Consider the parameter $P$ and a property $S$ associated with this parameter represented as $P_S$. For example, consider that the height of a tree (“H”) is my parameter and the property is the tree’s name. Therefore, $H_{T1}$ refers to the height of the tree named $T1$.
Now, consider that $H_{T1}$ depends on 2 inputs, named $A$ and $B$ respectively. So, for the above example we have:
$$ H_{T1}^{A,B} $$
My first concern is whether the above is a valid notation or should I have written something else like $H_{T1}(A,B)$?
I am then using the cases environment in order to define the different value that $H_{T1}$ takes for a number of differetn $A, B$ inputs:
$$ \begin{eqnarray} H_{T1}^{A,B} = \begin{cases} H_{T1}^{5,8} & \text{if $A=5$, $B=8$}\\ H_{T1}^{9,1} & \text{if $A=5$, $B=1$}\\ H_{T1}^{3,3} & \text{if $A=3$, $B=3$}\\ \end{cases} \end{eqnarray} $$
For a particular set of $A,B$ inputs, e.g. for the set $(A=3, B=3)$, $H_{T1}$ depends on $2$ extra inputs, named $C$ and $D$ respectively like that:
$$ \begin{eqnarray} H_{T1}^{3,3,C,D} = \begin{cases} H_{T1}^{3,3,1,1} & \text{if $A=3$, $B=3$, $C=1$, $D=1$}\\ H_{T1}^{3,3,4,1} & \text{if $A=3$, $B=3$, $C=4$, $D=1$}\\ H_{T1}^{3,3,6,2} & \text{if $A=3$, $B=3$, $C=6$, $D=2$}\\ \end{cases} \end{eqnarray} $$
Since only for $H_{T1}^{3,3}$ I have the 2 extra inputs, what would be the best way of writing that? I thought of writing something like that:
$$ \begin{eqnarray} H_{T1}^{A,B} = \begin{cases} H_{T1}^{5,8} & \text{if $A=5$, $B=8$}\\ H_{T1}^{9,1} & \text{if $A=5$, $B=1$}\\ H_{T1}^{3,3,C,D} & \text{if $A=3$, $B=3$}\\ \end{cases} \end{eqnarray} $$
where
$$ \begin{eqnarray} H_{T1}^{3,3,C,D} = \begin{cases} H_{T1}^{3,3,1,1} & \text{if $A=3$, $B=3$, $C=1$, $D=1$}\\ H_{T1}^{3,3,4,1} & \text{if $A=3$, $B=3$, $C=4$, $D=1$}\\ H_{T1}^{3,3,6,2} & \text{if $A=3$, $B=3$, $C=6$, $D=2$}\\ \end{cases} \end{eqnarray} $$
but I am not sure whether it’s correct or not.
Thanks in advance.