### 11.4.5 Chebyshev polynomial of the second kind

The Chebyshev polynomial of second kind U(n,x) is defined by:

U(n,x)= |
sin((n+1) arccosx)
sin(arccosx)

or equivalently:

sin((n+1)x)=U(n,cosx)sinx.

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
| d x.

The tchebyshev2 command finds the Chebyshev 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 Chebyshev polynomial of second kind of degree n.

#### Examples

tchebyshev2(3)

8 x3−4 x |  |  |  |  |  |  |  |  |  |

tchebyshev2(3,y)

8 y3−4 y |  |  |  |  |  |  |  |  |  |

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