Bailey pairs, Eichler integrals and unified Witten-Reshetikhin-Turaev invariants

In 1999, Lawrence and Zagier expressed the Witten-Reshetikhin-Turaev (WRT) invariant of the Poincar\'e homology sphere as the limiting value of the Eichler integral of a weight 3/2 modular form. Habiro's construction of the unified WRT invariant subsequently recast this result as an identity for a $q$-hypergeometric series at roots of unity. This motivated Hikami to prove analogous $q$-series identities involving the unified WRT invariants of certain Brieskorn homology spheres. Hikami also made several conjectures of a similar type for $q$-series with no apparent connection to quantum invariants. In this paper we use the Bailey pair machinery and a novel relation between incomplete quadratic Gauss sums with periodic coefficients to construct infinite families of identities between $q$-multisums at roots of unity and limiting values of Eichler integrals of weight 3/2 modular forms. These identities include all of Hikami's results and conjectures as well as a generalization of the result of Lawrence and Zagier.