  We study $K$-theoretic integrals over famed quiver moduli via wall-crossing
phenomena. We study the chainsaw quiver varieties, and consider generating
functions defined by two types of $K$-theoretic classes. In particular, we
focus on integrals over the handsaw quiver varieties of type $A_{1}$, and get
functional equations for each of them. We also give explicit formula for these
partition functions. In particular, we obtain geometric interpretation of
transformation formulas for multiple basic hypergeometric series including the
Kajihara transformation formula, and the one studied by
Langer-Schlosser-Warnaar and Halln\"as-Langman-Noumi-Rosengren.
