Symmetrized Chebyshev polynomials

Abstract

We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves alternate) result that their coefficients are non-negative. As a corollary we find that T n ( c cos ⁡ θ ) T_n(c \cos \theta ) and U n ( c cos ⁡ θ ) U_n(c \cos \theta ) are positive definite functions. We further show that a Central Limit Theorem holds for the coefficients of our polynomials.