A generalization of Zwegers' multivariable $\mu$-function

We introduce a one parameter deformation of Zwegers' multivariable $\mu$-function by applying iterations of the $q$-Borel summation method, which is also a multivariate analogue of the generalized $\mu$-function introduced by the authors. For this deformed multivariable $\mu$-function, we give some formulas, for example, forward shift formula, translation and $\mathfrak{S}_{N+1}$-symmetry. Further we mention modular formulas for the Zwegers' original multivariable $\mu$-function.