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\newcommand{\ds}[1]{\displaystyle{#1}}
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\newcommand{\raíz}[2][]{\,\sqrt[#1]{\vphantom{\large Un}\,#2\,}\,}
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\newcommand{\verts}[1]{\left\vert\, nº 1 \,\right\vert}$
Con $\ds{\verts{z}\ <\ 1}$:
\begin{align}
{\cal F}_{n}\pars{z}&\equiv\sum_{k\ =\ 1}^{\infty}\fermi_{n}\pars{k}z^{k}
=\sum_{k\ =\ 1}^{\infty}\sum_{i\ =\ 1}^{k}\fermi_{n - 1}\pars{i}z^{k}
=\sum_{i\ =\ 1}^{\infty}\fermi_{n - 1}\pars{i}\sum_{k\ =\ i}^{\infty}z^{k}
=\sum_{i\ =\ 1}^{\infty}\fermi_{n - 1}\pars{i}\,{z^{i} \over 1 - z}
\\[5mm]&={1 \over 1 - z}\sum_{i\ =\ 1}^{\infty}\fermi_{n - 1}\pars{i}z^{i}
\end{align}
\begin{align}
\imp&\quad{\cal F}_{n}\pars{z}={{\cal F}_{n - 1}\pars{z} \over 1 - z}
={{\cal F}_{n - 2}\pars{z} \over \pars{1 - z}^{2}}=\cdots
={{\cal F}_{1}\pars{z} \over \pars{1 - z}^{n - 1}}
\\[5mm]&={1 \over \pars{1 - z}^{n - 1}}\,
\sum_{k\ =\ 1}^{\infty}\ \overbrace{\fermi_{1}\pars{k}}^{\dsc{H_{k}}}z^{k}
=-\,{1 \over \pars{1 - z}^{n - 1}}\,{\ln\pars{1 - z} \over 1 - z}
=-\,{\ln\pars{1 - z} \over \pars{1 - z}^{n}}
\\[5mm]&=-\lim_{\mu\ \to\ -n}\partiald{\pars{1 - z}^{\mu}}{\mu}
=-\lim_{\mu\ \to\ -n}\partiald{}{\mu}
\sum_{k\ =\ 0}^{\infty}{\mu \choose k}\pars{-1}^{k}z^{k}
\\[5mm]&=-\lim_{\mu\ \to\ -n}\partiald{}{\mu}
\sum_{k\ =\ 1}^{\infty}{-\mu + k - 1\choose k}z^{k}\quad\imp\quad
\fermi_{n}\pars{k}=-\lim_{\mu\ \to\ -n}\partiald{}{\mu}
{-\mu + k - 1\choose k}
\end{align}
$$
\fermi_{n}\pars{k}={\Gamma\pars{k + n} \\Gamma\pars{k + 1}\Gamma\pars{n}}\,
\bracks{\Psi\pars{k + n} - \Psi\pars{n}}
$$
$$
\color{#66f}{\large\fermi_{n}\pars{n}}
={\Gamma\pars{2n} \over \Gamma\pars{n + 1}\Gamma\pars{n}}\,
\bracks{\Psi\pars{2n} - \Psi\pars{n}}
=\color{#66f}{\large{2n - 1 \elegir n}\bracks{\Psi\pars{2n} - \Psi\pars{n}}}
$$
