Tchebychev polynomial of the second kind: tchebyshev2

5.27.5 Tchebychev polynomial of the second kind: tchebyshev2

tchebyshev2 takes as argument an integer n and optionally a variable name (by default x).
tchebyshev2 returns the Tchebychev polynomial of second kind of degree n.
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))

Then U(n,x) satisfies 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

f(x)g(x)

√

1−x²

Input :

tchebyshev2(3)

Output :

8*x^3+-4*x

Input :

tchebyshev2(3,y)

Output :

8*y^3+-4*y

Indeed:

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