$$
\require{AMSmath}
x^n = \sum_{k} \genfrac{\{}{\}}{0}{0}{n}{k} x^{\underline{k}} \\
\Downarrow \\
y^n = \sum_{k} \genfrac{\{}{\}}{0}{0}{n}{k} x^{\frac{1}{2}\underline{k}}\quad \dagger y = \sqrt{x} \\
\Downarrow \\
x^{\frac{1}{2}\underline{k}} = \sqrt{x}(\sqrt{x}-1)(\sqrt{x}-2)\cdots (\sqrt{x}-k+1) \\
\Downarrow \\
\sqrt{x}(\sqrt{x}-1)(\sqrt{x}-2)\cdots (\sqrt{x}-k+1)
= \frac{\sqrt{x}!}{(\sqrt{x}-k)!} \\
\Downarrow \\
\frac{\sqrt{x}!}{(\sqrt{x}-k)!} = \frac{\Gamma(\sqrt{x}+1)}{\Gamma(\sqrt{x}-k+1)}
$$
令\(r = \frac{n}{N},\phi = z^{\frac{1}{N}}\)
$$
z^r = \sum_{k} \genfrac{\{}{\}}{0}{0}{r}{r-k} z^{\underline{r-k}} \\
\Downarrow \\
{\phi}^n = \sum_{k} \genfrac{\{}{\}}{0}{0}{r}{r-k} \frac{z!}{(z-(r-k))!} \\
\Downarrow \\
{\phi}^n = \sum_{k} \genfrac{\{}{\}}{0}{0}{r}{r-k} \frac{\Gamma({\phi}^N+1)}{\Gamma({\phi}^N-(\frac{n}{N}-k)+1)} = \sum_{k} \genfrac{\{}{\}}{0}{0}{r}{r-k} {{\phi}^N}^{\underline{\frac{n}{N}-k}}\\
\Downarrow \\
{\phi}^n = \sum_{j} \genfrac{\{}{\}}{0}{0}{n}{j}{\phi}^{\underline{j}}
$$
$$
\begin{align}
\sum_{k} \genfrac{\{}{\}}{0}{0}{r}{r-k}\frac{z!}{(z-(r-k))!} &= \sum_{k} \genfrac{\{}{\}}{0}{0}{r}{r-k}\frac{\Gamma(z+1)}{\Gamma(z-r+k+1)} \\
&= \sum_{k} \genfrac{\{}{\}}{0}{0}{r}{r-k} (r-k)! \binom{z}{r-k} \\
&= \sum_{k} \frac{1}{(r-k)!} \sum_{j}
(-1)^j \binom{r}{j} (r-j)^{r-k}(r-k)! \binom{z}{r-k} \\
&= \sum_{k} \sum_{j}
(-1)^j \binom{r}{j} (r-j)^{r-k} \binom{z}{r-k} \\
\end{align}
$$
$$
z! = \int_{0}^{\infty} t^z e^{-t}dt \\
\Downarrow \\
(z-k)! = \int_{0}^{\infty} t^{z-k} e^{-t} dt \\
\Downarrow \\
z^{\underline{k}} = \frac{\int_{0}^{\infty} t^z e^{-t}dt}{\int_{0}^{\infty} t^{z-k} e^{-t}dt} \\
\Downarrow \\
z^{\underline{k}}\int_{0}^{\infty} t^{z-k} e^{-t}dt = \int_{0}^{\infty} t^z e^{-t}dt, \quad z^{\underline{k}} = \frac{z!}{(z-k)!} \\
$$
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