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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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