∫ − 1 1 ( x 2 − 1 ) l d x = ∫ 0 π ( sin θ ) 2 l + 1 d θ {\displaystyle \int \limits _{-1}^{1}\left(x^{2}-1\right)^{l}dx=\int \limits _{0}^{\pi }\left(\sin \theta \right)^{2l+1}d\theta }
K k l m = δ k l ( − 1 ) l + m 2 2 l ( l ! ) 2 ( l + m 2 m ) ∫ − 1 1 ( x 2 − 1 ) l d 2 m d x 2 m [ ( 1 − x 2 ) m ] d 2 l d x 2 l [ ( 1 − x 2 ) l ] d x , {\displaystyle 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,}
d 2 m d x 2 m [ ( 1 − x 2 ) m ] = ( − 1 ) k ( 2 k ) ! . {\displaystyle {\frac {d^{2m}}{dx^{2m}}}\left[\left(1-x^{2}\right)^{m}\right]=(-1)^{k}\,(2k)!\,.}
K k l m = δ k l ( − 1 ) l + m 2 2 l ( l ! ) 2 ( l + m 2 m ) ∫ − 1 1 ( x 2 − 1 ) l ( − 1 ) l ( 2 l ) ! ( − 1 ) m ( 2 m ) ! d x = δ k l ( 2 l ) ! ( 2 m ) ! 2 2 l ( l ! ) 2 ( l + m 2 m ) ∫ − 1 1 ( x 2 − 1 ) l {\displaystyle 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}}
![{\displaystyle 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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0129e8557b32d04e7b1d03507fe2ee6f5a4793)
![{\displaystyle {\frac {d^{2m}}{dx^{2m}}}\left[\left(1-x^{2}\right)^{m}\right]=(-1)^{k}\,(2k)!\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4029aa0a05df374550dd9aa73a872546a68b133)
