The book mentioned in a comment, Sequences of Real Numbers, by Sîntămărian & Furdui, presents on page 11 the recursion $$x_{n+1}=x_n+\frac{n^{2\beta}}{x_1+x_2+\cdots+x_n}$$ and the derivation of the limit $$\lim_{n\rightarrow\infty}x_n/n^\beta=\sqrt{1+1/\beta}$$ as an application of the Stolz–Cesàro Theorem.
The OP asks where this recursion relations might appear in a research context. It appears as a discretization of the Emden–Fowler nonlinear differential equation, $$f''(t)=t^{p}[f(t)]^q.$$ A particular solution is $$f(t)=\lambda t^{(p+2)/(1-q)},\;\;\lambda=\left[\frac{(p+2)(p+q+1)}{(q-1)^2}\right]^{1/(q-1)}.$$ One readily checks that the asymptotic limit given above for $x_n=f'(n)$ is obtained for $p=2\beta$, $q=-1$.