Let $f(x)= |x|$ on $-1\leq x\leq 1$.
Then there is a Fourier-Legendre expansion
$f(x)$ = $\sum_{m=0}^{\infty} c_mP_m(x)$ where
$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} \left[(x^2-1)^n\right]$
$c_0 = ?$
$c_1 = ?$
and for all $m \geq 2$
$c_m = \frac {1}{m!} \ * ? * \frac {d^k}{dx^k} \textstyle\ (x^2 - 1)$ evaluated at $x = ?$
where
$k = ?$
$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\on}[1]{\operatorname{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ $\ds{\on{f}\pars{x}}$ is an even function of $\ds{x}$. Its expansion, in terms of Legendre Polynomials, is given by $\ds{\on{f}\pars{x} = \sum_{\ell = 0}^{\infty} a_{\ell}\on{P}_{2\ell\,}\pars{x}}$.
Then, \begin{align} &\int_{-1}^{1}\on{f}\pars{x} \on{P}_{2\ell\,}\pars{x}\,\dd x \\[5mm] = &\ \sum_{\ell\,' = 0}^{\infty} a_{\ell\,'}\int_{-1}^{1} \on{P}_{2\ell\,}\pars{x} \on{P}_{2\ell\,'\,}\pars{x}\,\dd x \\[5mm] & 2\int_{0}^{1}x\on{P}_{2\ell\,}\pars{x}\,\dd x = \sum_{\ell\,' = 0}^{\infty}a_{\ell\,'}\ {2\,\delta_{2\ell,\,2\ell\,'} \over 2\pars{2\ell} + 1} \\[5mm] & a_{\ell} = \pars{4\ell + 1} \int_{0}^{1}x\on{P}_{2\ell\,}\pars{x}\,\dd x \end{align} In addition, \begin{align} {1 \over \root{1 - 2xh + h^{2}}} & = \sum_{\ell = 0}^{\infty}\on{P}_{\ell}\pars{x}h^{\ell} \\ {1 \over \root{1 + 2xh + h^{2}}} & = \sum_{\ell = 0}^{\infty}\on{P}_{\ell}\pars{x} \,\pars{-1}^{\ell}\,h^{\ell} \end{align} \begin{align} &{1 \over \root{1 - 2xh + h^{2}}} + {1 \over \root{1 + 2xh + h^{2}}} \\[5mm] = &\ 2\sum_{\ell = 0}^{\infty} \on{P}_{2\ell}\pars{x}h^{2\ell} \end{align} and \begin{align} a_{\ell} & = \pars{2\ell + {1 \over 2}} \\[2mm] & \bracks{h^{2\ell}}\left(% \int_{0}^{1}{x \over \root{1 - 2xh + h^{2}}}\,\dd x \right. \\[2mm] & \phantom{\bracks{h^{2\ell}}\,\,\,\,\,} \left.+\int_{0}^{1} {x \over \root{1 + 2xh + h^{2}}}\,\dd x\right) \\[5mm] & = {2 \over 3}\pars{2\ell + {1 \over 2}}\bracks{h^{2\ell}} {\pars{1 + h^{2}}^{3/2} - 1 \over h^{2}} \\[5mm] & = {2 \over 3}\pars{2\ell + {1 \over 2}} \bracks{h^{2\ell}} \sum_{\ell\,' = 1}^{\infty} {3/2 \choose \ell\,'}h^{2\ell\,' - 2} \\[5mm] & = {2 \over 3}\pars{2\ell + {1 \over 2}} \bracks{h^{2\ell}} \sum_{\ell\,' = 0}^{\infty} {3/2 \choose \ell\,' + 1}h^{2\ell\,'} \\[5mm] & = {2 \over 3}\pars{2\ell + {1 \over 2}} {3/2 \choose \ell + 1} \end{align}
\begin{align} \verts{x} & = \bbx{{2 \over 3}\sum_{\ell = 0}^{\infty} \pars{2\ell + {1 \over 2}} {3/2 \choose \ell + 1}\on{P}_{2\ell\,}\pars{x}} \\[2mm] &\ x \in \pars{-1,1} \end{align}