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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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##### References:
 [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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