The colored Jones function is q-holonomic

Abstract

A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions. Using a general state sum definition of the colored Jones function of a link in 3–space, we prove from first principles that the colored Jones function is a multisum of a $q$–proper-hypergeometric function, and thus it is $q$–holonomic. We demonstrate our results by computer calculations.