# Alternative orthogonality relations between associated Legendre polynomials

The usual orthogonality relations quoted for associated Legendre polynomials is: $$\int_{-1}^{1}P_{n}^{m}(x)P_{n'}^{m}(x)dx=\frac{2(n+m)!}{(2n+1)(n-m)!}\delta_{n,n'}$$ However, I have come across some alternative forms that I don't see how to show. They are: $$\int_{-1}^{1}(xP_{n}^{m}(x)-P_{n-1}^{m}(x))P_{n'}^{m-1}(x)\frac{1}{\sqrt{1-x^{2}}}dx=-\frac{2(n+m-1)!}{(2n+1)(n-m)!}\delta_{n,n'}$$ $$\int_{-1}^{1}((n+m)P_{n-1}^{m}(x)-(n-m)xP_{n}^{m}(x))P_{n}^{m+1}(x)\frac{1}{\sqrt{1-x^{2}}}dx=-\frac{2(n+m+1)!}{(2n+1)(n-m-1)!}\delta_{n,n'}$$ $$\int_{-1}^{1}(-(n+1)xP_{n-1}^{1}(x)+(nx^{2}+1)P_{n}^{1}(x))P_{n'}(x)\frac{1}{\sqrt{1-x^{2}}}dx=-\frac{2n^{2}(n+1)}{(2n+1)(2n+3)}\delta_{n,n'-1}-\frac{2n(n+1)^{2}}{(2n-1)(2n+1)}\delta_{n,n'+1}$$ $$\int_{-1}^{1}(P_{n-1}^{1}(x)-xP_{n}^{1}(x))P_{n'}(x)dx=-\frac{2(n+1)}{(2n+1)(2n+3)}\delta_{n,n'-1}+\frac{2(n+1)}{(2n-1)(2n+1)}\delta_{n,n'+1}$$ It is possible that the last two also have a generalisation for other values of $$m$$, but I haven't managed to find them yet. One can see that they hold for $$n>0$$ by putting the following into Mathematica:

Table[Integrate[(x LegendreP[n, m, x] -
LegendreP[n - 1, m, x]) LegendreP[n1, m - 1, x]/
Sqrt[1 - x^2], {x, -1,
1}] - (-2 (n + m - 1)! KroneckerDelta[n,
n1]/((2 n + 1) (n - m)!)), {n, 1, 10}, {n1, 1, 10}, {m, 1,
10}] // MatrixForm
Table[Integrate[((n + m) LegendreP[n - 1, m, x] - (n - m) x LegendreP[
n, m, x]) LegendreP[n1, m + 1, x]/Sqrt[1 - x^2], {x, -1,
1}] - (-2 (n + m + 1)! KroneckerDelta[n,
n1]/((2 n + 1) (n - m - 1)!)), {n, 1, 10}, {n1, 1, 10}, {m, 1,
10}] // MatrixForm
Table[Integrate[(-(n + 1) x LegendreP[n - 1, 1,
x] + (n x^2 + 1) LegendreP[n, 1, x]) LegendreP[n1, x]/
Sqrt[1 - x^2], {x, -1,
1}] - (-2 n^2 (n + 1) KroneckerDelta[n,
n1 - 1]/((2 n + 1) (2 n + 3)) -
2 n (n + 1)^2 KroneckerDelta[n,
n1 + 1]/((2 n - 1) (2 n + 1))), {n, 1, 10}, {n1, 1,
10}] // MatrixForm
Table[Integrate[(LegendreP[n - 1, x] - x LegendreP[n, x]) LegendreP[
n1, x], {x, -1,
1}] - (-2 (n + 1) KroneckerDelta[n,
n1 - 1]/((2 n + 1) (2 n + 3)) +
2 (n + 1) KroneckerDelta[n, n1 + 1]/((2 n - 1) (2 n + 1))), {n,
1, 10}, {n1, 1, 10}] // MatrixForm


I'm really not sure how to go about proving these as I can't seem to simplify the expressions $$xP_{n}^{1}(x)-P_{n-1}^{1}(x)$$, $$-(n+1)xP_{n-1}^{1}(x)+(nx^{2}+1)P_{n}^{1}(x)$$ and $$P_{n-1}^{1}(x)-xP_{n}^{1}(x)$$ using any recurrence relations. I tried using the definitions: $$P_{n}^{m}(x)=\frac{(-1)^{m}}{2^{n}n!}(1-x^{2})^{m/2}\frac{d^{n+m}}{dx^{n+m}}(x^{2}-1)^n$$ but things got incredibly messy. Any help would be greatly appreciated.

• These follow from recurrence formulae. Your first two integrals are the sixth and seventh recurrence formulae in that link. Jun 29 '21 at 16:13