D'Alembert's functional equation and Chebyshev polynomials

Abstract

We consider D’Alembert’s functional equation (1) where the domain of the function $f$ is the additive group of the integers and the codomain is an arbitrary commutative ring with identity. We show that if $f(0) = 1$ then $f(n)$ is the value of the Chebyshev polynomial $T_{|n|}$ evaluated at $f(1)$.