Zur Transzendenz gewisser Reihen. (On the transcendence of certain series). (German) Zbl 0601.10025

Bundschuh, Peter; Pethö, Attila

Monatsh. Math. 104, 199-223 (1987).

W. M. Schmidt’s famous theorem on the simultaneous approximation of algebraic numbers is used to derive transcendence results concerning certain series of rational numbers. The denominators of these numbers are taken from finitely many linear recursive sequences with the same characteristic polynomial which in turn is the minimal polynomial of a PV number. These denominators have to satisfy a divisibility as well as a growth condition (and in an appendix of the paper the second author studies the connections between these two kinds of hypotheses). The nominators of the rational terms of the series must satisfy some growth conditions too.
One section of the present paper is devoted to the study of the implications of Mahler’s analytic transcendence method from 1929 to the arithmetical questions considered here.

MSC:

11J81	Transcendence (general theory)
11R06	PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11B37	Recurrences

Keywords:

simultaneous approximation of algebraic numbers; transcendence; series of rational numbers; linear recursive sequences; PV number

References:

[1]	Bieberach, L.: Analytische Fortsetzung. Berlin-Göttingen-Heidelberg: Springer. 1955.
[2]	Bundschuh, P.: Aufgabe 772. El. Math.32, 98-99 (1977).
[3]	Carmichael, R. D.: On sequences of integers defined by recurrence relations. Quart. J. Math.48, 343-372 (1920).
[4]	Hua, L. K., Wang, Y.: Applications of Number Theory to Numerical Analysis. Berlin-Heidelberg-New York: Springer. 1981. · Zbl 0465.10045
[5]	Lech, C.: A note on recurring series. Ark. Math.2, 417-421 (1953). · Zbl 0051.27801 · doi:10.1007/BF02590997
[6]	Lewis, D. J.: Diophantine equations:p-adic methods. In: Studies in Number Theory, pp. 25-75. MAA Studies in Math.6. 1969.
[7]	Mahler, K.: Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen. Math. Ann.101, 342-366 (1929). · JFM 55.0115.01 · doi:10.1007/BF01454845
[8]	Mahler, K.: A remark on recursive sequences. J. Math. Sci.1, 12-17 (1966). · Zbl 0158.29503
[9]	Mahler, K.: Lectures on transcendental numbers. 1969 Number Theory Institute (Proc. Sympos. Pure Math. Vol. XX, State Univ. New York, Stony Brook, N. Y., 1969). W. J. LeVeque, ed. pp. 248-274. Providence, R. I.: Amer. Math. Soc. 1971.
[10]	Mignotte, M.: An application of W. Schmidt’s theorem. Transcendental numbers and golden number. Fibonacci Quart.15, 15-16 (1977). · Zbl 0353.10025
[11]	Pethö, A.: Perfect powers in second order linear recurrences. J. Number Theory15, 5-13 (1982). · Zbl 0488.10009 · doi:10.1016/0022-314X(82)90079-8
[12]	Pethö, A.: Perfect powers in second order recurrences. In: Topics in Classical Number Theory (Budapest 1981). G. Halász, ed. pp. 1217-1227. Colloq. Math. Soc. János Bolyai 34. Amsterdam-New York: North-Holland. 1984.
[13]	Van der Poorten, A. J.: Some determinants that should be better known. J. Austral. Math. Soc. Ser. A21, 278-288 (1976). · Zbl 0343.15002 · doi:10.1017/S1446788700018577
[14]	Van der Poorten, A. J., Schlickewei, H. P.: The growth conditions for recurrence sequences. Macquarie Univ. Math. Report 82-0041. North Ryde, Australia. 1982.
[15]	Schinzel, A.: On two theorems of Gelfond and some of their applications. Acta Arith.13, 177-236 (1967). · Zbl 0159.07101
[16]	Schmidt, W. M.: Über simultane Approximation algebraischer Zahlen durch rationale. Acta Math.114, 159-209 (1965). · Zbl 0136.33802 · doi:10.1007/BF02391821
[17]	Schmidt, W. M.: Simultaneous approximation to algebraic numbers by rationals. Acta Math.125, 189-201 (1970). · Zbl 0205.06702 · doi:10.1007/BF02392334
[18]	Szegö, G.: Über Potenzreihen mit endlich vielen verschiedenen Koeffizienten. Sitzber. preuß. Akad. Wiss., Math.-phys. Kl.1922, 88-91 (=Collected PapersI, pp. 557-560). · JFM 48.0330.02
[19]	Zhu, Y., Wang, L., Xu, G.: On the transcendence of a class of series. Kexue Tongbao25, 1-6 (1980).