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2.1 Definition of the "h" function

According to its relationship with the confluent hypergeometric function in (1.5) , "h" can be specified as

\begin{equation*} h_{s}^{c}=\frac{{{\left( -c \right)}^{0}}}{\left( s+1 \right)}+\frac{{{\left( -c \right)}^{1}}} {\left( s+1 \right)\left( s+2 \right)}+\cdots+\frac{{{\left( -c \right)}^{n}}}{\prod\limits_{i=0}^{n} {\left( s+1+i \right)}},\text{ }n\to \infty. \tag{2.1} \end{equation*}

[Identities of (2.1)]

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*}

[Proof for (2.2)]

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 nth-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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