Tchebychev polynomial of second kind: tchebyshev2

### 2.26.5 Tchebychev polynomial of second kind: `tchebyshev2`

`tchebyshev2` takes as argument an integer n and optionnally a variable name (by default x).
`tchebyshev2` returns the Tchebychev polynomial of second kind of degree n.
The Tchebychev polynomial of second kind U(n,x) is defined by:

U(n,x)= |
sin((n+1).arccos(x))
sin(arccos(x))

or equivalently:

sin((n+1)x)=sin(x)*U(n,cos(x))

The U(n,x) verifies the recurrence relation:

U(0,x)=1, U(1,x)=2x, U(n,x)=2xU(n−1,x)−U(n−2,x)

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

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

Input :

`tchebyshev2(3)`

Output :

`8*x```^```3+-4*x`

Input :

`tchebyshev2(3,y)`

Output :

`8*y```^```3+-4*y`

Indeed:

sin(4.x)=sin(x)*(8*cos(x)3−4.cos(x))=sin(x)*U(3,cos(x))