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G&T Monographs |
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The colored Jones function is q-holonomic
Stavros Garoufalidis and Thang T Q Le
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Geometry & Topology 9 (2005)
1253–1293
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Abstract
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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.
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Keywords
holonomic functions, Jones polynomial,
Knots, WZ algorithm, quantum invariants, D–modules,
multisums, hypergeometric functions
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Mathematical Subject Classification
Primary: 57N10
Secondary: 57M25
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Publication
Received: 28 October 2004
Revised: 20 July 2005
Accepted: 3 July 2005
Published: 24 July 2005
Proposed: Walter Neumann
Seconded: Joan Birman, Vaughan Jones
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