#### Geometry & Topology, Vol. 9 (2005)
Paper no. 29, pages 1253--1293.

## The colored Jones function is q-holonomic

### Stavros Garoufalidis and Thang TQ Le

**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.
**Keywords**.
Holonomic functions, Jones polynomial, Knots, WZ algorithm, quantum invariants, D-modules, multisums, hypergeometric functions

**AMS subject classification**.
Primary: 57N10.
Secondary: 57M25.

**E-print:** `arXiv:math.GT/0309214`

**DOI:** 10.2140/gt.2005.9.1253

Submitted to GT on 28 October 2004.
(Revised 20 July 2005.)
Paper accepted 3 July 2005.
Paper published 24 July 2005.

Notes on file formats
Stavros Garoufalidis Thang TQ Le

School of Mathematics, Georgia Institute of Technology

Atlanta, GA 30332-0160, USA

Email: stavros@math.gatech.edu, letu@math.gatech.edu

URL: http://www.math.gatech.edu/~stavros

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