Do you know any research on finding closed forms of recursively-defined sequences?

This is an elementary problem, not a research topic. The recurrence is linear, and you begin by solving the homogeneous part,

$$U_{n+1}=aU_n,$$ that has the solution $U_n=ca^n$ where $c$ is an arbitrary constant.

Then as the RHS has the same shape as the homogeneous solution, you try the Ansatz $U_n=dna^n$:

$$U_{n+1}-aU_n=d(n+1)a^{n+1}-dna^{n+1}=ba^{n+1}$$ so that $d=b$.

The general solution is

$$U_n=(nb+c)a^n.$$

Search using phrases such as "difference equations" and "finite differences" and "nonlinear difference equations". Dozens, perhaps hundreds of books and tens of thousands of papers have been written on this topic in the past 300 some years. Indeed, George Boole (presently better known for Boolean logic stuff) wrote a well-known treatise on the subject in 1860 (1880 3rd edition).