06-1350/Some Equations by Kurlin

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Claim [Kurlin]. The function [math]\displaystyle{ f(\lambda,\mu) }[/math] defined by

[math]\displaystyle{ 1+2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda+\mu}-e^{-\lambda-\mu}}{2(\lambda+\mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}}+\frac{2\mu}{e^\mu-e^{-\mu}}-1\right) }[/math]

Satisfies

[math]\displaystyle{ f(\lambda,\mu)+e^\mu f(\mu,-\lambda-\mu)+e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda+\mu} \left(\frac{e^\mu-1}{\mu}+\frac{e^{-\lambda}-1}{\lambda}\right). }[/math]

If this class is successful, at some point in the future I will tell you why I care. But meanwhile a few questions to expose my ignorance of simple algebra. Note that [math]\displaystyle{ f }[/math] is is a rational function in its variables and their exponentials. Let's call such functions "tame". The equation [math]\displaystyle{ f }[/math] satisfies involves rational expressions in the variables and their exponentials and linear substitutions applied to [math]\displaystyle{ f }[/math]. Let us call such equations "tame". (The equation is also linear in [math]\displaystyle{ f }[/math]. You may or may not include this in the definition of "tame"). So we see that a tame equation has a tame solution.

• Is that always the case?
  • A prety example is the tame equation [math]\displaystyle{ e(x+y)=e(x)e(y) }[/math], whose solution is the "first" transcendental function [math]\displaystyle{ e^x }[/math].
  • Are we done defining "new" special functions after we've defined [math]\displaystyle{ e^x }[/math] or are there further ones that cannot be defined in terms of it?
  • If there are further ones, why haven't we heard about them? Or else, where do we read about them?
  • If there aren't further ones, this is a lovely "closure" property enjoyed by exponentials. How come I don't know it yet?

• Assuming not all tame equations have tame solutions, what was special about our equation, that lead to the existance of a tame solution? Can you write a very general condition on tame equations that will ensure the existance of a tame solution? In other words, was Kurlin simply lucky that his equation had a tame solution (and clever that he found it!), or are there some general rules that guarded him against the wilderness?

[Kurlin] ^  Compressed Drinfel'd Associators, V. Kurlin, arXiv:math.GT/0408398.