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, $$f''(t)=\frac{t^{2\beta}}{f(t)},$$ see https://math.stackexchange.com/q/3612204/87355 .