Quantized Chebyshev polynomials and cluster characters with coefficients

We introduce quantized Chebyshev polynomials as deformations of generalized Chebyshev polynomials previously introduced by the author in the context of acyclic coefficient-free cluster algebras. We prove that these quantized polynomials arise in cluster algebras with principal coefficients associated to acyclic quivers of infinite representation types and equioriented Dynkin quivers of type $\mathbb{A}$. We also study their interactions with bases and especially canonically positive bases in affine cluster algebras.