Twisting q-holonomic sequences by complex roots of unity

Stavros Garoufalidis; Christoph Koutschan

doi:10.1145/2442829.2442857

Twisting q-holonomic sequences by complex roots of unity

Stavros Garoufalidis
, Christoph Koutschan

Research Institute for Symbolic Computation

Research output: Chapter in Book/Report/Conference proceeding › Conference proceedings › peer-review

Abstract

A sequence f_n(q) is q-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in q and q^n. Our main theorems state that q-holonomicity is preserved under twisting, i.e., replacing q by w*q where w is a complex root of unity, and under the substitution q -> q^alpha where alpha is a rational number. Our proofs are constructive, work in the multivariate setting of \partial-finite sequences and are implemented in the Mathematica package HolonomicFunctions. Our results are illustrated by twisting natural q-holonomic sequences which appear in quantum topology, namely the colored Jones polynomial of pretzel knots and twist knots. The recurrence of the twisted colored Jones polynomial can be used to compute the asymptotics of the Kashaev invariant of a knot at an arbitrary complex root of unity.

Original language	English
Title of host publication	ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Editors	Joris von der Hoeven, Mark von Hoej
Publisher	ACM
Pages	179-186
Number of pages	8
ISBN (Print)	9781450312691
DOIs	https://doi.org/10.1145/2442829.2442857
Publication status	Published - 2012

Fields of science

101001 Algebra
101002 Analysis
101 Mathematics
102 Computer Sciences
102011 Formal languages
101009 Geometry
101013 Mathematical logic
101020 Technical mathematics
101025 Number theory
101012 Combinatorics
101005 Computer algebra
101006 Differential geometry
101003 Applied geometry
102025 Distributed systems

JKU Focus areas

Computation in Informatics and Mathematics

Access to Document

10.1145/2442829.2442857

Cite this

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title = "Twisting q-holonomic sequences by complex roots of unity",

abstract = "A sequence f\_n(q) is q-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in q and q\textasciicircum{}n. Our main theorems state that q-holonomicity is preserved under twisting, i.e., replacing q by w*q where w is a complex root of unity, and under the substitution q -> q\textasciicircum{}alpha where alpha is a rational number. Our proofs are constructive, work in the multivariate setting of \textbackslash{}partial-finite sequences and are implemented in the Mathematica package HolonomicFunctions. Our results are illustrated by twisting natural q-holonomic sequences which appear in quantum topology, namely the colored Jones polynomial of pretzel knots and twist knots. The recurrence of the twisted colored Jones polynomial can be used to compute the asymptotics of the Kashaev invariant of a knot at an arbitrary complex root of unity.",

author = "Stavros Garoufalidis and Christoph Koutschan",

year = "2012",

doi = "10.1145/2442829.2442857",

language = "English",

isbn = "9781450312691",

pages = "179--186",

editor = "\{Joris von der Hoeven, Mark von Hoej\}",

booktitle = "ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation",

publisher = "ACM",

}

Twisting q-holonomic sequences by complex roots of unity. / Garoufalidis, Stavros; Koutschan, Christoph.
ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation. ed. / Joris von der Hoeven, Mark von Hoej. ACM, 2012. p. 179-186.

Research output: Chapter in Book/Report/Conference proceeding › Conference proceedings › peer-review

TY - GEN

T1 - Twisting q-holonomic sequences by complex roots of unity

AU - Garoufalidis, Stavros

AU - Koutschan, Christoph

PY - 2012

Y1 - 2012

N2 - A sequence f_n(q) is q-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in q and q^n. Our main theorems state that q-holonomicity is preserved under twisting, i.e., replacing q by w*q where w is a complex root of unity, and under the substitution q -> q^alpha where alpha is a rational number. Our proofs are constructive, work in the multivariate setting of \partial-finite sequences and are implemented in the Mathematica package HolonomicFunctions. Our results are illustrated by twisting natural q-holonomic sequences which appear in quantum topology, namely the colored Jones polynomial of pretzel knots and twist knots. The recurrence of the twisted colored Jones polynomial can be used to compute the asymptotics of the Kashaev invariant of a knot at an arbitrary complex root of unity.

AB - A sequence f_n(q) is q-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in q and q^n. Our main theorems state that q-holonomicity is preserved under twisting, i.e., replacing q by w*q where w is a complex root of unity, and under the substitution q -> q^alpha where alpha is a rational number. Our proofs are constructive, work in the multivariate setting of \partial-finite sequences and are implemented in the Mathematica package HolonomicFunctions. Our results are illustrated by twisting natural q-holonomic sequences which appear in quantum topology, namely the colored Jones polynomial of pretzel knots and twist knots. The recurrence of the twisted colored Jones polynomial can be used to compute the asymptotics of the Kashaev invariant of a knot at an arbitrary complex root of unity.

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DO - 10.1145/2442829.2442857

M3 - Conference proceedings

SN - 9781450312691

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EP - 186

BT - ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

A2 - Joris von der Hoeven, Mark von Hoej, null

PB - ACM

ER -

Twisting q-holonomic sequences by complex roots of unity

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