I am trying to expand Associated Legendre polynomials of order m=2 into Chebyshev polynomials, shown as: $$P^{2}_{n}(x)=\sum_{k=0}^{n}a_{k}T_{k}(x), $$ where $P^{2}_{n}$ is Associated Legendre polynomials of order m=2, $T_{k}$ is Chebyshev polynomials, and $a_{k}$ are unknown coefficients.
As the Orthogonality of Chebyshev polynomial is $$\int_{-1}^{1}T_{j}(x)T_{k}(x)\frac{1}{\sqrt{1-x^2}}dx=\frac{\pi}{2}\delta_{jk},\quad j^{2}+k^{2}\neq0,$$ so $$a_{k}(n,k)=\frac{2}{\pi}\int_{-1}^{1}P^{2}_{n}(x)T_{k}(x)\frac{1}{\sqrt{1-x^2}}dx.$$ But i do not how to do this integral analytically.
I found a answer on how to expend Legendre polynomials into Chebyshev polynomials using the generation function. Still, I do not know how to generalize it to the Associated Legendre polynomials.
Can anyone help? Many thanks!