Different types of notation for recursive math? There are particular notations for arithmetic operations on sequences. Ultimately, it is recursive and we are able to shorthand this instead of writing $1+1+1+1+...$ etc.
One example of this is the summation operation:
$$ \sum_{i=0}^N a_i=a_0+a_1+...+a_N $$
Another example is the product operations:
$$ \prod_{i=0}^N a_i=a_0\times a_1\times ...\times a_N$$
My question is: Are there more types of this notation for recursive operation, whether it contains arithmetical meaning or not? I've been trying to look for a proper name for this by looking on Google typing, "Recursive mathematical operations" but I have not found anything. Maybe there's a different name of this algorithm that I am not aware of?

This is not used too commonly, but Gauss's notation for continued fractions is an example:
$$
\substack{n\\\Huge \text{K}\\i=1}\frac1{a_i}=\frac{1}{a_1+\frac1{a_2+\frac{1}{a_3+\ddots}}}
$$
