1. Some doubly exponential sequences, Fibonacci Quart. 11 (1973),

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1 References A.V. Aho and N.J.A. Sloane 1. Some doubly exponential sequences, Fibonacci Quart. 11 (1973), J.-P. Allouche and J. Shallit 1. The ring of k-regular sequences, Theoret. Compo Sci. 98 (1992), M. Amou 1. Algebraic independence of the values of certain functions at a transcendental number, Acta Arithmetica 59 (1991), An improvement of a transcendental measure of Galochikin and Mahler's S-numbers, J. Austral. Math. Soc. (Series A) 52 (1992), A. Baker 1. Transcendental number theory, Cambridge UP, K. Barre-Sirieix, G. Diaz, F. Gramain and G. Philibert 1. Une preuve de la conjecture de Mahler-Manin, Invent. math. 124 (1996), 1-9. A. Beardon 1. Iteration of rational functions, GTM 132, Springer, P.-G. Becker 1. Transcendence measure by Mahler's transcendence method, Bull. Austral. Math. Soc. 33 (1986), Masse fur algebraische Unabhangigkeit nach einer Methode von Mahler, Acta Arith. 50 (1988), Effective measures for algebraic independence of the values of Mahler type functions, Acta Arith. 58 (1991), Algebraic independence of the values of certain series by Mahler's method, Mh. Math. 4 (1992),

2 5. Transcendence of the values of functions satisfying generalized Mahler type functional equations, J. reine angew. Math. 440 (1993), k-regular power series and Mahler-type functional equations, J. Number Theory 49 (1994), Transcendence measures for the values of generalized Mahler functions in arbitrary characteristic, Publ. Math. Debrecen 45 (1994), P.-G. Becker and W. Bergweiler 1. Transcendency of local conjugacies in complex dynamics and trascendency of their values, Manu. Math. 81 (1993), P.-G. Becker and T. Topfer 1. Transcendency results for sums of reciprocals of linear recurrences, Math. Nachr. 168 (1994), P. Bundschuh and A. Petho 1. Zur Transzendenz gewisser Reihen, Mh. Math. 104 (1987), J.W.S. Cassels 1. Local fields, Cambridge UP, P.M. Cohn 1. Algebraic numbers and algebraic functions, Chapman & Hall, M. Dekking 1. Transcendence des nombres de Thue-Morse, C.R. Acad. Sci. Paris 285 (1977), M. Dekking, M. Mendes France and A.J. van der Poorten 1. Folds! I, II, III, Math. Intelligencer 4 (1983), , , Y.Z. Flicker 1. Algebraic independence by a method of Mahler, J. Austral. Math. Soc. Ser. A 27 (1979),

3 J.N. Franklin and S.W. Golomb 1. A function-theoretic approach to the study of nonlinear recurring sequences, Pac. J. Math. 56 (1975), A.a. Galochikin 1. Transcendence measure of values offunctions satisfying certain functional equations, Math. Notes 27 (1980), F.R. Gantmacher 1. Applications of the theory of matrices, New York, Intersciences, I.J. Good 1. A reciprocal series of Fibbonacci numbers, Fibonacci Quart. 12 (1974), 346. F. Gramain, M. Mignotte and M. Waldschmidt 1. Valeurs algebriques de fonctions analytiques, Acta Arith. 47 (1986), E. Heeke 1. Uber analytische Funktionen und die Verteilung von Zahlen mod Eines, Hamb. Abh. 1 (1921), N. Koblitz 1. p--adic numbers, p--adic analysis and zeta functions, GTM 58, Springer, K.K. Kubota 1. On the algebraic independence of holomorphic solutions of certain functional equations and their values, Math. Ann. 227 (1977), On a transcendental problem of K. Mahler, Canad, J. Math. 29 (1977), Linear functional equations and algebraic independence, Transcendence Theory: Advances and Applications, ed. by A. Baker and D.W. Masser, Academic Press (1977),

