Chebyshev polynomials of the first kind

The Chebyshev polynomial of first kind T(n,x) is defined by

T(n,x)=cos(n arccosx)

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

f(x)g(x)

√

1−x²

dx.

The tchebyshev1 command finds the Chebyshev 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 Chebyshev polynomial of first kind of degree n.

tchebyshev1(4)

8 x⁴−8 x²+1

tchebyshev1(4,y)

8 y⁴−8 y²+1

Indeed, cos(4x)=Re((cosx+i sinx)⁴)=cos⁴ x−6cos² x (1−cos² x)+((1−cos² x))²=T(4,cos(x)).