Quantum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -theory of projective spaces and confluence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -difference equat

Givental’s <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -theoretical <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> </mml:math> -function can be used to reconstruct genus zero <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -theoretical Gromov–Witten invariants. We view this function as a fundamental solution of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -difference system. In the case of projective spaces, we show that we can use the confluence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -difference systems to obtain the cohomological <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> </mml:math> -function from its <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -theoretic setting.