Tchebychev polynomials of first kind: tchebyshev1

### 2.26.4 Tchebychev polynomials of first kind: `tchebyshev1`

`tchebyshev1` takes as argument an integer n and optionnally 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 verify 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))