  The Coulomb branch indices of Argyres-Douglas theories on $L(k,1)\times
S^{1}$ are recently identified with matrix elements of modular transforms of
certain $2d$ vertex operator algebras in a particular limit. A one parameter
generalization of the modular transformation matrices of $(2N+3,2)$ minimal
models are proposed to compute the full Coulomb branch index of
$(A_{1},A_{2N})$ Argyres-Douglas theories on the same space. Morever, M-theory
construction of these theories suggests direct connection to the refined
Chern-Simons theory. The connection is made precise by showing how the modular
transformation matrices of refined Chern-Simons theory are related to the
proposed generalized ones for minimal models and the identification of Coulomb
branch indices with the partition function of the refined Chern-Simons theory.
