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On explicit estimates for linear forms in the values of a class of E-functions

Published online by Cambridge University Press:  17 April 2009

Xu Guangshan  and
Wang Lianxiang
Xu Guangshan
Affiliation:
School of Mathematics and Physics, Macquarie University, North Ryde, New South Wales 2113, Australia.
Wang Lianxiang
Affiliation:
School of Mathematics and Physics, Macquarie University, North Ryde, New South Wales 2113, Australia.
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Abstract

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We apply methods of Mahler to obtain explicit lower bounds for certain combinations of E-functions satisfying systems of linear differential equations as studied by Makarov. Our results sharpen and generalise earlier results of Mahler, Shidlovskii, and Väänänen.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 25 , Issue 1 , February 1982 , pp. 37 - 69
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Mahler, Kurt, “On a paper by A. Baker on the approximation of rational powers of e”, Acta Arith. 27 (1975), 61–87.CrossRefGoogle Scholar
[2]Mahler, Kurt, Lectures on transcendental numbers (Lecture Notes in Mathematics, 546. Springer-Verlag, Berlin, Heidelberg, New York, 1976).CrossRefGoogle Scholar
[3]Маяаров, Ю.Н. [Yu.N. Makarov], “Об оценяах меры линейной независимости значений E-фянкций” [On the estimates of the measure of linear independence for the values of E-functions], Vestnik Moskov. Univ. Ser. I Mat. Meh. 2 (1978), 3–12.Google ScholarPubMed
[4]Schneider, Theodor, Einführung in die Transzendenten Zahlen (Die Grundlehren der Mathematischen Wissenschaften, 81. Springer-Verlag, Berlin, Göttingen, Heidelberg, 1957).CrossRefGoogle Scholar
[5]Шидловский, А.В. [A.B. Šidlovskiĭ], “О трансцендентности и алгебсаичесной независимости значений целых функций некоторых нлассов” [Transcendence and algebraic independence of the values of entire functions of certain classes], Moskov. Gos. Univ. Uč. Zap. 186 (1959), 11–70.Google ScholarPubMed
[6]Шидловский, А.Б. [A.B. Šidlovskii˘], “О кримерии алгебраической независимости значений одного класса целЫх функций” [A criterion for algebraic independence of the values of a class of entire functions], Izv. Akad. Nauk SSSR Ser. Mat. 23 (1959), 35–66; Amer. Math. Soc. Transl. (2) 22 (1962), 339–370.Google ScholarPubMed
[7]Shidlovskii, A.B., “On the estimates of the algebraic independence measures of the values of E-functions”, J. Austral. Math. Soc. Ser. A 27 (1979), 385–407.CrossRefGoogle Scholar
[8]Väänänen, Keijo, “On lower estimates for linear forms involving certain transcendental numbers”, Bull. Austral. Math. Soc. 14 (1976), 161–179.CrossRefGoogle Scholar
[9]Väänänen, Keijo, “On linear forms of the values of one class of E-functions”, Acta Univ. Oulu. Ser. A Sci. Rerum Natur. No. 41 Math. No. 12 (1976), 19PP.Google Scholar
[10]Shan, Xu Guang and Rui, Yu Kun, “Some diophantine inequalities involving a class of Siegel's E-functions”, Kexue Tongbao 24 (1979), 481–486.Google Scholar
[11]Rui, Yu Kun and Shan, Xu Guang, “A note on a theorem of Baker and Mahler”, Acta Math. Sinica 22 (1979), 487–494.Google Scholar