I have a function represented in a basis of 2D Legendre polynomials, $$ f(x,y) = \sum_{n=0}^N c_n P_n(x,y) $$ where $P_n(x,y)$ is a 2D Legendre polynomial given by $$ P_n(x,y) = P_l(x)P_m(y) $$ where $0\leq l \leq L$ and $0 \leq m \leq M$ and $n$ follows the sequence $$ \begin{array}{c|c} n & (l,m)\\ \hline 0 & (0,0)\\ 1 & (0,1)\\ 2 & (0,2) \\ \vdots & \vdots\\ & (0,M)\\ & (1,0)\\ \vdots & \vdots \\ N & (L,M) \end{array} $$ I want to know if there is a formula for how my set of coefficients change under a rotation applied to $(x,y)$? In other words, if $(x',y') = (x \cos(\phi) - y\sin(\phi),x \sin(\phi)+ y\cos(\phi))$, then my polynomial would now be $$ g(x,y) = \sum_{n=1}^N c_n P_n(x',y') = \sum_{n=1}^N c_{n}' P_{n}(x,y) $$
Is there a convenient way to determine what my $c_n'$ values are?