pandas iteratively update column values
The important point about these kinds of problems that you need to know is that you're on a paradox spot right now. This means that you're on a point that you want to take advantage of both vectorization and non-vectorization like threading or parallelization.
In such situation you can try one/some of the following options:

Change the type of your data structure.
Rethink your problem and see if it's possible to solve this entirely in a Vectorized way (preferably)
Simply use a non-vectorized-based approach but sacrifice something else like memory.
I have a pandas Series like the following:
a = pd.Series([a1, a2, a3, a4, ...]) 
and I want to create another pandas Series based on the following rule:
b = pd.Series(a1, a2+a1**0.8, a3 + (a2 + a1**0.8)**0.8, a4 + (a3 + (a2 + a1**0.8)**0.8)**0.8, ...).
This is doable using iteration,  but I have a large dataset (millions of records) and I must perform operation for thousands of times (for optimization purposes). I need to do this operation very fast. Is there any possible way for me to realize this by using pandas or numpy built-in functions?