Editing Chebyshev polynomials (section)

===Composition and divisibility properties===
The trigonometric definitions of {{math|''T''<sub>''n''</sub>}} and {{math|''U''<sub>''n''</sub>}} imply the composition or nesting properties:<ref>{{citation|last1=Rayes|first1=M. O.|last2=Trevisan|first2=V.|last3=Wang|first3=P. S.|title=Factorization properties of chebyshev polynomials|journal=Computers & Mathematics with Applications|volume=50|issue=8–9|year=2005|pages=1231–1240|doi=10.1016/j.camwa.2005.07.003|doi-access=free}}</ref>
<math display="block">\begin{align}
T_{mn}(x) &= T_m(T_n(x)),\\
U_{mn-1}(x) &= U_{m-1}(T_n(x))U_{n-1}(x).
\end{align}
</math>
For {{math|''T''<sub>''mn''</sub>}} the order of composition may be reversed, making the family of polynomial functions {{math|''T''<sub>''n''</sub>}} a [[commutative]] [[semigroup]] under composition.

Since {{math|''T''<sub>''m''</sub>(''x'')}} is divisible by {{mvar|x}} if {{mvar|m}} is odd, it follows that {{math|''T''<sub>''mn''</sub>(''x'')}} is divisible by {{math|''T''<sub>''n''</sub>(''x'')}} if {{mvar|m}} is odd. Furthermore, {{math|''U''<sub>''mn''−1</sub>(''x'')}} is divisible by {{math|''U''<sub>''n''−1</sub>(''x'')}}, and in the case that {{mvar|m}} is even, divisible by {{math|''T''<sub>''n''</sub>(''x'')''U''<sub>''n''−1</sub>(''x'')}}.