# Is the orthogonality between Associated Legendre polynomials preserved on an interval [-a,a]

So I am aware of the orthogonality between the Associated Legendre polynomials on the interval $[-1,1]$, that is:

$$\int_{-1}^{1}P^m_kP^m_ldx\propto\delta_{k,l}$$

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:

$$\int^{a}_{-a}P^m_kP^m_ldx\propto\delta_{k,l}$$

for $[-a,a]\subset [-1,1]$. Perhaps this is true for $\lim_{a\rightarrow0}$?

Any help would be appreciated.

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 $m=1$ $$\left( \begin{array}{ccc} k & l &f(k,l,1) \\ 1 & 1 & \frac{2 a^3}{3} \\ 1 & 2 & 0 \\ 1 & 3 & a^5-a^3 \\ 1 & 4 & 0 \\ 1 & 5 & \frac{9 a^7}{4}-\frac{7 a^5}{2}+\frac{5 a^3}{4} \\ 1 & 6 & 0 \\ 1 & 7 & \frac{143 a^9}{24}-\frac{99 a^7}{8}+\frac{63 a^5}{8}-\frac{35 a^3}{24}\\ 2 & 2 & \frac{9 a^5}{10}-a^3+\frac{a}{2} \\ 2 & 3 & 0 \\ 2 & 4 & \frac{15 a^7}{8}-\frac{25 a^5}{8}+\frac{13 a^3}{8}-\frac{3 a}{8} \\ 2 & 5 & 0 \\ 2 & 6 & \frac{77 a^9}{16}-\frac{21 a^7}{2}+\frac{63 a^5}{8}-\frac{5 a^3}{2}+\frac{5 a}{16} \\ 2 & 7 & 0 \\ 3 & 3 & \frac{25 a^7}{14}-3 a^5+\frac{3 a^3}{2} \\ 3 & 4 & 0 \\ 3 & 5 & \frac{35 a^9}{8}-\frac{77 a^7}{8}+\frac{57 a^5}{8}-\frac{15 a^3}{8} \\ 3 & 6 & 0 \\ 3 & 7 & \frac{195 a^{11}}{16}-33 a^9+\frac{261 a^7}{8}-14 a^5+\frac{35 a^3}{16}\\ 4 & 4 & \frac{1225 a^9}{288}-\frac{75 a^7}{8}+\frac{111 a^5}{16}-\frac{15 a^3}{8}+\frac{9 a}{32} \\ 4 & 5 & 0 \\ 4 & 6 & \frac{735 a^{11}}{64}-\frac{1995 a^9}{64}+\frac{987 a^7}{32}-\frac{427 a^5}{32}+\frac{155 a^3}{64}-\frac{15 a}{64} \\ 4 & 7 & 0 \end{array} \right)$$
• If we consider the case $a\rightarrow 0$, it seems the $k=l$ terms vanish slower than the $k\neq l$ terms. Do you think the the $k=l$ terms can be normalized while the $k\neq l$ terms can be made to vanish? Jul 28 '16 at 16:16
• In the proof of orthogonality between Associated Legendre polynomials, the domain of integration [-1,1], allow for the boundary limits to vanish once you carry out integration by parts $l+m$ times. It seems this domain makes the proof easy, but whether this is required, I am unsure. I was curious if there was some theorem I was not aware of .. Otherwise yes, that would be the most straightforward way to do it. See: link Jul 28 '16 at 16:21