On factorization of the solutions of second-order linear differential equations. (English) Zbl 1027.34093

Zheng, Jianhua; He, Yuzan

J. Math. Soc. Japan 53, No. 4, 835-845 (2001).

The authors prove the following results: (i) Let \(f(z)= \psi(e^{z})\exp(\Phi(z)+ dz)\), where \(d\) is not a rational number, \(\psi(\zeta)\) is a polynomial having at least a nonzero simple zero, and \(\Phi(z)\) is a nonconstant periodic entire function of finite order. Then \(f(z)\) is prime.
(ii) Let \(f(z)\) be a solution to the equation \[ w''+ A(e^{z})w = 0, \quad\text{where }A\in C[z],\;\deg A \geq 2, \] with the properties that \(0\) is not a Picard exceptional value of \(f(z)\) and that the exponent of convergence for the zeros of \(f(z)\) is finite. If \(\gamma = \sqrt(-b_{0})\) is not a rational number, then \(f(z)\) is prime \((b_{0}= A(0))\).

Reviewer: Nikolay Vasilye Grigorenko (Kyïv)

MSC:

34M05

Entire and meromorphic solutions to ordinary differential equations in the complex domain

Keywords:

factorization; solutions; second-order linear differential equations