So I am aware of the orthogonality between the Associated Legendre polynomials on the interval $[-1,1]$, that is:
\begin{equation} \int_{-1}^{1}P^m_kP^m_ldx\propto\delta_{k,l} \end{equation}
where $\delta_{k,l}$ is the kronecker delta function (I am only interested in the case where the upper indices of the Legendre polynomials are equal, but feel free to also discuss the opposite case as well). However, what I'm after is whether the following is true:
\begin{equation} \int^{a}_{-a}P^m_kP^m_ldx\propto\delta_{k,l} \end{equation}
for $[-a,a]\subset [-1,1]$. Perhaps this is true for $\lim_{a\rightarrow0}$?
Any help would be appreciated.
This is not an answer but it is too long for a comment.
Let us consider $$f(k,l,m)=\int_{-a}^a P_k(x){}^m \,P_l(x){}^m \, dx$$ and let just compute a few values for $m=1$ $$\left( \begin{array}{ccc} k & l &f(k,l,1) \\ 1 & 1 & \frac{2 a^3}{3} \\ 1 & 2 & 0 \\ 1 & 3 & a^5-a^3 \\ 1 & 4 & 0 \\ 1 & 5 & \frac{9 a^7}{4}-\frac{7 a^5}{2}+\frac{5 a^3}{4} \\ 1 & 6 & 0 \\ 1 & 7 & \frac{143 a^9}{24}-\frac{99 a^7}{8}+\frac{63 a^5}{8}-\frac{35 a^3}{24}\\ 2 & 2 & \frac{9 a^5}{10}-a^3+\frac{a}{2} \\ 2 & 3 & 0 \\ 2 & 4 & \frac{15 a^7}{8}-\frac{25 a^5}{8}+\frac{13 a^3}{8}-\frac{3 a}{8} \\ 2 & 5 & 0 \\ 2 & 6 & \frac{77 a^9}{16}-\frac{21 a^7}{2}+\frac{63 a^5}{8}-\frac{5 a^3}{2}+\frac{5 a}{16} \\ 2 & 7 & 0 \\ 3 & 3 & \frac{25 a^7}{14}-3 a^5+\frac{3 a^3}{2} \\ 3 & 4 & 0 \\ 3 & 5 & \frac{35 a^9}{8}-\frac{77 a^7}{8}+\frac{57 a^5}{8}-\frac{15 a^3}{8} \\ 3 & 6 & 0 \\ 3 & 7 & \frac{195 a^{11}}{16}-33 a^9+\frac{261 a^7}{8}-14 a^5+\frac{35 a^3}{16}\\ 4 & 4 & \frac{1225 a^9}{288}-\frac{75 a^7}{8}+\frac{111 a^5}{16}-\frac{15 a^3}{8}+\frac{9 a}{32} \\ 4 & 5 & 0 \\ 4 & 6 & \frac{735 a^{11}}{64}-\frac{1995 a^9}{64}+\frac{987 a^7}{32}-\frac{427 a^5}{32}+\frac{155 a^3}{64}-\frac{15 a}{64} \\ 4 & 7 & 0 \end{array} \right)$$