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).

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 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 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 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

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