\over 1+ q^3+}\, {q^4\over 1+ q^5+}\cdots \] with \(|q|< 1\). Ramanujan discovered an identity with an infinite product, \[ K(q)= q^{1/2} {(q; q^8)_\infty(q^7; q^8)_\infty\over (q^3; q^8)_\infty(q^5; q^8)_\infty}, \] where \((a; q)_\infty= \prod^\infty_{n=0} (1- aq^n)\). Therefore, by putting \(q= e^{2\pi iz}\), the function \(K(q)\) is intimately related to modular forms, in particular to Jacobi’s theta functions and to Dedekind’s eta function \(\eta(z)= q^{1/24}(q; q)_\infty\), and there are many relations of the type of modular equations, connecting \(K(q)\), \(K(q')\) and products of Jacobi and Dedekind functions. Some recent ones are due to H. H. Chan and S.-S. Huang [Ramanujan J. 1, No. 1, 75–90 (1997; Zbl 0905.11008)] and to K. R. Vasuki and B. R. Srivatsa Kumar [J. Comput. Appl. Math. 187, No. 1, 87–95 (2006; Zbl 1081.11004)]. In the paper under review the authors add more identities of this type. They are used to find explicit algebraic values for \(K(q)\) and some other modular functions at special points \(q= e^{-\pi\sqrt{n/k}}\) with positive integers \(n\), \(k\).
This extends work by the authors cited above and by J. Yi [Ph.D. thesis, Univ. of Ill. (2001), and J. Math. Anal. Appl. 292, No. 2, 381–400 (2004; Zbl 1060.11027)]. We reproduce two examples: \[ K\left(e^{2\pi}\right)= \sqrt{{2^{3/4}+ 2^{1/4}-1\over 2^{3/4}+ 3\cdot 2^{1/4}+ 1}},\qquad K\left(e^{-3\pi/\sqrt{2}}\right)= \sqrt{{2^{1/4}(\sqrt{3}- \sqrt{2})(2- \sqrt{3})- 1\over 2^{1/4} (\sqrt{3}+ \sqrt{2})(2+ \sqrt{3})+ 1}}. \]

MSC:
11A55 Continued fractions
11F20 Dedekind eta function, Dedekind sums
33D90 Applications of basic hypergeometric functions
11F27 Theta series; Weil representation; theta correspondences
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[5] Baruah ND, Saikia N (2006) On explicit evalautions of Ramanujan-Selberg continued fraction. Int J Math Math Sci: Art. ID 54901: 1–15 · Zbl 1154.11008
[16] Yi J (2001) Construction and application of modular equations. PhD thesis, University of Illionis at Urbana-Champaign
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