Indefinite Integration of the Gamma Integral
and Related Statistical Applications
3.2 Arithmetic Rule of the "$h$" Function
As a basic function, the "$h$" function has many interesting algebraic properties. The following are
arithmetic rules, including addition, subtraction, multiplication, and division.
\begin{equation*}
h_{s}^{p+q}=\sum\limits_{i=0}^{\infty }{\frac{{{q}^{i}}}{i!}\frac{{{\partial }^{\left( i \right)}}h_{s}^{p}}
{\partial {{p}^{i}}}}.\tag{3.3}
\end{equation*}
[Proof for (3.3)]
\begin{equation*}
h_{s}^{p-q}=\sum\limits_{i=0}^{\infty }{\frac{{{\left( -q \right)}^{i}}}{i!}\frac{{{\partial }^{\left( i \right)}}h_{s}^{p}}{\partial
{{p}^{i}}}}.\tag{3.4}
\end{equation*}
[Proof for (3.4)]
\begin{equation*}
h_{s}^{pq}=h_{s}^{p}+p\left( 1-q \right)\sum\limits_{i=0}^{\infty }{\frac{{{\left( -pq \right)}^{i}}}{\prod\limits_{j=0}^{i}{\left( s+1+i \right)}}{{h}_{s+1+i}^{p}}}.\tag{3.5}
\end{equation*}
[Proof for (3.5)]
\begin{equation*}
h_{s}^{p{{q}^{-1}}}=h_{s}^{p}+p\left( 1-{{q}^{-1}} \right)\sum\limits_{i=0}^{\infty }{\frac{{{\left( -p{q}^{-1}
\right)}^{i}}}{\prod\limits_{j=0}^{i}{\left( s+1+i \right)}}{{h}_{s+1+i}^{p}}}.\tag{3.6}
\end{equation*}
[Proof for (3.6)]
In the above formulas, $p$ and $q$ are interchangeable when performing addition and multiplication.
Welcome all math lovers!
