Polynomial Fitting from Chebyshev Coefficients

I have been reading Numerical recipes about how to create a power series approximation to a function once you have a Chebyshev approximation to the function. However it is still very unclear to me how one can go from the coefficients $c_k$ in $\Sigma c_k T_k - c_0/2$ to the $d_k$ in $\Sigma d_k x_k$.

I understand just how one can get the $c_k$'s and I am very confident that I can get those correctly, but I'm getting confused about how to implement this so called Clenshaw recurrence.

Any help is greatly appreciated.

Thanks.

One way to get the $d_k$ is to expand each Chebyshev polynomial in powers, and then take linear combinations.
You can evaluate a Chebyshev sum at any pareticular $x$ in a Horner-like fashion by using the 3-term recurrence to eliminate recursively the highest term until you end up with a linear function. By differentiating the resulting recurrence you also get the values of the first derivative.