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.

  • $\begingroup$ Is this a cohomology thing? $\endgroup$ – Steve Huntsman Sep 21 '10 at 23:25
  • $\begingroup$ Are these not Puiseux series? $\endgroup$ – Gerry Myerson Sep 21 '10 at 23:46
  • $\begingroup$ @Steve: The original context is that the graph of $\gamma$ should be the stable manifold for a (non-hyperbolic) fixed point of a certain map, namely $f(x,y) = (x+x^{1+\varepsilon} - y, y-y^{1+\varepsilon})$. So I don't know that I'd call it a cohomology thing, but it's definitely from dynamical systems. $\endgroup$ – Vaughn Climenhaga Sep 21 '10 at 23:59
  • $\begingroup$ @Gerry: Based on a quick glance at the Wikipedia article on Puiseux series, it looks as though they're required to have only rational exponents, whereas in my setting the exponents can be any real numbers. It may or may not be the case that an appropriate coordinate change would make all my exponents rational -- I'm not sure. $\endgroup$ – Vaughn Climenhaga Sep 22 '10 at 0:11
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    $\begingroup$ For transseries, see my expository introduction... math.ohio-state.edu/~edgar/preprints/trans_begin $\endgroup$ – Gerald Edgar Sep 22 '10 at 2:32

The series Σanenz 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.



"They are a generalization of Puiseux series ".

" They allow for arbitrary exponents of the indeterminate so long as the set supporting them forms a well-ordered subset of the value group"


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