In this question, I ask about equations like $$ g(n) = g^{(n)}(a) \tag{1} $$ and $$ h(n) = \frac{ h^{(n)}(0) }{ n! }. \tag{2} $$
Equation $(1)$ is discussed here for $a=0$, and here for the general case where $a \in \mathbb{R}$. Also, equation $(2)$ is looked into in this question and over here. Through Kevin Dietrich's suggestion and some research, I've learned that equations like this are called higher-order delay differential equations.
I am curious whether there is any literature on this topic that covers such equations extensively. I know there are books on the subject, but the ones I've found fall short on what I am looking for.
For instance, the following book by T. Erneux discusses mostly lower-order delay differential equations, like $\frac{dy}{dt} = ky (t - \tau). $ I am also curious what happens in higher-order cases.
Then there is also a class of papers that delve into higher-order DDEs, but they mostly go into numerical properties or long time behaviour. Examples include this article and this one.
What I am looking for, are sources that delve into (a) meromorphic or analytic solutions to higher-order DDEs (this article provides a rare instance of a relevant source in this sense), and (b) look mostly at higher-order DDEs for functions at (positive) integer-valued arguments, like those in equation $(1)$ when $a \in \mathbb{Z}_{\geq 0}$, and $(2)$.
Question: are there any resources on analytic or meromorphic solutions to higher-order delay differential equations at integer-valued arguments?
