Indefinite Integration of the Gamma Integral
and Related Statistical Applications
2.1 Definition of the "$h$" function
According to its relationship with the confluent hypergeometric function in (1.5)
, "$h$" can be specified as
We factorize the gamma integral into a product of the power function, the exponential function, and the "$h$" function
\begin{equation*}
g \left( s,c,u \right) ={{u}^{s}}{\exp(cu)}h_{s-1}^{cu}. \tag{2.2}
\end{equation*}
Every "$h$" function is an infinite series, in which the base parameter $s$ and the power parameter $c$ determine its functional value. Base parameter $s$ defines the starting number of the factorial term in the denominator. Power parameter $c$ defines the negative power term in the numerator. While $s$ and $c$ can be any real number, we temporarily exclude the case where $s$ is a negative integer and will return to it in the next section.
Given a constant $c$, $h_{s}^{c}$ is a convergent series. As the number of the expansion terms increases, the factorial denominator will eventually overpower the numerator, and the incremental value becomes infinitesimal. Furthermore, $h_{s}^{c}$ will approach $0$ as $s$ departs from $0$, either in the positive or negative direction, because the absolute value of the denominator becomes larger and larger. Computationally, using an $n$th-order expansion to calculate $h_{s}^{c}$ possesses an error of $O\left( {{c}^{n+1}} \right)$ as $c$ approaches 0
\begin{equation*}
h_{s}^{c}=\sum\limits_{i=0}^{n}{\frac{{{\left(-c \right)}^{i}}}{\prod\limits_{j=0}^{i}{\left( s+1+j \right)}}}+O\left( {{c}^{n+1}} \right). \end{equation*}
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