I wonder if it may be a worthwhile exploration.
This question asks if we can create a certain type of power series. The idea is that we start out
$1y+$
$(1+x)y^2+$
$(1+x+x^2+x^3)y^3+$
etc.
Now I was thinking that for a starting $x$ value, we can perform a procedure that allows us to modify $x$ while calculating out the correct value.
We start with $(1)y$. Call this [1]. Then we multiply what we have by $(1+x^{1/2})y$. This gives us $(1+x^{1/2})y^2$. Now we rewrite/recalculate $x$ as $x^2$. Then we have $(1+x)y^2$. Call this [2]. Now we can add this calculation to the old calculation to give us
$(1)y + (1+x)y^2$
Now, once again, we use the same procedure we used on [1] to give us [2], but this time we use it on [2] to give us [3]. The procedure repeats ad infinitum, like a calculus or integral or something. I wonder if this has ever been explored. I also wonder what some of the major obstacles are that make this method tough.
So my question is, is there a math that has been explored allowing rewriting of variables in some shape or form, similar to this? If so, I'd like an answer to describe where I can learn more about it or about attempts at it.