Let a 2D Legendre Polynomial be defined as a product of 1D Legendre polynomials, $V_n^m(x,y) = P_n(x)P_m(y)$. I want to evaluate the integral,
$ \int_{x=-1}^1 \int_{y=-\sqrt{1-x^2}}^{\sqrt{1-x^2}} V_n^m(x,y)V_p^r(x,y)dxdy $.
Of course this can be written as,
$ \int_{x=-1}^1 \int_{y=-\sqrt{1-x^2}}^{\sqrt{1-x^2}} P_n(x)P_p(x)P_m(y)P_r(y)dxdy $.
I am aware that a product of Legendre polynomials can be written as a series of Legendre polynomials, and I think that that will be necessary twice to evaluate the integral. I'm not sure though how to evaluate the first integral over y with functional bounds. Obviously if we let $u=\frac{y}{\sqrt{1-x^2}}$ then the bounds of the integral become $u=-1..1$, but then the integrand becomes $P_m(\sqrt{1-x^2}u)P_r(\sqrt{1-x^2}u)/\sqrt{1-x^2}$ and I don't think that helps in using orthogonality.
I'm kind of at a loss as to how to evaluate this so any help would be greatly appreciated.