Question
Write $(10.7)$ with $m$ replaced by $-m ;$ then use Problem 7 to show that$$P_{I}^{-m}(x)=(-1)^{m} \frac{(l-m) !}{(l+m) !} P_{l}^{\prime \prime}(x)$$Comment: This shows that $(10.7)$ is a solution of $(10.1)$ whe? $m$ is negative.
Step 1
7)$, which gives us: $$ P_{l}^{-m}(x)=\frac{1}{2^{l} l !}\left(1-x^{2}\right)^{-m / 2} \frac{d^{l-m}}{d x^{l-m}}\left(x^{2}-1\right)^{l} $$ Show more…
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Write (10.7) with $m$ replaced by $-m ;$ then use Problem 7 to show that $$P_{l}^{-m}(x)=(-1)^{m} \frac{(l-m) !}{(l+m) !} P_{l}^{m}(x)$$ Comment: This shows that (10.7) is a solution of (10.1) when $m$ is negative.
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