Bulletin of the
Korean Mathematical Society BKMS

We consider the delay differential equations \begin{equation*} b(z)w(z+1)+c(z)w(z-1)+a(z)\frac{w'(z)}{w^k(z)}=\frac{P(z,w(z))}{Q(z,w(z))}, \end{equation*} where $k\in\{1,2\}$, $a(z)$, $b(z)\not\equiv 0$, $c(z)\not\equiv 0$ are rational functions, and $P(z,w(z))$ and $Q(z,w(z))$ are polynomials in $w(z)$ with rational coefficients satisfying certain natural conditions regarding their roots. It is shown that if this equation has a non-rational meromorphic solution $w$ with hyper-order $\rho_{2}(w)<1$, then either $\deg_w(P)=\deg_w(Q)+1\leq 3$ or $\max\{\deg_w(P),\deg_w(Q)\}\leq 1$. In addition, it is shown that in the case $\max\{\deg_w(P),\deg_w(Q)\}= 0$ the equations above can have such a solution, with an additional zero density requirement, only if the coefficients of the equation satisfy certain strict conditions.

Keywords: Delay differential equations, Painlev\'e equation, Nevanlinna theory, meromorphic, hyper-order

MSC numbers: Primary 30D35; Secondary 34K40, 34M55

Supported by: The second author thanks the support of the China Scholarship Council (No. 2020063300 38).