Editing Chebyshev polynomials (section)

===Products of Chebyshev polynomials===
The Chebyshev polynomials of the first kind satisfy the relation:
<math display="block">T_m(x)\,T_n(x) = \tfrac{1}{2}\!\left(T_{m+n}(x) + T_{|m-n|}(x)\right)\!,\qquad \forall m,n \ge 0,</math>
which is easily proved from the [[List of trigonometric identities#Product-to-sum and sum-to-product identities|product-to-sum formula]] for the cosine:
<math display="block">2 \cos \alpha \, \cos \beta = \cos (\alpha + \beta) + \cos (\alpha - \beta).</math>
For {{math|1=''n'' = 1}} this results in the already known recurrence formula, just arranged differently, and with {{math|1=''n'' = 2}} it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest {{mvar|m}}) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:
<math display="block">\begin{align}
  T_{2n}(x) &= 2\,T_n^2(x)           - T_0(x) &&= 2 T_n^2(x) - 1, \\
T_{2n+1}(x) &= 2\,T_{n+1}(x)\,T_n(x) - T_1(x) &&= 2\,T_{n+1}(x)\,T_n(x) - x, \\
T_{2n-1}(x) &= 2\,T_{n-1}(x)\,T_n(x) - T_1(x) &&= 2\,T_{n-1}(x)\,T_n(x) - x .
\end{align}</math>

The polynomials of the second kind satisfy the similar relation:
<math display="block"> T_m(x)\,U_n(x) = \begin{cases}
\frac{1}{2}\left(U_{m+n}(x) + U_{n-m}(x)\right),  & ~\text{ if }~ n \ge m-1,\\
\\
\frac{1}{2}\left(U_{m+n}(x) - U_{m-n-2}(x)\right), & ~\text{ if }~ n \le m-2.
\end{cases} </math>
(with the definition {{math|''U''<sub>−1</sub> ≡ 0}} by convention ). They also satisfy:
<math display="block"> U_m(x)\,U_n(x) = \sum_{k=0}^n\,U_{m-n+2k}(x) = \sum_\underset{\text{ step 2 }}{p=m-n}^{m+n} U_p(x)~.</math>
for {{math|''m'' ≥ ''n''}}.
For {{math|1=''n'' = 2}} this recurrence reduces to:
<math display="block"> U_{m+2}(x) = U_2(x)\,U_m(x) - U_m(x) - U_{m-2}(x) = U_m(x)\,\big(U_2(x) - 1\big) - U_{m-2}(x)~,</math>
which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether {{mvar|m}} starts with 2 or 3.