**Motivation:**
For the sake of concreteness, I'll state a very particular context, but my question is a little more general. I'm trying to find a function $\gamma\colon [0,\delta) \to [0,\delta')$ that satisfies the following functional equation:
$$
\gamma(y) + \gamma(y)^{1+\varepsilon} - y = \gamma(y - y^{1+\varepsilon}).
$$
Here $0<\varepsilon<1$, and I'm most interested in the behaviour of $\gamma$ for very small values of $y$. I've persuaded myself that $\gamma(y) = y^{1/(1+\varepsilon)}(1+o(1))$, but I'd really like an exact solution, which I imagine would have to come in the form of a power series $\gamma(y) = \sum_n c_n y^{a_n}$. Due to the form of the equation, though, the exponents $a_n$ aren't going to be integers, and unless I'm mistaken, they won't even all be integer multiples of some fixed $a_0$, so I can't get back to "regular" power series by doing a simple change of coordinates. This motivates my question...

**Question:**
Can anyone suggest a reference on dealing with power series where the exponents take non-integer values (and are not all integer multiples of some fixed exponent)? Or suggest a paper where such power series are used (for any purpose)? Ideally I'd like to see how similar functional equations have been solved, but any references at all would be appreciated.