Suppose we are approximating a function $f$ with a Legendre series of order $N$, namely $$ f(x) \approx \sum_{n=0}^N c_n P_n(x) \equiv f_N(x) $$ where $P_n(x)$ is the $n^{th}$ Legendre polynomial and $$ c_n = \int_{-1}^1 dx f(x) P_n(x) . $$ Now, the above approximation can be rewritten as $$ f_N(x) = \sum_{n=0}^N c'_n x^n . $$ I would like to know if there is a closed equation to relate the coefficients $\{c'_n\}_{n=0}^N$ in terms of the $\{c_n\}_{n=0}^N$.
In this question I found the (undocumented) relation $$ P_n(x) = \frac{1}{2^n} \sum_{k=0}^{\left[ \frac{n}{2} \right]} (-1)^k \frac{(2n-2k)!}{k!(n-k)!(n-2k)!} x^{n-2k} $$ which would then make $$ c'_n = \frac{1}{2^n} \sum_{k=0}^{\left[ \frac{n}{2} \right]} (-1)^k \frac{(2n-2k)!}{k!(n-k)!(n-2k)!} c_{n-2k} $$ at least for even $n$.
I was also wondering whether the same request could be answered for Chebyshev (or other orthogonal) polynomials replacing Legendre polynomials in the above question.