In Simon's book Harmonic Analysis, example 3.5.12 shows:
Fix $x \in \mathbb{R}^{\nu}$ (with $\left.\nu \geq 3\right), x \neq 0$ and define $f$ on $\{y:| y|<| x |\}$ by $$ f(y)=|x-y|^{-(\nu-2)}. $$ Then we have$,$ $$ |x-y|^{-(\nu-2)}=\sum_{d=0}^{\infty} \frac{|y|^{d}}{|x|^{d+\nu-2}} \alpha_{\nu, d} Z_{d}\left(\frac{y}{|y|}, \frac{x}{|x|}\right) \label{1}\tag{$\ast$} $$ where $\alpha_{\nu, d}$ is a constant which is dependent on $\nu, d$ and the $Z_{d}$ (the zonal harmonic of degree d) term at $y=0$ is interpreted as 1 if $d=0$ and is irrelevant if $d \geq 1$.
My Question is$,$ for fixed $x \in \mathbb{R}^{\nu}$ (with $\left.\nu \geq 3\right), x \neq 0$ and function $f(y)=|x-y|^{-\lambda}$ on $\{y|| y|<| x |\}$ with $\lambda>0,$ can we obtain a similar formula like \eqref{1} $?$
Any reference and suggestion will be greatly appreciated. Thanks!
