Tchebychev polynomials of the first kind: tchebyshev1

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

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

The tchebyshev1 command finds the Tchebychev polynomials of the first kind.

- tchebyshev1 takes one mandatory argument and one optional argument:
  - n, an integer.
  - Optionally x, a variable name (by default x).
- tchebyshev1(n ⟨,x⟩) returns the Tchebychev polynomial of first kind of degree n.

#### Examples

- Input:
  tchebyshev1(4)
  Output:

  8 x4−8 x2+1

- Input:
  tchebyshev1(4,y)
  Output:

  8 y4−8 y2+1

  Indeed

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