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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.

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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 $m=1$ $$\left( \begin{array}{ccc} k & l &f(k,l,1) \\ 1 & 1 & \frac{2 a^3}{3} \\ 1 & 2 & 0 \\ 1 & 3 & a^5-a^3 \\ 1 & 4 & 0 \\ 1 & 5 & \frac{9 a^7}{4}-\frac{7 a^5}{2}+\frac{5 a^3}{4} \\ 1 & 6 & 0 \\ 1 & 7 & \frac{143 a^9}{24}-\frac{99 a^7}{8}+\frac{63 a^5}{8}-\frac{35 a^3}{24}\\ 2 & 2 & \frac{9 a^5}{10}-a^3+\frac{a}{2} \\ 2 & 3 & 0 \\ 2 & 4 & \frac{15 a^7}{8}-\frac{25 a^5}{8}+\frac{13 a^3}{8}-\frac{3 a}{8} \\ 2 & 5 & 0 \\ 2 & 6 & \frac{77 a^9}{16}-\frac{21 a^7}{2}+\frac{63 a^5}{8}-\frac{5 a^3}{2}+\frac{5 a}{16} \\ 2 & 7 & 0 \\ 3 & 3 & \frac{25 a^7}{14}-3 a^5+\frac{3 a^3}{2} \\ 3 & 4 & 0 \\ 3 & 5 & \frac{35 a^9}{8}-\frac{77 a^7}{8}+\frac{57 a^5}{8}-\frac{15 a^3}{8} \\ 3 & 6 & 0 \\ 3 & 7 & \frac{195 a^{11}}{16}-33 a^9+\frac{261 a^7}{8}-14 a^5+\frac{35 a^3}{16}\\ 4 & 4 & \frac{1225 a^9}{288}-\frac{75 a^7}{8}+\frac{111 a^5}{16}-\frac{15 a^3}{8}+\frac{9 a}{32} \\ 4 & 5 & 0 \\ 4 & 6 & \frac{735 a^{11}}{64}-\frac{1995 a^9}{64}+\frac{987 a^7}{32}-\frac{427 a^5}{32}+\frac{155 a^3}{64}-\frac{15 a}{64} \\ 4 & 7 & 0 \end{array} \right)$$

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  • $\begingroup$ 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? $\endgroup$ – Decebalus Jul 28 '16 at 16:16
  • $\begingroup$ Interesting comment ! Le me think about it. It is late here and I need some food and rest. If I find anything of interest, I'll post it. Who knows ? Cheers :-) $\endgroup$ – Claude Leibovici Jul 28 '16 at 16:21
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Surely a simple answer to this question is to carry out the orthogonality calculation with limits a = 1, -a =-1 in the integral and observe if there is anything special about the values +-1. If there is not then using +-a as the limits should work. This is based on the Wiki page for Associated Legendre Polynomials.

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  • $\begingroup$ 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 $\endgroup$ – Decebalus Jul 28 '16 at 16:21

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