Okazaki and Smith discovered many elegant formulas expressing some matrix integrals as some celebrated $q$-series such as the Rogers--Ramanujan functions or Jacobi theta functions. These integrals arise as Wilson line defect half-indices of 3d $\mathcal{N}=2$ supersymmetric ${\rm SU}(N)$ Chern-Simons theories. We evaluate them by carefully calculating the constant terms of some infinite products. Along the way we use some crucial facts about antisymmetric multivariate formal Laurent series. Consequently, we prove three general conjectures of Okazaki and Smith which provide explicit formulas for half indices of the ${\rm SU}(N)_{-N-k}$ ($k=0,1/2,1$) Chern-Simons theories. During the process, we extend these ${\rm SU}(N)$ formulas to include one additional parameter. Furthermore, we generalize the ${\rm SU}(N)_{-N-1/2}$ and ${\rm SU}(N)_{-N-1}$ conjectures by calculating the corresponding half-indices of Wilson lines of arbitrary charge. As a special instance of our generalizations, we also confirm the ${\rm SU}(3)_{-4}$ conjecture of Okazaki and Smith.