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# Mathematical Research Letters

## Volume 24 (2017)

### Number 3

### From state integrals to $q$-series

Pages: 781 – 801

DOI: https://dx.doi.org/10.4310/MRL.2017.v24.n3.a8

#### Authors

#### Abstract

It is well-known to the experts that multi-dimensional state integrals of products of Faddeev’s quantum dilogarithm which arise in Quantum Topology can be written as finite sums of products of basic hypergeometric series in $q = e^{2\pi i \tau}$ and $\tilde{q} = e^{- 2 \pi i / \tau}$. We illustrate this fact by giving a detailed proof for a family of onedimensional integrals which includes state-integral invariants of $4_1$ and $5_2$ knots.

#### Keywords

state-integrals, $q$-series, quantum dilogarithm, Euler triangular numbers, Nahm equation, gluing equations, $4_1$, $5_2$

#### 2010 Mathematics Subject Classification

Primary 57N10. Secondary 33F10, 39A13, 57M25.

S.G. was supported in part by grant DMS-0805078 of the US National Science Foundation.

R.K. was supported in part by the Swiss National Science Foundation.

Received 29 April 2013

Published 1 September 2017