zbMATH — the first resource for mathematics

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 $$a_i$$.

MSC:
 11J70 Continued fractions and generalizations 11J81 Transcendence (general theory)
Full Text:
References:
 [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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.