## 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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### References:

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