On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is $$\sum_{\alpha}\frac{z^{\alpha}\left(r1\right)^{\alpha}}{\gamma_{\alpha}}=\sum_{k=0}^{\infty}\frac{z_{1}^{k}r^{k}}{N\left(k\right)}?$$ Here $\alpha=\left(\alpha_{1},...,\alpha_{n}\right)$ is a multi-index, so $z^{\alpha}=z_{1}^{\alpha_{1}}...z_{n}^{\alpha_{n}}$. Did author used here something like $0^{0}=1$? Please if someone knows how can we prove this to tell me. Thank you.
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2$\begingroup$ Using the convention $0^0=1$ when writing power series is fairly standard. Is that your only concern? If not, you might want to explain what $\gamma_\alpha$, $N(k)$ etc. are. $\endgroup$– Michael RenardyCommented Feb 22, 2015 at 14:48
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$\begingroup$ Thank you for answer. Yes. That was my only concern. But can we really use that? I learned that $0^{0}$ is undefined so that bothered me here. Thanks again. $\endgroup$– AlemCommented Feb 22, 2015 at 14:55
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