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We study the asymptotic expansion of the colored Jones polynomial (theMelvin-Morton expansion) using a recursion formula for the deframed universalweight system for the $sl(2)$ Lie algebra. Combined with the formula for theuniversal weight system for the Lie superalgebra $gl(1|1)$ (which correspondsto the Alexander-Conway knot polynomial) this formula gives a very short proofof the Melvin-Morton conjecture relating the colored Jones invariant and theAlexander-Conway polynomial of knots.

Universal weight systems and the Melvin-Morton expansion of the colored Jones knot invariant