Quantized Chebyshev polynomials and cluster characters with coefficients

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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 A. We also study their interactions with bases and especially canonically positive bases in affine cluster algebras.