Zeros of solutions of higher-order linear differential equations (Q2720029)

The authors investigate the exponent of convergence of the zero-sequence of any nontrivial solution to the higher-order equation NEWLINE\[NEWLINEf^{(k)}+ a_{k-1}(z) f^{(k-1)}+\cdots+ a_1(z)f'+ (Q_1(z) e^{p_1(z)}+ Q_2(z) e^{p_2(z)}+ Q(z)) f=0,NEWLINE\]NEWLINE where \(p_1(z)= \zeta_1 z^n+\cdots\), \(p_2(z)= \zeta_2 z^n+\cdots\) are nonconstant polynomials, \(Q_1(z)(\not\equiv 0)\), \(Q_2(z)(\not\equiv 0)\), \(Q(z)\) and \(a_j(z)\), \(j= 1,\dots, k-1\), are entire functions and their orders are less than \(n\). They prove that the exponent is infinite when \(\zeta_2/\zeta_1\) is not real. This result improve \textit{K. Ishizaki} and \textit{K. Tohge's} result on the second-order differential equation \(f''+ (e^{p_1(z)}+ e^{p_2(z)}+ Q(z)) f=0\) [J. Math. Anal. Appl. 206, No. 2, 503-517 (1997; Zbl 0877.34009)].