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I would like to approximate a recurrence relation of an arbitrary type $$x_{n+1}=f(x_n), \quad x_n\in \mathbb{C}$$ with a differential equation of an $k$th order. The sequence $x_n$ is assumed to be bounded.

One of the solutions could be to use a Taylor expansion for $x_{n+1}$: $$ x_{n+1}=x_{n}+x_{n}'+\frac{1}{2!}x_{n}''+\frac{1}{3!}x_{n}'''+...+\frac{1}{k!}x_n^{(k)}, $$ The resulting differential equation would look like this: $$ x+x'+...+\frac{1}{k!}x^{(k)}=f(x). $$ The method kind of works, but I do not know what initial conditions would be appropriate for the differential equation and if there are any better ways to approximate a recurrence relation with a differential equation. Does anyone know if this topic was researched by anyone?

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