When reading the GARCH modeling part of book Analysis of Financial Time Series, I read the following statement. In specific, I do not understand how does the author get that statement marked with yellow. How to get the relationship between the fractional difference with the ACF of $u_t$.
using binomial expansion the fractional difference operator $(1-L)^d$ can be rewritten as $$(1-L)^d = \sum_{j=0}^\infty a^d_j L^j$$. One can show that the coefficients $a_j^d$ satisfy $$a_j^d \sim \frac{1}{\Gamma(d)} j^{-1-d} \qquad\text{for } j\rightarrow\infty.$$ As a result, the acf of $u_t$ shows the same asymptotic behavior. For more details, see for example Time Series: Theory and Methods by Brockwell and Davis.