The relationship between Legendre Polynomials and monomial basis polynomials 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.
 A: I might as well: there is an algorithm, due to Salzer, for converting to and from different orthogonal polynomial expansions. (I previously talked about this algorithm here.) If you intend to do this entire business of conversions, I believe an algorithm might be more useful to you than an integral expression.
What follows are a few relevant Mathematica routines. Even though I used Mathematica, I hope that the algorithm is still transparent and easily translatable (the indexing starts from 1; some adjustment is necessary if the indexing in your language starts at 0):
monomialToLegendre[pc_?VectorQ] := 
 Module[{n = Length[pc] - 1, lc, q},
  lc[1] = pc[[n]]; lc[2] = Last[pc];
  Do[
   q[1] = lc[1];
   lc[1] = pc[[n - k + 1]] + lc[2]/3;
   Do[
    q[j] = lc[j];
    lc[j] = j lc[j + 1]/(2 j + 1) + (j - 1) q[j - 1]/(2 j - 3);
    , {j, 2, k - 1}];
   q[k] = lc[k];
   lc[k] = (k - 1) q[k - 1]/(2 k - 3);
   lc[k + 1] = k q[k]/(2 k - 1);
   , {k, 2, n}];
  Table[lc[k], {k, n + 1}]]

One could use Salzer again for converting from the Legendre to the monomial basis. However, it is a bit clearer (and gives rise to essentially the same method) to treat the problem as one of Taylor/Maclaurin expansion of the polynomial expressed in terms of Legendre polynomials. One can then use Clenshaw's algorithm for the purpose:
legendreToMonomial[lc_?VectorQ] := 
 Module[{n = Length[lc] - 1, pc, v = 0, w, z},
  pc[1] = Last[lc]; pc[_] = z[_] = 0;
  Do[
   w = pc[1];
   pc[1] = lc[[j]] - v z[1];
   z[1] = w;
   Do[
    w = pc[i];
    pc[i] = (2 j - 1) z[i - 1]/j - v z[i];
    z[i] = w;
    , {i, 2, n - j + 2}];
   v = (j - 1)/j;
   , {j, n, 1, -1}];
  Table[pc[j], {j, n + 1}]]

As an example, to demonstrate the identity
$$5P_0(x)-4P_1(x)-2P_2(x)+3P_3(x)+P_4(x)=\frac{35}{8}x^4+\frac{15}{2}x^3-\frac{27}{4}x^2-\frac{17}{2}x+\frac{51}{8}$$
monomialToLegendre[legendreToMonomial[{5, -4, -2, 3, 1}]] ==
 {5, -4, -2, 3, 1}
True

legendreToMonomial[monomialToLegendre[{51/8, -17/2, -27/4, 15/2, 35/8}]] 
 == {51/8, -17/2, -27/4, 15/2, 35/8}
True

A: I am assuming your domain is $(-1,1)$ else you would need to do some rescaling of the following argument. The main thing to exploit is that the Legendre polynomials are orthogonal i.e. $$\displaystyle \int_{-1}^{1} P_m(x) P_n(x) = \frac{2}{2n+1} \delta_{mn}.$$ Using this we get that $$g_n = \left(\frac{2n+1}{2} \right) \sum_{m=0}^{M-1} \left( h_m \int_{-1}^{1} x^m P_n(x) \right)$$
