I'm trying to prove by induction on the multi-index $\alpha$, that, $$\sum\limits_{j=\frac{|\alpha|}{2}}^{|\alpha|}\sum\limits_{|\beta|=2j-|\alpha|}c_{\beta}x^{\alpha}[m_z(x)]^{j+1}=D^{\alpha}m_z(x)$$ where $m_z(x)=z(z-4\pi^2|x|^2)^{-1}, z \in \{u \in \mathbb{C}-\mathbb{R}_+;Re z \geq 0\}, x \in \mathbb{R}^n$ and $c_\beta$ are complex constants which do not depend on $z$. Somebody can help me or show some references?
2026-03-26 01:14:46.1774487686
A derivative identity, for any multi-index.
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