×

On the transcendence of some power series. (English) Zbl 1366.11089

Summary: In this study, we consider some power series with rational coefficients and investigate transcendence of the values of these series for Liouville number arguments. It is proved that these values are either a Liouville number or a rational number under certain conditions.

MSC:

11J81 Transcendence (general theory)
11J17 Approximation by numbers from a fixed field

References:

[1] Liouville J: Sur des classes trés étendues de quantités dont la valeur n’est ni algébrique, ni méme reductible à des irrationnelles algébriques.C. R. Math. Acad. Sci. Paris 1844, 18:883-885. 910-911
[2] Cantor G: Über eine Eigenschaft des Inbegriffs aller reelen algebraishen Zahlen.J. Reine Angew. Math. 1974, 77:258-262.
[3] Baker A: Transcendental Number Theory. Cambridge University Press, Cambridge; 1975. · Zbl 0297.10013 · doi:10.1017/CBO9780511565977
[4] Sprindzhuk VG: Classical Diophantine Equations in Two Unknowns. Nauka, Moscow; 1982. (In Russian) · Zbl 0523.10008
[5] Mahler K: Zur Approximation der Exponentialfunktion und des Logarithmus, I, II.J. Reine Angew. Math. 1932, 166:118-150. · JFM 58.0207.01
[6] Maillet E: Sur la classification des irrationelles.C. R. Math. Acad. Sci. Paris 1906, 143:26-28. · JFM 37.0255.01
[7] Perna A: Sui numeri transcendenti in generale e sulla loroconstruzione in base al criterio di Liouville.G. Mat. Battaglini 1914, 52:305-365. · JFM 45.0276.03
[8] Morduchai-Boltovskoj D: On transcendental numbers with successive approximations defined by algebraic equations.Rec. Math. Moscou 1934, 41:221-232. · Zbl 0010.15301
[9] Koksma JF: Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen.Monatshefte Math. Phys. 1939, 48:176-189. · Zbl 0021.20804 · doi:10.1007/BF01696176
[10] Bugeaud Y: Approximation by Algebraic Numbers. Cambridge University Press, Cambridge; 2004. [Cambridge Tracts in Mathematics 160] · Zbl 1055.11002
[11] Wirsing E: Approximation mit algebraischen Zahlen beschr änkten Grades.J. Reine Angew. Math. 1961, 206:67-77. · Zbl 0097.03503
[12] Leveque WJ: On Mahler’sU-numbers.J. Lond. Math. Soc. 1953, 28:220-229. · Zbl 0053.36203 · doi:10.1112/jlms/s1-28.2.220
[13] Oryan MH: Über gewisse Potenzreihen, die für algebraische Argumente Werte aus Der Mahlerschen Unterklassen [InlineEquation not available: see fulltext.] nehmen. Istanbul Univ. Fen Fak. Mecm., Seri A 1980, 45:1-42. · Zbl 0665.10022
[14] Oryan MH: Über gewisse Potenzreihen, deren Funktionswerte für Argumente aus der Menge der Liouvilleschen ZahlenU-Zahlen vom Grade ≤ m sind.Istanbul Univ. Fen Fak. Mecm., Seri A 1990, 47:15-34. · Zbl 0706.11036
[15] Saradha N, Tijdeman R: On the transcendence of infinite sums of values of rational functions.J. Lond. Math. Soc. 2003,67(3):580-592. · Zbl 1045.11051 · doi:10.1112/S0024610702003988
[16] Yuan P, Li J: On the transcendence of some infinite sums.J. Lond. Math. Soc. 2009,80(2):431-445. · Zbl 1293.11082 · doi:10.1112/jlms/jdp029
[17] Mulase M, Penkava M: Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over Q.Asian J. Math. 1998, 2:875-920. · Zbl 0964.30023
[18] Leveque WJ: Topics in Number Theory, vol. II. Addison-Wesley, Reading (1956) · Zbl 0070.03803
[19] Perron O: Irrationalzahlen. Walter de Gruyter, Berlin; 1960. · Zbl 0090.03202 · doi:10.1515/9783110836042
[20] Schneider T: Einführung in die transzendenten Zahlen. Springer, Berlin; 1957. · Zbl 0077.04703 · doi:10.1007/978-3-642-94694-3
[21] Mahler K: Lectures on Transcendental Numbers. Springer, Berlin; 1976. · Zbl 0332.10019
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.