Question

I am currently doing filter designs and stumbled across this mathematical problem which I cannot understand. I was hoping for some insight from experts around this field to help me with this.

$\sum_{m=0}^{M-1}\ h_{m}\ x^{m} = \sum_{m=0}^{M-1}\ g_{m}\ P_{m}\left(x\right)$

where $P_{m}\left(x\right)$ is the Legendre polynomials and $x^{m}$ is the conventional polynomials.

I want to know the relationship between $g_{m}$ and $h_{m}$, what is the best approach? Is there a closed form solution for $h_{m}$?

Thanks in advance, and apologies if it is a silly question.