### 5.27.4 Tchebychev polynomials of the first kind: tchebyshev1

tchebyshev1 takes as argument an integer n and optionally a variable name (by default x).
tchebyshev1 returns the Tchebychev polynomial of first kind of degree n.
The Tchebychev polynomial of first kind T(n,x) is defined by

T(n,x)= cos(n arccos(x))

and satisfy the recurrence relation:

T(0,x)=1, T(1,x)=x, T(n,x)=2xT(n−1,x)−T(n−2,x)

The polynomials T(n,x) are orthogonal for the scalar product

<f,g>= | ∫ |
+1
−1
f(x)g(x)
√ |
1−x2
| dx

Input :

tchebyshev1(4)

Output :

8*x`^`4+-8*x`^`2+1

Input :

tchebyshev1(4,y)

Output :

8*y`^`4+-8*y`^`2+1

Indeed

cos( 4 x) | = | Re((cos(x)+i sin(x))4)
| = | cos(x)4−6.cos(x)2 (1−cos(x)2)+((1−cos(x)2)2
| = | T(4,cos(x))