Functional equations of Nekrasov functions proposed by Ito, Maruyoshi, and Okuda

We prove functional equations of Nekrasov partition functions for $A_{1}$-singularity, suggested by Ito, Maruyoshi, and Okuda. Our proof is given by the computation similar to that from a recent paper by the author. This is the method by Nakajima and Yoshioka based on the theory of wall-crossing formula developed by Mochizuki.