 
 
 
11.4.5  Chebyshev polynomial of the second kind
The Chebyshev polynomial of second kind U(n,x) is defined by:
| U(n,x)= | | sin((n+1) arccosx) |  |  |  | sin(arccosx) | 
 | 
or equivalently:
| sin((n+1)x)=U(n,cosx)sinx. | 
These satisfy 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
The tchebyshev2
command finds the Chebyshev polynomials of
the first kind.
- 
tchebyshev2 takes one mandatory argument and one
optional argument:
- 
n, an integer.
- Optionally x, a variable name (by default x).
 
- tchebyshev2(n  ⟨,x⟩) returns
the Chebyshev polynomial of second kind of degree n.
Examples
Indeed, sin(4 x)=sin(x) (8 cos(x)3−4 cos(x))=sin(x) U(3,cos(x)).
 
 