4 J.H. Loxton 1. A method of Mahler in transcendence theory and some of its applications, Bull. Austral. Math. Soc. 29 (1984), Automata and transcendence, New advances in transcendence theory, ed. by A. Baker, Cambridge UP (1988), J.H. Loxton and A.J. van der Poorten 1. On algebraic functions satisfying a class of functional equations, Aequationes Math. 14 (1976), A class of hypertranscendental functions, Aequationes Math. 16 (1977), Arithmetic properties of certain functions in several variables, J. Number Theory 9 (1977), Arithmetic properties of certain functions in several variables II, J. Austral. Math. Soc. Ser.A 24 (1977), Arithmetic properties of certain functions in several variables III, Bull Austral. Math. Soc. 16 (1977), Transcendence and algebraic independence by a method of Mahler, Transcendence Theory: Advances and applications, ed. by A. Baker and D.W. Masser, Academic Press (1977), Algebraic independence properties of the Fredholm series, J. Austral. Math. Soc. Ser. A 26 (1978), Arithmetic properties of the solutions of a class of functional equations, J. reine angew. Math. 330 (1982), Arithmetic properties of automata: regular sequences, J. reine angew. Math. 392 (1988), K. Mahler 1. Arithmetische Eigenschaften der Losungen einer Klasse von Funktionalgleichungen, Math. Ann. 101 (1929), Uber das Verschwinden von Potenzreihen mehrerer Veranderlichen in speziellen Punktfolgen, Math. Ann. 103 (1930),

5 3. Arithmetische Eigenschaften einer Klasse transzendental- transzendenter Funktionen, Math. Z. 32 (1930), On the generating function of the integers with a missing digit, J. Indian Math. Soc. 15A (1951), Remarks on a paper by W. Schwarz, J. Number Theory 1 (1969), Lectures on transcendental numbers (Summer Insitute on Number Theory at Stony Brook, 1969), Amer. Math. Soc. Symp. 20, On the coefficients of the 2 n-th transformation polynomial for j(w), Acta Arith. 21 (1972), On the coefficients of the transformation polynomials for the modular function, Bull. Austral. Math. Soc. 10 (1974), On the transcendency of the solutions of a special class of functional equations, Bull. Austral. Math. Soc. 14 (1976), Corrigendum to "On the transcendency of the solutions of a special class offunctional equations", Bull. Austral. Math. Soc. 15 (1976), On a class of non-linear functional equations connected with modular functions, J. Austral. Math. Soc. Ser.A 22 (1976), Lectures on transcendental numbers, LNM 546, Springer, On a special function, J. Number Theory 12 (1980), On a special non-linear functional equation, Philos. Trans. Roy. Soc. London Ser.A 378 (1981), Fifty years as a mathematician, J. Number Theory 14 (1982), On the analytic solution of certain functional and difference equations, Proc. Roy. Soc. London Ser.A 389 (1983), On two analytic functions, Acta Arith. XLIX (1987),

6 D.W. Masser 1. A vanishing theorem for power series, Invent. math. 67 (1982), Algebraic independence properties of the Heeke-Mahler series, preprint. H. Matsumura 1. Commutative algebra, Benjamin, M. Mendes France and A.J. van der Poorten 1. Arithmetic and analytic properties of paper folding sequence, Bull. Austral. Math. Soc. 24 (1981), W.M. Miller 1. Transcendence measures by a method of Mahler, J. Austral. Math. Soc. Ser. A 32 (1982), S.M. Molchanov 1. Estimates for the measure of transcendence in the Mahler method, Diophantine Approximations (Russian), Mascov, Gos. Univ., Mekh. Math. Fak., Moscow 1985, Yu. V. Nesterenko 1. Estimates for the orders of zeros of functions of a certain class and applications in the theory of transcendental numbers, Izv. Akad, Nau k SSSR Ser. Mat. 41 (1977), ; English transl. in Math. USSR Izv. 11 (1977), Estimates for the characteristic function of a prime ideal, Mat. Sb. 123 (165) (1984), 11-34; English transl. in Math. USSR Sb. 51 (1985), On algebraic independence of algebraic powers of algebraic numbers, Mat. Sb. 123 (165) (1984), ; English transl. in Math. USSR Sb. 51 (1985), On a measure of the algebraic independence of the values of certain functions, Mat. Sb. 128 (170) (1985); English transl, in Math. USSR Sb. 56 (1987),

7 Keiji Nishioka 1. A note on differentially algebraic solution of first order linear difference equations, Aequationes Math. 27 (1984), Algebraic function solutions of a certain class of functional equations, Arch. Math. 44 (1985), Keiji Nishioka and Kumiko Nishioka 1. Algebraic independence of elementary functions and its applications to Masser's vanishing theorem, Aequationes math. 40 (1990), Algebraic independence of functions satisfying a certain type of functional equations, Funkcialaj Ekvacioj 37 (1994), Kumiko Nishioka 1. On a problem of Mahler for transcendency of function values, J. Austral. Math. Soc. Ser. A 33 (1982), On a problem of Mahler for transcendency of function values II, Tsukuba J. Math. 7 (1983), Proof of Masser's conjecture on the algebraic independence of values of Liouville series, Proc. Japan Academy 62 Ser. A (1986), Algebraic independence of certain power series of algebraic numbers, J. Number Theory 23 (1986), Algebraic independence of three Liouville series. Arch. Math. 47 (1986), Conditions for algebraic independence of certain power series of algebraic numbers, Compositio Math. 62 (1987), Algebraic independence of certain power series, Sminaire de Theorie des Nombres de Paris (1987/1988), Evertse theorem in Algebraic independence, Arch. Math. 53 (1989), On an estimate for the orders of zeros of Mahler type functions, Acta Arith. 56 (1990), New approach in Mahler's method, J. reine angew. Math. 407 (1990),

