06-1350/Some Equations by Kurlin

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

$12\lambda \mu f(\lambda ,\mu )={\frac {e^{\lambda \mu }-e^{-\lambda -\mu }}{2(\lambda \mu )}}\left({\frac {2\lambda }{e^{\lambda }-e^{-\lambda }}}{\frac {2\mu }{e^{\mu }-e^{-\mu }}}-1\right)$

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 $e(xy)=e(x)e(y)$, 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.