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I know addition and multiplication are well defined operations on formal power series. Now say you have two formal power series $F(x),G(x)\in R[[x]]$, with $R\supset\mathbb{Q}$ is the coefficient ring.

Is there a way to define $F(x)^{G(x)}$? Is there a standard well defined definition for this operation that hopefully satisfies the usual exponent laws? Thanks.

Chelsea Dirks
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    No, even $x^x$ doesn't make sense in the power series ring ! – Georges Elencwajg Feb 14 '12 at 21:55
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    At least, $F(x)$ probably needs to have a nonzero constant term, since otherwise you expect something like a logarithmic singularity (think of the expression as $\exp(G(x)\ln F(x))$). But then, I would suspect (but don't know) that this standard formula will provide a positive answer. – Harald Hanche-Olsen Feb 14 '12 at 22:00
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    It interesting to note that $${\left( {1 + x + \frac{{{x^2}}}{{2!}} + \frac{{{x^3}}}{{3!}} + \cdots } \right)^x} = 1 + x^2 + \frac{{{{\left( {{x^2}} \right)}^2}}}{{2!}} + \frac{{{{\left( {{x^2}} \right)}^3}}}{{3!}} + \cdots $$ i.e $(e^x)^x = e^{x^2}$ – Pedro Feb 18 '12 at 23:30

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Composition of formal power series can be done via the power-series version of Faà di Bruno's formula. In that way one can find the series for $\log_e F(x)$. Then one can say $$ F(x)^{G(x)} = e^{G(x)\log_e F(x)}. $$ Multiplication of formal power series is a well-known operation. Exponentiation can be done via the exponential formula.

Michael Hardy
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