Let $$S(x,n) = \sum_{d|n} x^d, \quad n \in \Bbb N. $$

Do these sums appear in the literature? What are they called if they do and what is known about them?

To clarify, note that this sum is **not** the same as the generalized divisor function
$$ \sigma_x(n) = \sum_{d|n}d^x.$$
The function $f(n) = n^x$ is an arithmetic function for any constant $x$ (in the sense that $f(pq) = f(p)f(q)$ for primes $p,q$), so the method of Möbius inversion may be applied to study $\sigma_x(n)$.
In constrast, $f(n) = x^n$ is not arithmetic when $x\neq 1$ or $0$, which suggests the functions $S(x,n)$ may require the use of other less-common techniques to understand their behavior.