06-1350/Some Equations by Kurlin

06-1350/Navigation Panel   # Week of... Links 1 Sep 11 About, Tue, Thu 2 Sep 18 Tue, Kurlin(P), Thu 3 Sep 25 Tue, Photo, Thu 4 Oct 2 HW1, Tue, Thu 5 Oct 9 Tue(P), Thu(P) 6 Oct 16 HW2, Tue(P), Thu(P) 7 Oct 23 Tue(P), Thu 8 Oct 30 HW3, Tue, Thu 9 Nov 6 Tue ($\nu$), Thu 10 Nov 13 Tue, Thu 11 Nov 20 HW4(P), Thu 12 Nov 27 Thu 13 Dec 4 Syzygies in Asymptote, Final Jan 8 Grades Note. (P) means "contains a problem that Dror cares about". Add your name / see who's in! On to 07-1352

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.