8 11. Algebraic independence measures of the values of Mahler functions, J. reine angew. Math. 420 (1991), Mahler functions and transcendental numbers (in Japanes), Seminar on Math. Sci. 17, Keio Univ., Note on a paper by Mahler, Tsukuba J. Math. 17 (1993), Algebraic independence by Mahler's method and S-unit equations, Compositio Math. 92 (1994), Algebraic independence of Mahler functions and their values, Tohoku Math. J. 48(1996), A new proof of Masser's vanishing theorem, to appear III Proc. A.M.S Algebraic independence of reciprocal sums of binary recurrences, to appear in Mh. Math.. Kumiko Nishioka, 1. Shiokawa and J. Tamura 1. Arithmetic properties of certain power series, J. Number Theory 42 (1992), Kumiko Nishioka and T. Topfer 1. Transcendence measures and nonlinear functional equations of Mahler type, Arch. Math. 57 (1991), A.J. van der Poorten 1. Substitution, automata, functional equations and functions algebraic over a finite field, Contemporary Mathematics 9 (1982), Obituary, Kurt Mahler, J. Austral. Math. Soc. (Ser. A) 51 (1991), B. Rande 1. Equations fonctionnelles de Mahler et fonctions elementaires, Groupe de travail en Theorie Analytique des Nombres, Paris 1991, see also These, Universite de Bordeaux,

9 T. Tanaka 1. Algebraic independence of certain numbers defined by linear recurrences, Keio Science and Technology Reports Vol. 47, No.2 (1994), Algebraic independence of the values of power series generated by linear recurrences, Acta Arith. 74(1996), T. Topfer 1. An Axiomatization of Nesterenko's method and applications on Mahler functions, J. Number Theory, 49 (1994), An Axiomatization of Nesterenko's method and applications on Mahler functions II, Compo Math. 95 (1995), Algebraic independence of the values of generalized Mahler functions, Acta Arith. LXX (1995), Zero order estimates for functions satisfying generalized functional equations of Mahler type, to appear in Acta Arith.. P. Turan 1. On a new method of analysis and its applications, John Wiley & Sons, M. Waldschmidt 1. Nombres transcendants, LNM 402, Springer, N. Ch. Wass 1. Algebraic independence of the values at algebraic points of a class of functions considered by Mahler, Dissertationes Mathematicae 303 (1990). A. Weil 1. Foundation of algebraic geometry, Amer. Math. Soc., O. Zariski and P. Samuel 1. Commutative algebra II, GTM 29, Springer

10 Index algebraic function field of one variable p. 9 algebraic independence measure p. 137 algebraic integer p. 1 algebraic number p. 1 algebraic number field p. 4 Baker's theorem p..57, 62 characteristic root of a linear recurrence p..55 continued fraction p. 44 degree of a linear recurrence p..5.5 density p..58 derivation p. 4.5 difference field p. 162 differential field p. 4.5 differential mod ule p. 166 em bedding theorem p.,50 exponent of a primary ideal p. 119 exponential type p. 47 Fibonacci number p. 1:3 field of constants p. 46 fundamental inequality p, 2 Hecke p. 43 height of an ideal p. 118 Hilbert's seventh problem p. 3 index p. 91 irreducible system p. 164 Kronecker's theorem p lattice point p. 14 leading part of a linear recurrence p. 5,5 linear recurrence p..5.5 Liouville inequality p. 2 Mandelbrot set p. 26 minimal polynomial p. 1 modular invariant p. 26 Morse-Thue sequence p p-adic exp function p..50 p-adic log function p. 50 p-adic number field p. 50 q-kernel p q-regular p reducible system p. 162 Riemann's formula p. 10 Rudin-Shapiro sequence p Siegel's lemma p. 14 Skolern-Lech-Mahler's theorem p..53 transcendence type p. 137 transcendental number p. 1 Turari's theorem p..59 unmixed ideal p

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