Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (3.16)

\begin{align*}
\int_{{}}^{\left( n \right)}{h_{s}^{c}ds}&=\frac{-1}{\left( n-1 \right)!}\sum\limits_{i=1}^{n-1}{\binom {n-1} {n-i}{{s}^{\left( n-i \right)}}{{w}_{i}}}\left( \sum\limits_{j=i}^{n-1}{\frac{1}{j}} \right)\label{eq:q32} \\
&+\frac{{\exp(-c)}}{\left( n-1 \right)!}\left( \sum\limits_{i=0}^{\infty }{\frac{{{c}^{i}}}{i!}{{\left( s+1+i \right)}^{n-1}}\ln \left( s+1+i \right)} \right), \notag
\end{align*}Proof:
We start by taking the first-, second-, and third-order antiderivatives,
\begin{align*}
\int{h_{s}^{c}}ds=&{\exp(-c)}\sum\limits_{i=0}^{\infty }{\frac{{{c}^{i}}}{i!}\ln \left( s+1+i \right)}, \\
\int_{{}}^{\left( 2 \right)}{h_{s}^{c}ds}=&-s+{\exp(-c)}\sum\limits_{i=0}^{\infty }{\frac{{{c}^{i}}}{i!}\left( s+1+i \right)\ln \left( s+1+i \right)}, \\
\int_{{}}^{\left( 3 \right)}{h_{s}^{c}ds}=&-\frac{3{{s}^{2}}}{4}-\frac{s}{2}-\frac{cs}{2}+\frac{{\exp(-c)}}{2}\sum\limits_{i=0}^{\infty }{\frac{{{c}^{i}}}{i!}{{\left( s+1+i \right)}^{2}}\ln \left( s+1+i \right)}.
& \cdots
\end{align*}
Repeating the same process of integration by finding the proper constant, we can conclude the proof.
$\square$

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