Editing Legendre polynomials (section)

=== In trigonometry ===

The trigonometric functions {{math|cos ''nθ''}}, also denoted as the [[Chebyshev polynomials]] {{math|''T<sub>n</sub>''(cos ''θ'') ≡ cos ''nθ''}}, can also be multipole expanded by the Legendre polynomials {{math|''P<sub>n</sub>''(cos ''θ'')}}. The first several orders are as follows:
<math display="block">\begin{alignat}{2}
T_0(\cos\theta)&=1           &&=P_0(\cos\theta),\\[4pt]
T_1(\cos\theta)&=\cos  \theta&&=P_1(\cos\theta),\\[4pt]
T_2(\cos\theta)&=\cos 2\theta&&=\tfrac{1}{3}\bigl(4P_2(\cos\theta)-P_0(\cos\theta)\bigr),\\[4pt]
T_3(\cos\theta)&=\cos 3\theta&&=\tfrac{1}{5}\bigl(8P_3(\cos\theta)-3P_1(\cos\theta)\bigr),\\[4pt]
T_4(\cos\theta)&=\cos 4\theta&&=\tfrac{1}{105}\bigl(192P_4(\cos\theta)-80P_2(\cos\theta)-7P_0(\cos\theta)\bigr),\\[4pt]
T_5(\cos\theta)&=\cos 5\theta&&=\tfrac{1}{63}\bigl(128P_5(\cos\theta)-56P_3(\cos\theta)-9P_1(\cos\theta)\bigr),\\[4pt]
T_6(\cos\theta)&=\cos 6\theta&&=\tfrac{1}{1155}\bigl(2560P_6(\cos\theta)-1152P_4(\cos\theta)-220P_2(\cos\theta)-33P_0(\cos\theta)\bigr).
\end{alignat}</math>

This can be summarized for <math>n>0</math> as

<math>
T_n(x)=2^{2n-n'}\hat n!\sum_{t=0}^{\hat n} (n-2t+1/2)
\frac{(n-t-1)!}{2^{2t}t!(n-1)!}
\times
\frac{(-1)\cdot 1\cdot 3\cdots (2t-3)}{(1+2n')(3+2n')\cdots (2n-2t+1)}P_{n-2t}(x)
.
</math>

where <math>\hat n\equiv \lfloor n/2\rfloor</math>,
<math>n'\equiv \lfloor (n+1)/2\rfloor</math>, and where the products
with the steps of two in the numerator and denominator are 
to be interpreted as 1 if the are empty, i.e., if the last factor is smaller than the
first factor.


Another property is the expression for {{math|sin (''n'' + 1)''θ''}}, which is
<math display="block">\frac{\sin (n+1)\theta}{\sin\theta}=\sum_{\ell=0}^n P_\ell(\cos\theta) P_{n-\ell}(\cos\theta).</math>