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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





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.

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Substitution comes in various guises throughout most of mathematics and has no single branch devoted to it. What you're doing in particular looks like it could be expressed as the action of a power series in the operator $S:f(x)\mapsto (1+xy)f(x^2)$. – anon Jan 31 '12 at 15:50
@anon: Thanks for the information. I don't really understand how your operator $S$ would work, though. Where can I find out more about this action/operator? – Matt Groff Jan 31 '12 at 15:59
Why did you multiply by $(1+x^{1/2})y$ rather than $(1+x)y$? – Henry Jan 31 '12 at 16:44
@Henry: So that the procedure can be repeated correctly by recalculating $x$. If I multiply $(1+x)y^2$ by $(1+x)y$ instead, I'd get $(1+2x+x^2)y^3$. The procedure would then yield incorrect results. – Matt Groff Jan 31 '12 at 16:54
While it is true that substitution appears in many areas of mathematics, there is indeed one branch of mathematics that deals with the theory of substitution: the theory of rewriting systems. It's a border area of mathematics and computer science. – Manolito Pérez Jan 31 '12 at 17:17

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