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.

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper29.abs.html

References in corpus (3)

• Skein-theoretical derivation of some formulas of Habiro
• Difference equation of the colored Jones polynomial for torus knot
• The C-polynomial of a knot

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• Double affine Hecke algebras and generalized Jones polynomials
• The AJ-conjecture and cabled knots over the figure eight knot
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• Elliptic Quantum Curves of Class ${\mathcal{S}}_k$
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• The slope conjecture for graph knots
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• q-holonomic formulas for colored HOMFLY polynomials of 2-bridge links
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• A Geometric Quantisation view on the AJ-conjecture for the Teichmüller TQFT
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• Algebra of quantum $\mathcal{C}$-polynomials
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