In this paper we investigate the finite N exact values of the S3 partition function of the \( \mathcal{N} \) = 4 super Yang-Mills theory with one adjoint hypermultiplet and Nf fundamental hypermultiplets, which describes N M2-branes on \( {\mathbb{C}}^2\times {\mathbb{C}}^2/{\mathbb{Z}}_{N_{\textrm{f}}} \) , with mass and FI deformations. We claim that the grand canonical sum of the partition function obeys a bilinear difference relation with respect to the shifts of the mass parameters of the fundamental hypermultiplets, which results in a new recursion relation for the partition function with respect to N. As an application, we also determine the analytic expression for the leading 1/N non-perturbative correction to the free energy of these models, which would correspond holographically to the contribution from an M2-brane wrapped on a 3d volume in the internal space of AdS4×S7/ \( {\mathbb{Z}}_{N_{\textrm{f}}} \) .
