Wall-crossing between stable and co-stable ADHM data

We prove formula between Nekrasov partition functions defined from stable and co-stable ADHM data for the plane following method by Nakajima and Yoshioka (Kyoto J Math 51(2):263–335, 2011) based on the theory of wall-crossing formula developed by Mochizuki (Donaldson type invariants for algebraic surfaces: transition of moduli stacks, Lecture notes in mathematics, vol 1972, Springer, Berlin, 2009). This formula is similar to conjectures by Ito et al. J High Energy Phys 2013(5):045, 2013, (4.1), (4.2)] for \(A_{1}\) singularity.