Stavros Garoufalidis and Jeff Geronimo
Abstract: In this paper we develop an asymptotic analysis for formal and actual solutions of $q$-difference equations, under a regularity assumption. In particular, evaluations of solutions to regular $q$-difference equations have an exponential growth rate which can be computed from the $q$-difference equation. The motivation for the paper comes from the Hyperbolic Volume Conjecture, which states that a specific evaluation of the colored Jones function has an exponential growth rate, which is proportional to the volume of the knot complement. The connection of the Hyperbolic Volume Conjecture with the paper comes from the fact that the colored Jones function of a knot is a solution of a q-difference equation, as was proven by TTQ. Le and the author.
Key words: q-difference equations, asymptotics, knots, colored Jones function, Hyperbolic Volume Conjecture, $\e$-difference equations, Birkhoff, Trjitzinsky, WKB, approximation schemes, recursion relations.
Notes: 25 pages, 3 figures.