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So far, I've encountered three questions here on MSE that involve a function evaluated at a natural argument $n$ on the one left side of the equation, and the same function that has been differentiated $n$ times and evaluated at a constant argument. Examples include the following equations:

  1. In this question, the titular equation is discussed: $$ f(n) = f^{(n)}(0). \tag{1} $$ Here, $n \in \mathbb{N}$ or $\mathbb{N}_{>0}$.
  2. The question above is generalized over here, and asks about real-analytic functions $g(\cdot)$ that satisfy $$ g(n) = g^{(n)}(a) \tag{2} $$ for any $a \in \mathbb{R}$.
  3. Finally, the following question leads to the equation $$ h(n) = \frac{ h^{(n)}(0) }{ n! }, \tag{3} $$ as pointed out by user Dan in the comment section. The same question was also posed earlier.

All of these equations are of the form $$p(n) = p^{(n)}(a) \ q(n) \tag{*} $$ for some function $q: \mathbb{N} \to \mathbb{Q} \setminus \{ 0 \} $ and some $p: \mathbb{R} \to \mathbb{R}$. They look a bit like differential equations, but it seems they're also different compared to them.

Questions: do equations of the form $(*)$ have a name? Are they mentioned and described in the literature somewhere?

Max Muller
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  • They all fall under the bracket of functional differential equations. More specifically, they involve finding functions with self-similar derivatives or generating functions. – Jam Sep 27 '23 at 09:49
  • @Jam: Yes. I do hope though that there's a more specific name for this type of equations within the big field of functional differential equations. And I wonder whether they have been studied before -- all references are welcome – Max Muller Sep 27 '23 at 18:39
  • That's a linear delay differential equation (DDE) with a single discrete delay at $x = 0$: $$ \begin{align} f\left( x + n \right) &= \frac{\operatorname{d}^{n}f\left( x \right)}{\operatorname{d}x^{n}}\ g\left( x + n \right) &= \frac{\operatorname{d}^{n}g\left( x + a \right)}{\operatorname{d}x^{n}}\ \end{align} $$ If $n$ is not an integer then you could think of it as a linear fractional delay differential equation (FDDE) with a single discrete delay at $x = 0$. Does that answer the question / do you mean that?? – Kevin Dietrich Sep 30 '23 at 14:44
  • @KevinDietrich Hmm yes, it does seem to be related. However, I don't see any higher order derivates on de DDE page you linked to -- do these also appear in the study on DDEs? – Max Muller Oct 01 '23 at 10:47

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