Tchebychev polynomial of the second kind: tchebyshev2

### 5.30.5 Tchebychev polynomial of the second kind: tchebyshev2

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))

These satisfy 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

The tchebyshev2 command finds the Tchebychev polynomials of the first kind.

- tchebyshev2 takes one mandatory argument and one optional argument:
  - n, an integer.
  - Optionally x, a variable name (by default x).
- tchebyshev2(n ⟨,x⟩) returns the Tchebychev polynomial of second kind of degree n.

#### Examples.

- Input:
  tchebyshev2(3)
  Output:

  8 x3−4 x

- Input:
  tchebyshev2(3,y)
  Output:

  8 y3 − 4y

  Indeed:

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