The colored Jones function is q-holonomic

Garoufalidis, Stavros¹ ; Le, Thang T Q¹

¹ School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160, USA

Geometry & topology, Tome 9 (2005) no. 3, pp. 1253-1293.

Résumé

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.

Détail

DOI : 10.2140/gt.2005.9.1253

Keywords: holonomic functions, Jones polynomial, Knots, WZ algorithm, quantum invariants, $D$–modules, multisums, hypergeometric functions

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