Séries de Engel et fractions continuées
Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 1, p. 37-68

Relations between continued fraction expansion and Engel's series of a real number are investigated. Product of matrices corresponding to these expansions leads to transducers which convert the continued fraction expansion of any irrational number to its Engel's series and reciprocally. Finally, new results about Lucas numbers, Fredholm numbers and various transcendental numbers with bounded or unbounded partial quotients are obtained.

Le thème de ce travail est la conversion entre le développement en fraction continuée d'un nombre réel et son développement en série de Engel. Chacun d'eux peut se traduire en terme de produits matriciels, produits qui sont à l'origine d'algorithmes, exprimés sous la forme de transducteurs, permettant de calculer un des développements à partir de l'autre. Cette méthode fournit des résultats nouveaux sur les nombres de Lucas, les nombres de Fredholm et sur toute une variété de nombres transcendants, à quotients partiels bornés ou non.

@article{JTNB_2000__12_1_37_0,
     author = {Liardet, Pierre and Stambul, Pierre},
     title = {S\'eries de Engel et fractions continu\'ees},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {12},
     number = {1},
     year = {2000},
     pages = {37-68},
     zbl = {1007.11045},
     mrnumber = {1827837},
     language = {fr},
     url = {http://www.numdam.org/item/JTNB_2000__12_1_37_0}
}
Liardet, Pierre; Stambul, Pierre. Séries de Engel et fractions continuées. Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 1, pp. 37-68. http://www.numdam.org/item/JTNB_2000__12_1_37_0/

[A-D-Q-Z] J.-P. Allouche, J.L. Davison, M. (Queffélec, L.Q. Zamboni, Transcendence of sturmian or morphic continued fractions. préprint 1999, pp. 26. | Zbl 0998.11036

[A-L-M-P-S] J.-P. Allouche, A. Lubiw, M. Mendès France, A.J. Van Der Poorten, J.O. Shallit, Convergents of folded continued fractions. Acta Arithmetica 77 (1996), 77-96. | MR 1404978 | Zbl 0848.11004

[BI-Me] A. Blanchard, M. Mendès France, Symétrie et transcendance. Bull. Sci. Math. 106 (1982), 325-335, | MR 680277 | Zbl 0492.10027

[Bo] E. Borel, Sur les développements unitaires normaux. C.R.A.S Paris 225 (1947), 773. | MR 23007 | Zbl 0029.15303

[Da] J.L. Davison, 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. | MR 1123082 | Zbl 0693.10028

[De-Po-Me] M. Dekking, A.J. Van Der Poorten, M. Mendès France. Folds ! Math. Intell. 4 (1982), 130-138, 173-181, 190-195. | MR 684028 | Zbl 0493.10001

[E-R-S] P. Erdös, A. Rényi, P. Szüsz, On Engel's and Sylvester series. Ann. Univ. Sci. Budapest, Sectio Math. 1 (1957), 7-12. | MR 102496 | Zbl 0107.27002

[Kmo] M. Kmosek, Rozwinieçie niektórych liczb niewymiernych na ulamki lancuchowe. Thèse (en polonais), Uniwersytet Warszawski, Varsovie, (1979).

[Kö] G. Köhler, Some More Predictable Continued Fractions. Mh. Math. 89, (1980), 95-100. | MR 572885 | Zbl 0419.10010

[La] S. Lang, Diophantine Geometry. Interscience Publishers (1962). | MR 142550 | Zbl 0115.38701

[Li-St] P. Liardet, P. Stambul, Algebraic computations with continued fractions. Journal of Number Theory 73 (1998), 92-121. | MR 1654886 | Zbl 0929.11066

[Lu] E. Lucas, Théorie des Nombres. Gauthier-Villars (1891). | JFM 23.0174.02

[Me-Sh] M. Mendès France, J.O. Shallit, Wire Bending. Journal of Combinatorial Theory Series A 50 (1989), 1-23. | MR 978063 | Zbl 0663.10056

[Pe] O. Perron, Irrationalzahlen. De Gruyter, Berlin et Leipzig, deuxième édition (1939), 116-122. | JFM 65.0192.02 | MR 115985

[Qu] M. (Queffélec, Transcendance des fractions continues de Thue-Morse, J. Number Theory 73 (1998), 201-211. | MR 1658023 | Zbl 0920.11045

[Sc] W. Schmidt, On simultaneous approximations of two algebraic numbers by rationals. Acta Math. 119 (1967), 27-50. | MR 223309 | Zbl 0173.04801

[Sh1] J.O. Shallit, Real numbers with bounded partial quotients: a survey. The Mathematical Heritage of Friedrich Gauss, G. M. Rassias, editor, World Scientific Publishing (1991) .

[Sh2] J.O. Shallit, Simple continued fractions for some irrational numbers. J. Number Theory 11 (1979), 209-217. | MR 535392 | Zbl 0404.10003

[Sh3] J.O. Shallit, Simple continued fractions for some irrational numbers II. J. Number Theory 14 (1982), 228-231. | MR 655726 | Zbl 0481.10005

[Sh4] J.O. Shallit, Explicit descriptions of some continued fractions. Fibonacci Quart. 20 (1982), 77-81. | MR 660766 | Zbl 0472.10012

[Si] W. Sierpinski, Elementary Theory of Numbers. Institute of Math. of Polish Acad. of Sciences (1964). | MR 175840 | Zbl 0122.04402

[Ta] J. Tamura, Explicit formulae for Cantor series representing quadratic irrationals. Number theory and combinatorics, Japan, World Scientific Publishing Co. (1984), 369-381. | MR 827796 | Zbl 0608.10013

[VdP] A.J. Van Der Poorten, An introduction to continued fractions. Diophantine Analysis, J.H. Loxton and A.J. van der Poorten, editors, Cambridge University press (1986), 99-138. | MR 874123 | Zbl 0596.10008