06-1350/Some Equations by Kurlin

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Claim [Kurlin]. The function f(\lambda,\mu) defined by

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)

Satisfies

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).

The claim is, of course, very easily verifiable and if this class is successful, at some point in the future I will tell you why I care.

Typically long equations have no closed form solutions. Yet note that Kurlin first derived the equation and only then found its solution. Was he merely lucky that a solution existed, or was there some deeper principle guarding him from failure? I don't believe in luck and I prefer to believe in principles. Yet, at what level did the principle arise? Is it that all solvable equations of this kind have closed-form solutions, or was there an a priori reason why the math that Kurlin was doing lead him to an equation with a closed form solution?

Here are a few concrete questions to expose my ignorance of simple algebra. Note that f is is a rational function in its variables and their exponentials. Let's call such functions "tame". The equation f satisfies involves rational expressions in the variables and their exponentials and linear substitutions applied to f. Let us call such equations "tame". (The equation is also linear in f. You may or may not wish to include this in the definition of "tame"). So we see that this particular tame equation has a tame solution.

  • Is that always the case?
    • A pretty example is the tame equation e(x+y)=e(x)e(y), whose solution is the "first" transcendental function e^x.
    • Are we done defining "new" special functions after we've defined e^x 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 existence of a tame solution? Can you write a very general condition on tame equations that will ensure the existence 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.