Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers
From some of the deep results of \textit{Yu. V. Nesterenko} [Sb. Math. 187, 1319-1348 (1996); translation from Mat. Sb. 187, 65-96 (1996; Zbl 0898.11031)] on the transcendence of modular functions, the present authors derive two kinds of applications. (i) If \(RR(q)\) denotes for any \(q\in{\mathbb{C}}\) with \(| q|< 1\) the Rogers-Ramanujan continued fraction \(1+{{q| }\over 1}+ {{q^2| }\over{| 1}}+ \ldots\), then \(RR(q)\) is transcendental for each algebraic \(q\) with \(0 < | q| < 1\). (ii) Let \(\alpha,\beta\) be algebraic numbers with \(| \alpha| > 1 > | \beta| \), and denote \(U_n:= (\alpha^n-\beta^n)/(\alpha-\beta)\), \(V_n:=\alpha^n+\beta^n\) for \(n = 1,2,\ldots\) . If \(\beta = \alpha^{-1}\), then the sums \(\sum U_n^{-2s}\), \(\sum V_n^{-s}\) are transcendental for any \(s = 1,2,\ldots\), and if \(\beta=-\alpha^{-1}\) the same holds for \(\sum U_n^{-2s}\), \(\sum V_n^{-2s}\), \(\sum U^{-s}_{2n-1}\), where all sums run over \(n = 1,2,\ldots\) . Some corollaries concerning the Fibonacci and Lucas sequences are obvious. Reviewer's remark: From (i) and Ramanujan's wonderful formula \(e^{-2\pi/5} RR(e^{-2\pi})= (\gamma^2+1)^{1/2}- \gamma\) with \(\gamma:= (1+\sqrt{5})/2\) one gets again the transcendence of \(e^\pi\).
scientific article; zbMATH DE number 1160948
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