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On the question of whether f″+ e−zf′ + B(z)f = 0 can admit a solution f ≢ 0 of finite order

Published online by Cambridge University Press:  14 November 2011

Gary G. Gundersen
Gary G. Gundersen
Affiliation:
Department of Mathematics, University of New Orleans, New Orleans, Louisiana 70148, U.S.A
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Synopsis

We show that if B(z) is either (i) a transcendental entire function with order (B)≠1, or (ii) a polynomial of odd degree, then every solution f≠0 to the equation f″ + e−zf′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 1-2 , 1986 , pp. 9 - 17
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Amemiya, I. and Ozawa, M.. Non-existence of finite order solutions of wʺ + e–zWˊ + Q(z)w = 0. Hokkaido Math. J. 10 (1981) Sp., 1–17.Google Scholar
2Bank, S. B. and Laine, I.. Representations of solutions of periodic second order linear differential equations. Reine Angew. Math. 344 (1983), 1–21.Google Scholar
3Frei, M.. Über die subnormalen Lösungen der Differentialgleichung wʺ + e–zwˊ + (konst.)w = 0. Comment. Math. Helv. 36 (1962), 1–8.CrossRefGoogle Scholar
4Hayman, W. K.. Meromorphic Functions (Oxford: Clarendon Press, 1964).Google Scholar
5Hille, E.. Lectures on Ordinary Differential Equations (Reading, Mass.: Addison-Wesley, 1969).Google Scholar
6Hille, E.. Ordinary Differential Equations in the Complex Domain (New York: Wiley and Sons, 1976).Google Scholar
7Marden, M.. Logarithmic derivative of an entire function. Proc. Amer. Math. Soc. 28 (1971), 513–518.CrossRefGoogle Scholar
8Markushevich, A. I.. Theory of Functions of a Complex Variable, Vol. II, translated by Silverman, R. A. (Englewood Cliffs, N.J.: Prentice-Hall, 1965).Google Scholar
9Ozawa, M.. On a solution of wʺ + e–zwˊ + (az + b)w = 0. Kodai Math. J. 3 (1980), 295–309.Google Scholar
10Wittich, H.. Subnormale Losungen der Differentialgleichung wʺ + p(ez)wˊ + q(ez)w = 0.Nagoya Math. J. 30 (1967), 29–37.Google Scholar