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2.26.4  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>=
+1


−1
f(x)g(x)
1−x2
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))

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