Quantum $K$-theory of projective spaces and confluence of $q$-difference equations

Givental's $K$-theoretical $J$-function can be used to reconstruct genus zero $K$-theoretical Gromov--Witten invariants. We view this function as a fundamental solution of a $q$-difference system. In the case of projective spaces, we show that we can use the confluence of $q$-difference systems to obtain the cohomological $J$-function from its $K$-theoretic analogue. This provides another point of view to one of the statements of Givental--Tonita's quantum Hirzebruch--Riemann--Roch theorem. Furthermore, we compute connection numbers in the equivariant setting.