Editing Chebyshev polynomials (section)

===Orthogonality===
Both {{mvar|T<sub>n</sub>}} and {{mvar|U<sub>n</sub>}} form a sequence of [[orthogonal polynomials]]. The polynomials of the first kind {{mvar|T<sub>n</sub>}} are orthogonal with respect to the weight:
<math display="block">\frac{1}{\sqrt{1 - x^2}},</math>
on the interval {{closed-closed|−1, 1}}, i.e. we have:
<math display="block">\int_{-1}^1 T_n(x)\,T_m(x)\,\frac{\mathrm{d}x}{\sqrt{1-x^2}} = 
\begin{cases}
0             & ~\text{ if }~ n \ne m, \\[5mu]
\pi           & ~\text{ if }~ n=m=0, \\[5mu]
\frac{\pi}{2} & ~\text{ if }~ n=m \ne 0.
\end{cases}</math>

This can be proven by letting {{math|1=''x'' = cos ''θ''}} and using the defining identity {{math|1=''T''<sub>''n''</sub>(cos ''θ'') = cos(''nθ'')}}.

Similarly, the polynomials of the second kind {{mvar|U<sub>n</sub>}} are orthogonal with respect to the weight:
<math display="block">\sqrt{1-x^2}</math>
on the interval {{closed-closed|−1, 1}}, i.e. we have:
<math display="block">\int_{-1}^1 U_n(x)\,U_m(x)\,\sqrt{1-x^2} \,\mathrm{d}x =
\begin{cases}
0             & ~\text{ if }~ n \ne m, \\[5mu]
\frac{\pi}{2} & ~\text{ if }~ n = m.
\end{cases}</math>

(The measure {{math|{{sqrt|1 − ''x''<sup>2</sup>}} d''x''}} is, to within a normalizing constant, the [[Wigner semicircle distribution]].)

These orthogonality properties follow from the fact that the Chebyshev polynomials solve the [[Chebyshev equation|Chebyshev differential equations]]:
<math display="block">\begin{align}
(1 - x^2)T_n'' - xT_n' + n^2 T_n &= 0, \\[1ex]
(1 - x^2)U_n'' - 3xU_n' + n(n + 2) U_n &= 0,
\end{align}</math>which are [[Sturm–Liouville problem|Sturm–Liouville differential equations]]. It is a general feature of such [[differential equation]]s that there is a distinguished orthonormal set of solutions. (Another way to define the Chebyshev polynomials is as the solutions to [[Sturm–Liouville problem|those equations]].)

The {{mvar|T<sub>n</sub>}} also satisfy a discrete orthogonality condition:
<math display="block">\sum_{k=0}^{N-1}{T_i(x_k)\,T_j(x_k)} =
\begin{cases}
0           & ~\text{ if }~ i \ne j, \\[5mu]
N           & ~\text{ if }~ i = j = 0, \\[5mu]
\frac{N}{2} & ~\text{ if }~ i = j \ne 0,
\end{cases} </math>
where {{mvar|N}} is any integer greater than {{math|max(''i'', ''j'')}},{{sfn|Mason|Handscomb|2002}} and the {{math|''x''<sub>''k''</sub>}} are the {{mvar|N}} [[Chebyshev nodes]] (see above) of {{math|''T''<sub>''N''&thinsp;</sub>(''x'')}}:
<math display="block">x_k = \cos\left(\pi\,\frac{2k+1}{2N}\right) \quad ~\text{ for }~ k = 0, 1, \dots, N-1.</math>

For the polynomials of the second kind and any integer {{math|''N'' > ''i'' + ''j''}} with the same Chebyshev nodes {{math|''x''<sub>''k''</sub>}}, there are similar sums:
<math display="block">\sum_{k=0}^{N-1}{U_i(x_k)\,U_j(x_k)\left(1-x_k^2\right)} =
\begin{cases}
0           & \text{ if }~ i \ne j, \\[5mu]
\frac{N}{2} & \text{ if }~ i = j,
\end{cases}</math>
and without the weight function:
<math display="block">\sum_{k=0}^{N-1}{ U_i(x_k) \, U_j(x_k) } =
\begin{cases}
0                         & ~\text{ if }~ i \not\equiv j \pmod{2}, \\[5mu]
N \cdot (1 + \min\{i,j\}) & ~\text{ if }~ i \equiv j\pmod{2}.
\end{cases} </math>

For any integer {{math|''N'' > ''i'' + ''j''}}, based on the {{mvar|N}} zeros of {{math|''U''<sub>''N''&thinsp;</sub>(''x'')}}:
<math display="block">y_k = \cos\left(\pi\,\frac{k+1}{N+1}\right) \quad ~\text{ for }~ k=0, 1, \dots, N-1,</math>
one can get the sum:
<math display="block">\sum_{k=0}^{N-1}{U_i(y_k)\,U_j(y_k)(1-y_k^2)} =
\begin{cases}
0 & ~\text{ if } i \ne j, \\[5mu]
\frac{N+1}{2} & ~\text{ if } i = j,
\end{cases}</math>
and again without the weight function:
<math display="block">\sum_{k=0}^{N-1}{U_i(y_k)\,U_j(y_k)} =
\begin{cases}
0  & ~\text{ if }~ i \not\equiv j \pmod{2}, \\[5mu]
\bigl(\min\{i,j\} + 1\bigr)\bigl(N-\max\{i,j\}\bigr)  & ~\text{ if }~ i \equiv j\pmod{2}.
\end{cases}</math>