Tchebychev polynomials of first kind: tchebyshev1

tchebyshev1 takes as argument an integer n and optionnally a variable name (by default x).
tchebyshev1 returns the Tchebychev polynomial of first kind of degree n.
The Tchebychev polynomial of first kind T(n, x) is defined by

T(n, x) = cos(n.arccos(x))

and verify 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 > = $$\int_{{-1}}^{{+1}}$$$${\frac{{f(x)g(x)}}{{\sqrt{1-x^2}}}}$$dx

Input :

tchebyshev1(4)

Output :

8*x^4+-8*x^2+1

Input :

tchebyshev1(4,y)

Output :

8*y^4+-8*y^2+1

Indeed

cos(4.x) = Re((cos(x) + i.sin(x))⁴)

= cos(x)⁴ -6.cos(x)².(1 - cos(x)²) + ((1 - cos(x)²)²

= T(4, cos(x))