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

Gundersen, Gary G.

Proc. R. Soc. Edinb., Sect. A 102, 9-17 (1986).

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

Keywords:

second order differential equation; transcendental entire function; infinite order

References:

[1]	DOI: 10.1007/BF02566887 · Zbl 0115.06904
[2]	Bank, Reine Angew. Math. 344 pp 1– (1983)
[3]	Amemiya, Hokkaido Math. J. 10 pp 1– (1981)
[4]	DOI: 10.2996/kmj/1138036197 · Zbl 0463.34028
[5]	Hayman, Meromorphic Functions (1964)
[6]	Markushevich, Theory of Functions of a Complex Variable (1965)
[7]	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