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On the question of whether \(f''+e^{-z}f'+B(z)f=0\) can admit a solution \(f\not\equiv 0\) of finite order. (English) Zbl 0598.34002
The author proves the following results in this paper. Theorem 1. If B(z) is a transcendental entire function with order (B)\(\neq 1\), then every solution \(f\not\equiv 0\) to the DE \(f''+e^{-z}f'+B(z)f=0\) has infinite order. For the differential equation (1) \(f''+e^{-z}f'+Q(z)f=0\) where Q(z) is a polynomial, the following theorems are proved: Theorem 2. If Q(z) is a polynomial of odd degree, then every solution \(f\not\equiv 0\) to equation (1) has infinite order. Theorem 3. Let \(Q(z)=q_ nz^ n+...+q_ 0\) be a polynomial of even degree \(n\geq 2\). If either (i) \(n=2+4k\) \((k=0,1,2,...)\) and \(q_ n\) is not a positive real number, or (ii) \(n=4k\) \((k=1,2,3,...)\) and \(q_ n\) is not a negative real number, then every solution \(f\not\equiv 0\) to equation (1) has infinite order.
Reviewer: P.N.Bajaj

MSC:
34M99 Ordinary differential equations in the complex domain
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[1] DOI: 10.1007/BF02566887 · Zbl 0115.06904 · doi:10.1007/BF02566887
[2] Bank, Reine Angew. Math. 344 pp 1– (1983)
[3] Amemiya, Hokkaido Math. J. 10 pp 1– (1981) · doi:10.14492/hokmj/1381758109
[4] DOI: 10.2996/kmj/1138036197 · Zbl 0463.34028 · doi:10.2996/kmj/1138036197
[5] Hayman, Meromorphic Functions (1964)
[6] Markushevich, Theory of Functions of a Complex Variable (1965)
[7] DOI: 10.1090/S0002-9939-1971-0276470-0 · doi:10.1090/S0002-9939-1971-0276470-0
[8] Hille, Ordinary Differential Equations in the Complex Domain (1976)
[9] Hille, Lectures on Ordinary Differential Equations (1969)
[10] Wittich, Nagoya Math. J. 30 pp 29– (1967) · Zbl 0219.34005 · doi:10.1017/S0027763000012320
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