A Malmquist Type Theorem for a Class of Delay Differential Equations

We show that if the following delay differential equation of rational coefficients wk(z) n-ary sumation mu=1se mu(z)w(z+c mu)+a(z)w(n)(z)w(z)= n-ary sumation i=0pai(z)wi n-ary sumation j=0qbj(z)wj w<^>k(z)\sum _{\mu =1}<^>se_\mu (z)w(z+c_\mu )+a(z)\frac{w<^>{(n)}(z)}{w(z)}= \frac{\sum _{i=0}<^>pa_i(z)w<^>i}{\sum _{j=0}<^>qb_j(z)w<^>j} \end{aligned}$$\end{document}admits a transcendental entire solution w of hyper-order less than one, then it reduces into a delay differential equation of rational coefficients wk(z) n-ary sumation mu=1se mu(z)w(z+c mu)+a(z)w(n)(z)w(z)=1w(z) n-ary sumation i=0k+2Ai(z)wi(z), w<^>k(z)\sum _{\mu =1}<^>se_\mu (z) w(z+c_\mu )+a(z)\frac{w<^>{(n)}(z)}{w(z)}=\frac{1}{w(z)}\sum _{i=0}<^>{k+2}A_{i}(z)w<^>i(z), \end{aligned}$$\end{document}which improves some known theorems obtained most recently by Zhang and Huang. Some examples are constructed to show that our results are accurate.

Automated summary

The study focuses on the properties of integer solutions of delay differential equations with rational coefficients, and demonstrates the accuracy of the results by simplifying the equation form.