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We give a new reconstruction method of big quantum  K -ring based on the  q -difference module structure in quantum  K -theory [ 12 ,  13 ]. The  q -difference structure yields commuting linear operators  A    i ,com  on the  K -group as many as the Picard number of the target manifold. The genus-zero quantum  K -theory can be reconstructed from the  q -difference structure at the origin  t =0 if the  K -group is generated by a single element under the actions of  A    i ,com . This method allows us to prove the convergence of the big quantum  K -rings of certain manifolds, including the projective spaces and the complete flag manifold  Fl  3 .

Reconstruction and Convergence in Quantum K -Theory via Difference Equations