Duverney, Daniel; Nishioka, Keiji; Nishioka, Kumiko; Shiokawa, Iekata Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers. (English) Zbl 0902.11029 Proc. Japan Acad., Ser. A 73, No. 7, 140-142 (1997). From some of the deep results of 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 \(e^\pi\). Reviewer: P.Bundschuh (Köln) Cited in 2 ReviewsCited in 17 Documents MSC: 11J81 Transcendence (general theory) 11J91 Transcendence theory of other special functions 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11J70 Continued fractions and generalizations Keywords:transcendence; Rogers-Ramanujan continued fraction; Fibonacci sequence; Lucas sequence Citations:Zbl 0898.11031 PDFBibTeX XMLCite \textit{D. Duverney} et al., Proc. Japan Acad., Ser. A 73, No. 7, 140--142 (1997; Zbl 0902.11029) Full Text: DOI Online Encyclopedia of Integer Sequences: Decimal expansion of sum of reciprocals of squares of Fibonacci numbers. Decimal expansion of sum of reciprocals of squares of Lucas numbers. Decimal expansion of Sum_{n>=1} 1/Fibonacci(2*n-1). Decimal expansion of Sum_{n>=1} 1/A000032(2*n). References: [1] B. C. Berndt: Ramanujan’s Notebooks Part III. Springer (1991). · Zbl 0733.11001 [2] J. M. Borwein and P. B. Borwein : Pi and the AGM–A study in analytic number theory and computational complexity. Johon Wiley (1987). · Zbl 0611.10001 [3] D. Duverney, Ke. Nishioka, Ku. Nishioka, and I. Shiokawa: Transcendence of Jacobi’s theta series. Proc. Japan Acad., 72A, 202-203 (1996). · Zbl 0884.11030 · doi:10.3792/pjaa.72.202 [4] D. Duverney, Ke. Nishioka, Ku. Nishioka, and I. Shiokawa: Transcendence of Jacobi’s theta series and related results. Number Theory-Diophantine, Computational and Algebraic Aspects, Proc. Cont. Number Theory Eger 1996 (eds. K. Gyory, A. Petho, and V. T. Sos). W. de Gruyter (to appear). · Zbl 0938.11039 [5] T. Matala-Aho: On Diophantine approximations of the Rogers-Ramanujan continued fraction. J. Number Theory, 45 (2), 215-227 (1983). · Zbl 0790.11051 · doi:10.1006/jnth.1993.1073 [6] Yu. V. Nesterenko : Modular functions and transcendence problems. C. R. Acad. Sci. Paris, ser. 1, 322, 909-914 (1996). · Zbl 0859.11047 [7] Yu. V. Nesterenko : Modular functions and transcendence problems. Math. Sb., 187 (9), 65-96 (1996) (Russiam); English transl. Sbornik Math. 187 (9-10), 1319-1348 (1996). · Zbl 0898.11031 · doi:10.1070/SM1996v187n09ABEH000158 [8] C. F. Osgood: The diophantine approximation of certain continued fractions. Proc. Amer. Math. Soc, 3, 1-7 (1977). · Zbl 0256.10018 · doi:10.2307/2039026 [9] I. Shiokawa: Rational approximation to the Rogers-Ramanujan continued fractions. Acta Arith., L, 23-30 (1988). · Zbl 0655.10031 [10] I. J. Zucker : The summation of series of hyperbolic functions. SIAM J. Math. Anal., 10, 192-206 (1979). · Zbl 0411.33001 · doi:10.1137/0510019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.