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Engel series and continued fractions. (Séries de Engel et fractions continuées.) (French) Zbl 1007.11045
An Engel series \(]a_1, a_2, \ldots [\) is an expansion \(x=\sum_{n=1}^\infty 1/a_1a_2\cdots a_n\) in integers \(a_{i+1}\geq a_i\) and \(a_1\geq 2\). In this interesting and instructive paper, the authors provide a detailed analysis and explanation of algorithms and examples giving a continued fraction expansion \([b_1, b_2, \ldots ]\) as an Engel series expansion and conversely. The underlying principle is that both the convergents (truncations) of a continued fraction and the partial sums of an Engel series are given by unimodular matrices and that those matrices factor into matrix products respectively displaying the partial quotients \(b_i\) or the quotients \(a_i\).

11J70 Continued fractions and generalizations
11J81 Transcendence (general theory)
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[1] Allouche, J.-P., Davison, J.L., (Queffélec, M., Zamboni, L.Q., Transcendence of sturmian or morphic continued fractions. préprint 1999, pp. 26. · Zbl 0998.11036
[2] Allouche, J.-P., Lubiw, A., Mendès france, M., Van der Poorten, A.J., Shallit, J.O., Convergents of folded continued fractions. Acta Arithmetica77 (1996), 77-96. · Zbl 0848.11004
[3] Blanchard, A., Mendès France, M., Symétrie et transcendance. Bull. Sci. Math.106 (1982), 325-335, · Zbl 0492.10027
[4] Borel, E., Sur les développements unitaires normaux. C.R.A.S Paris225 (1947), 773. · Zbl 0029.15303
[5] Davison, J.L., A class of transcendental numbers with bounded partial quotients. Number Theory and Applications (Banff, AB, 1988) ; NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 265; Kluwer Acad. Publ.Dordrecht (1989), 365-371. · Zbl 0693.10028
[6] Dekking, M., van der Poorten, A.J., Mendès, M.France. Folds ! Math. Intell.4 (1982), 130-138, 173-181, 190-195. · Zbl 0493.10001
[7] Erdös, P., Rényi, A., Szüsz, P., On Engel’s and Sylvester series. Ann. Univ. Sci. Budapest, Sectio Math.1 (1957), 7-12. · Zbl 0107.27002
[8] Kmosek, M., Rozwinieçie niektórych liczb niewymiernych na ulamki lancuchowe. Thèse (en polonais), Uniwersytet Warszawski, Varsovie, (1979).
[9] Köhler, G., Some More Predictable Continued Fractions. Mh. Math.89, (1980), 95-100. · Zbl 0419.10010
[10] Lang, S., Diophantine Geometry. Interscience Publishers (1962). · Zbl 0115.38701
[11] Liardet, P., Stambul, P., Algebraic computations with continued fractions. Journal of Number Theory73 (1998), 92-121. · Zbl 0929.11066
[12] Lucas, E., Théorie des Nombres. Gauthier-Villars (1891). · JFM 23.0174.02
[13] Mendès france, M., Shallit, J.O., Wire Bending. Journal of Combinatorial Theory Series A 50 (1989), 1-23. · Zbl 0663.10056
[14] Perron, O., Irrationalzahlen. De Gruyter, Berlin et Leipzig, deuxième édition (1939), 116-122. · JFM 65.0192.02
[15] (Queffélec, M., Transcendance des fractions continues de Thue-Morse, J. Number Theory73 (1998), 201-211. · Zbl 0920.11045
[16] Schmidt, W., On simultaneous approximations of two algebraic numbers by rationals. Acta Math.119 (1967), 27-50. · Zbl 0173.04801
[17] Shallit, J.O., Real numbers with bounded partial quotients: a survey. The Mathematical Heritage of Friedrich Gauss, G. M. Rassias, editor, World Scientific Publishing (1991) .
[18] Shallit, J.O., Simple continued fractions for some irrational numbers. J. Number Theory11 (1979), 209-217. · Zbl 0404.10003
[19] Shallit, J.O., Simple continued fractions for some irrational numbers II. J. Number Theory14 (1982), 228-231. · Zbl 0481.10005
[20] Shallit, J.O., Explicit descriptions of some continued fractions. Fibonacci Quart.20 (1982), 77-81. · Zbl 0472.10012
[21] Sierpinski, W., Elementary Theory of Numbers. Institute of Math. of Polish Acad. of Sciences (1964). · Zbl 0122.04402
[22] Tamura, J., Explicit formulae for Cantor series representing quadratic irrationals. Number theory and combinatorics, Japan, World Scientific Publishing Co. (1984), 369-381. · Zbl 0608.10013
[23] Van der Poorten, A.J., An introduction to continued fractions. Diophantine Analysis, J.H. Loxton and A.J. van der Poorten, editors, Cambridge University press (1986), 99-138. · Zbl 0596.10008
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