The coefficients of the Chebyshev polynomials (what ApproxFun gives you, and the coefficients of S in your equation 9) are not the same as the coefficients of the monic polynomials (your C2k).

Furthermore, it looks like that paper may be using the exact minimax polynomial fit, whereas ApproxFun uses a (very good) approximation thereof. In any case, I would suggest comparing ApproxFun’s coefficients to those of S_2k in equation (9).

In general, there is no need to compute the coefficients of a polynomial in a monic basis. You can evaluate a polynomial directly from its coefficients in a Chebyshev basis using a Clenshaw recurrence.

It is possible to compute the monomial expansion from the Chebyshev one if you use high precision (BigFloat). Unfortunately I don’t think this is currently implemented in an easy-to-use way, though it wouldn’t be too hard.

I am not sure, but is there a chance you’re mixing the polynomials and the coefficients?
Namely, and I vaguely remember, Chebyshev Polynomials are the basis and the coefficients define the linear combination.

So in @blackeneth’s problem {S}_{2K} \left( x \right) are the Chebyshev Polynomials but the code he calculated are the coefficients of {S}_{2k} \left( x \right) but not itself (As implied from your answer: " and the coefficients of S in your equation 9").

So ApproxFun gives the coefficients for Chebyshev Polynomials while his problem his the coefficients of the polynomial of the form {x}^{2k} . In his problem they derive it by expanding the coefficients of {x}^{2k} from the equation 9.

So, in order to get {C}_{2k} @blackeneth needs to put the definition of Chebyshev Polynomials (Not the coefficients) in (9) and create the form in (10) with some basic algebra (Hopefully).