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Coefficients of differentially algebraic series

Part of: Series expansions

Published online by Cambridge University Press:  09 April 2009

Vichian Laohakosol ,
Kannika Kongsakorn  and
Patchara Ubolsri
Vichian Laohakosol
Affiliation:
Department of Mathematics Kasetsart UniversityBangkok 10900, Thailand
Kannika Kongsakorn
Affiliation:
Department of Mathematics Kasetsart UniversityBangkok 10900, Thailand
Patchara Ubolsri
Affiliation:
Department of Mathematics Kasetsart UniversityBangkok 10900, Thailand
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Abstract

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An Eisenstein-like criterion is proved for power series with algebraic coefficients satisfying algebraic differential equations of a certain general kind. The proof is elementary and the result extends earlier results of Hurwitz, Pólya and Popken

MSC classification

Secondary: 30B10: Power series (including lacunary series)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 3 , June 1990 , pp. 402 - 412
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Hurwitz, A., ‘Sur le développement des fonctions satisfaisant à une équation différentielle algébrique’, Ann. Sci. Ècole Norm. Sup. (4) 6 (1889), 327–332.CrossRefGoogle Scholar
[2]Kakeya, S., ‘On a power series with rational coefficients satisfying an algebraic differential equation’, Sci. Rep. Tôhoku Imp. Univ. 4 (1915), 7–20.Google Scholar
[3]Kongsakorn, K. and Laohakosol, V., ‘A generalization of Eisenstein's theorem’, Bull. Inst. Math. Acad. Sinica 14 (1986), 149–162.Google Scholar
[4]Laohakosol, V., Kongsakorn, K. and Ubolsri, P., ‘Denominators in the coefficients of power series satisfying linear differential equations’, to appear.Google Scholar
[5]Lipschitz, L. and Rubel, L. A., ‘A gap theorem for power series solutions of algebraic differential equations’, Amer. J. Math. 108 (1986), 1193–1214.CrossRefGoogle Scholar
[6]Mahler, K., ‘On formal power series as integrals of algebraic differential equations’, Rend. Accad. Naz. Lincei 50 (1971), 76–89.Google Scholar
[7]Mahler, K., ‘Lectures on transcendental numbers’, Lecture Notes in Math., Vol. 546, (Springer-Verlag, Berlin-Heidelberg-New York, 1976).Google Scholar
[8]Pincherle, S., ‘Sur la nature arithmétique des coefficients des séries intégrales des équations différentielles linéaires’, J. Reine Angew. Math. 103 (1888), 84–86.CrossRefGoogle Scholar
[9]Pólya, G., ‘Über das Anwachsen von ganzen Funktionen; die einer Differentialgleichung genugen,’ Vierteljahrschr. Naturforsch. Gesellschaft Zurich 61 (1916), 531–545.Google Scholar
[10]Pólya, G. and Szegö, G., Problems and theorems in analysis, Vol. II, (Springer-Verlag, New York-Heidelberg-Berlin, 1976).CrossRefGoogle Scholar
[11]Popken, J., Über arithmetische Eigenschaften Analytischer Funktionen, (North-Holland, Amsterdam, 1935).Google Scholar
[12]Popken, J., ‘An arithmetical property of a class of Dirichlet series,’ Indag. Math. 7 (1945/1946), 105–122.Google Scholar
[13]Sibuya, Y. and Sperber, S., ‘Arithmetic properties of power series solutions of algebraic differential equations,’ Ann. of Math. 113 (1981), 111–157.CrossRefGoogle Scholar