Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (2.5)

$$\begin{equation*} h_{k-r}^{c}={\exp(-c)}\sum\limits_{i=0}^{\infty }{\left\{ \frac{{{c}^{i}}}{i!}\left[ \sum\limits_{j=0}^{\infty }{\frac{{{r}^{j}}}{{{\left( k+i+1 \right)}^{j+1}}}} \right] \right\}}. \end{equation*}$$ Proof:
Performing long division for ${1}/{\left[ \left( k+1+i \right)-r \right]}$, we derive
$$\begin{equation*} \frac{1}{\left( k+1+i \right)-r}=\frac{{{r}^{0}}}{\left( k+1+i \right)}+\frac{{{r}^{1}}}{{{\left( k+1+i \right)}^{2}}}+\frac{{{r}^{2}}}{{{\left( k+1+i \right)}^{3}}}+\cdots. \end{equation*}$$
Therefore,
$$\begin{align*} {{c}^{k+1-r}}{\exp(c)}h_{k-r}^{c}&={\exp(-c)}\sum\limits_{i=0}^{\infty }{\frac{{{c}^{i}}}{i!\left( k+1+i-r \right)}}\\ &={\exp(-c)}\sum\limits_{i=0}^{\infty }{\left\{ \frac{{{c}^{i}}}{i!}\left[ \sum\limits_{j=0}^{\infty }{\frac{{{r}^{j}}}{{{\left( k+i+1 \right)}^{j+1}}}} \right] \right\}}. \end{align*}$$
$\square$

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