Stavros Garoufalidis and TTQ Le
Abstract: A function of continuous variables is called holonomic if it satisfies a maximally overdetermined system of linear differential equations 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 $q$-proper-hypergeometric function, and thus it is $q$-holonomic. We demonstrate our results by computer calculations.
Key words: Holonomic functions, Jones polynomial, Knots, WZ algorithm, quantum invariants, q-holonomic functions, D-modules, colored Jones function, multisums.
Notes: 21 pages, 8 figures.