  We study the Wilson line defect half-indices of 3d $\mathcal{N}=2$
supersymmetric $SU(N)$ Chern-Simons theories of level $k\le -N$ with Neumann
boundary conditions for the gauge fields, together with 2d Fermi multiplets and
fundamental 3d chiral multiplets to cancel the gauge anomaly. We derive some
exact results and also make some conjectures based on expansions of the
$q$-series. We find several interesting connections with special functions
known in the literature, including Rogers-Ramanujan functions for which we
conjecture integral representations, and the appearance of Appell-Lerch sums
for certain Wilson line half-index grand canonical ensembles which reveal an
unexpected appearance of mock modular functions. We also find intriguing
$q$-difference equations relating half-indices to Wilson line half-indices.
Some of these results also have a description in terms of a dual theory with
Dirichlet boundary conditions for the vector multiplet in the dual theory.
