Abstract
Applying a result obtained in [TK], we investigate the zero-distribution of fundamental solutions of some linear differential equations with at least one transcendental coefficient, and compare the results with some of the well-known results on equations with the polynomial coefficients. Results corresponding to Frank’s theorem [F] on exceptional fundamental sets and a theorem due to Frank and Hellerstein [F-H] on non-homogeneous equations are obtained.
The author was supported in part by the Grant-in-Aid for General Scientific Research from the Ministry of Education, Science and Culture 11640164 (N. Toda).
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References
Ahlfors, L., The theory of meromorphic curves, Acta Soc. Sci. Fenn. N.S.A., III (1941), 1–31.
Bank, S. B., On the frequency of complex zeros of solutions of certain differential equations, Kodai Math. J., 15 (1992), 165–184.
Brüggemann, F., On the zeros of fundamental systems of linear differential equations with Polynomial coefficients, Complex Variables Theory Appl., 15 (1990), 159–166.
Frank, G., Picardsche Ausnahmewerte bei Lösungen linearer Differen-tialgleichungen, Manuscripta Math., 2 (1970), 181–190.
Frank, G. and Hellerstein, S., On meromorphic solutions of nonhomogeneous linear differential equations with polynomial coefficients, Proc. London Math. Soc. (3), 53 (1986), 407–428.
Hayman, W. K., Meromorphic Functions, Clarendon Press, Oxford, 1964.
Laine, I., Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin-New York, 1993.
Langley, J., On Hayman’s alternative, Mathematika, 32 (1985), 139–146.
Lang, S., Introduction to Complex Hyperbolic Spaces, Springer-Verlag, New York-Berlin-Heidelberg-London-Paris-Tokyo, 1987.
Rieppo, J., Differential fields and complex differential equations, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertations, No. 118, 1998.
Steinmetz, N., Linear differential equations with exceptional fundamental sets, Analysis, 11 (1991), 119–128.
Toda, N., On the frequency of zeros of solutions of higher order linear differential equations with entire coefficients, Complex Variables Theory Appl., 23 (1993), 249–255.
Tohge, K., Logarithmic derivatives of meromorphic or algebroid solutions of some homogeneous linear differential equations, Analysis, 19 (1999), 273–297.
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© 2000 Kluwer Academic Publishers
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Tohge, K. (2000). On Meromorphic Solutions of Linear Differential Equations with at Least One Transcendental Coefficients. In: Begehr, H.G.W., Gilbert, R.P., Kajiwara, J. (eds) Proceedings of the Second ISAAC Congress. International Society for Analysis, Applications and Computation, vol 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0269-8_47
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DOI: https://doi.org/10.1007/978-1-4613-0269-8_47
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