Quantized Chebyshev polynomials and cluster characters with coefficients

Abstract

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.

Atomic bases of cluster algebras of types A and Ã

“…In Section 2, we prove that cluster monomials form the atomic basis in a cluster algebra of type A. In Section 3, we give a combinatorial interpretation to a conjectural formula provided in [14] for the atomic basis in a cluster algebra of typeÃ. Finally, we prove in Section 4 that this conjectural atomic basis is indeed the atomic basis in typeÃ.…”

Generic variables in acyclic cluster algebras

“…For further details concerning these polynomials, especially in the context of cluster algebras, one can refer to [15].…”

Differential equation description and Chebyshev approximation of linear time‐invariant circuits

“…Equation 20 is an approximation of the Mth order CSs representation. C is a matrix that is made up of M × N submatrices, and every submatrix is the expanded matrix that is defined by Equation 16.…”

“…17 The redundant CSs 18 or Chebyshev expansions 19 of the desired degree are cut off to find the Nth order polynomial approximating with the largest feasible leading coefficients. 20 The truncation errors of the CSs can be evaluated by the coefficients of the CSs. 21 Chebyshev polynomials form a complete orthogonal system.…”