Chebyshev polynomial

The Chebyshev polynomials of first kind are defined by the simpleformula

$$T_n(x)=\cos (nt),$$

where $x=\cos t$.

It is an example of a trigonometric polynomial.

This can be seen to be a polynomial by expressing $\cos (kt)$ as a polynomial of $\cos (t)$, by using the formula for cosine of angle-sum:

$\cos (1t)$$=$$\cos (t)$

$\cos (2t)$$=$$\cos (t)\cos (t)-\sin (t)\sin (t)=2{(\cos (t))}^2-1$

$\cos (3t)$$=$$4{(\cos (t))}^3-3\cos (t)$

$\mathrm{⋮}$

So we have

$T_0(x)$$=$$1$

$T_1(x)$$=$$x$

$T_2(x)$$=$$2x^2-1$

$T_3(x)$$=$$4x^3-3x$

$\mathrm{⋮}$

These polynomials obey the recurrence relation:

$$T_{n+1}(x)=\mathrm{\hspace{0.33em}2}x{T}_n(x)-T_{n-1}(x)$$

for $n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots }$

Related are the Chebyshev polynomials of the second kind that aredefined as

$$U_{n-1}(\cos t)=\frac{\sin (nt)}{\sin (t)},$$

whichcan similarly be seen to be polynomials through either a similar process as theabove or by the relation $U_{n-1}(t)=n{T}_n^{\prime }(t)$.

The first few are:

$U_0(x)$$=$$1$

$U_1(x)$$=$$2x$

$U_2(x)$$=$$4x^2-1$

$U_3(x)$$=$$8x^3-4x$

$\mathrm{⋮}$

The same recurrence relation also holds for $U$:

$$U_{n+1}(x)=\mathrm{\hspace{0.33em}2}x{U}_n(x)-U_{n-1}(x)$$

for $n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots }$.