User:Dnessett/Sandboxes/Second Sandbox

$$\int\limits_{-1}^{1}\left( x^{2} -1\right) ^{l} dx = \int\limits_{0}^{\pi}\left( \sin \theta \right) ^{2l+1} d\theta $$

$$ K_{kl}^{m} =\delta_{kl} \; \frac{(-1)^{l+m} }{2^{2l}\, (l!)^{2}} \binom{l+m}{2m} \int\limits_{-1}^{1}\left( x^{2} -1\right) ^{l} \frac{d^{2m} }{dx^{2m} } \left[ \left( 1-x^{2} \right) ^{m} \right] \frac{d^{2l} }{dx^{2l} } \left[ \left( 1-x^{2} \right) ^{l} \right] dx, $$

$$ \frac{d^{2m}}{dx^{2m} } \left[ \left( 1-x^{2} \right) ^{m} \right] = (-1)^{k}\, (2k)! \, . $$

$$ K_{kl}^{m} =\delta_{kl} \; \frac{(-1)^{l+m} }{2^{2l}\, (l!)^{2}} \binom{l+m}{2m} \int\limits_{-1}^{1}\left( x^{2} -1\right) ^{l} (-1)^{l}\ (2l)!\ (-1)^{m}\ (2m)!\ dx\ =\ \delta_{kl} \; \frac{\ (2l)!\ (2m)! }{2^{2l}\, (l!)^{2}} \binom{l+m}{2m} \int\limits_{-1}^{1}\left( x^{2} -1\right) ^{l} $$