In the DE packages, NASA gives us a series of coefficients for the Chebyshev approximation. As I understand it, those approximations are given by a series of polynomials $T_n$ for the interval $[-1,1]$. Then we consider $f(t)$ to be $\sum_i a_i T_i(\tau)$ (where $\tau$ is the normalised time). What is $a_i$ and what are its values?
Then we consider an approximation using a Taylor series defined as $f(t)=\sum_i b_i\tau^i$. What is $b_i$? Is it a Bessel function?
The last two relations represent the Clenshaw algorithm. Here I don't exactly understand how it works. Also does the number of 13 coefficients have something to do with Runge's Phenomenon?
Below are Eqs 3.52 through 3.59, from section 3.3.3 (Chebychev Approximation) in the book Satellite Orbits: Models, Methods and Applications* by De Oliver Montenbruck and Eberhard Gill Springer 2000,
ISBN 978-3-642-58351-3 which is viewable in books.google.com