D'Alembert's functional equation and Chebyshev polynomials
Author:
Davison, Thomas M.K.
xmlui.dri2xhtml.METS-1.0.item-citation: Annales Academiae Paedagogicae Cracoviensis. 4, Studia Mathematica 1 (2001), s. [31]-38
xmlui.dri2xhtml.METS-1.0.item-iso: en
Date: 2001
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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 $f(1)$.