**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.



Power series with non-integer exponents
The series Σane-λnz are called (generalized) Dirichlet series, and are special cases of the Laplace transform of a discrete measure. For t=e-z you get the power series with fractional exponents you are asking about. See Widder's book "The Laplace transform" for more details.


