  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 one-dimensional integrals which
includes state-integral invariants of 4_1 and 5_2 knots.
