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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.

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