I fail finding a proof (even in MathWorld, in my Mathematic dictionary or on the Web) for the following property of Chebyshev polynomials:

(T_i o T_j)(x) = (T_j o T_i)(x) = T_ij(x) when x is in ] -inf ; + inf [

Example :

T_2(x) = 2x^2-1

T_3(x) = 4x^3-3x

T_3(T_2(x)) = T_2(T_3(x)) = T_6(x)

When x is in ]-1 ; +1[ , it seems easy to use x=cosA and T_n(cosA) = cos(nA) .

But how to do when x is any real ?

Thanks,

Tony