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2.26.5  Tchebychev polynomial of second kind: tchebyshev2

tchebyshev2 takes as argument an integer n and optionnally 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))

The U(n,x) verifies 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
dx 

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)) 

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