# How can we take a power series and multiply each term, i.e. $c_n x^n$ by $y^n$?

In other words, given a power series $f(x)$, is there an alternative to taking $\lim_{x\to{x y}}f(x)$? I ask this because I thought that there may be a way to replace the limit by integration, or some other operation that I know how to do repeatedly.

To attempt to clarify slightly, I plan on taking this limit repeatedly in a "loop", along with a few other operations. I have seen repeated integration done in a book, e.g. Keith B. Oldham's and Jerome Spanier's The Fractional Calculus, published by Dover Publications, Inc.

For an example, suppose I have a generating function: $$f(x) = 15x^3 + 27x^5 + 300x^9$$ I want $$f(xy) = 15x^3y^3 + 27x^5y^5 + 300x^9y^9$$

Again, I'm hoping for one or more answers that show some generalized methods to go from a general function $f(x)$ to $f(xy)$.

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I don't see what this has to do with limits. The notation $\lim_{x\to xy}$ is misleading. The process of substitution, replacing $x$ by $xy$, is the algebraic operation of composing two functions. It can be done repeatedly if you want: in the second line, you can replace $x$ with $xy$ again, or replace $x$ with $xz$, or both $x$ and $y$ with $xz$ and $yz$, etc - depends on what you want to get out of this process.
Indeed, it looks like a notational confusion to me. The notation “$x\mapsto xy$” looks like “$x\to xy$”, but the two should always be distinguished. –  Lubin Jul 4 '12 at 21:19