  We propose novel functional equations for the BPS partition functions of 6d
(1,0) SCFTs, which can be regarded as an elliptic version of
Gottsche-Nakajima-Yoshioka's K-theoretic blowup equations. From the viewpoint
of geometric engineering, these are the generalized blowup equations for
refined topological strings on certain local elliptic Calabi-Yau threefolds. We
derive recursion formulas for elliptic genera of self-dual strings on the
tensor branch from these functional equations and in this way obtain a
universal approach for determining refined BPS invariants. As examples, we
study in detail the minimal 6d SCFTs with SU(3) and SO(8) gauge symmetry. In
companion papers, we will study the elliptic blowup equations for all other
non-Higgsable clusters.
