In this paper we introduce refined canonical stable Grothendieck polynomials and their duals with two infinite sequences of parameters. These polynomials unify several generalizations of Grothendieck polynomials including canonical stable Grothendieck polynomials due to Yeliussizov, refined Grothendieck polynomials due to Chan and Pflueger, and refined dual Grothendieck polynomials due to Galashin, Liu, and Grinberg. We give Jacobi--Trudi-like formulas, combinatorial models, Schur expansions, Schur positivity, and dualities of these polynomials.
Comment: 34 pages. This is the first part of a revised version of the previous manuscript