Blowup relations on $\mathbb{C}^2/\mathbb{Z}_2$ from Nakajima–Yoshioka blowup relations

Using an elementary algebraic approach, from the Nakajima–Yoshioka blowup relations, we obtain bilinear relations for the Nekrasov partition functions arising in the study of the tau functions of quantum $q$-Painlevé equations. In addition, using this approach, we prove certain symmetry relations for Nekrasov partition functions modified by a Chern–Simons term.