Homogeneous linear discrete delay differential equation with constant coefficients

Definition

A linear discrete delay differential equation with constant coefficients is a particular kind of delay differential equation described as follows:

$\sum _{k=0}^{r}\left(\sum _{j=1}^{m_{k}}A_{k,j}{\frac {d^{k}(x(t-\tau _{k,j}))}{dt^{k}}}\right)=0$

where $m_{k}$ are nonnegative integers for $k\in \{0,1,\dots ,r\}$ and $A_{k,j}$ are real numbers while $\tau _{k,j}$ are nonnegative real numbers.

Procedure for finding smooth solutions

We construct a characteristic equation as follows:

$\sum _{k=0}^{r}\left(\sum _{j=1}^{m_{k}}A_{k,j}z^{k}e^{-z\tau _{k,j}}\right)=0$