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#### 8.75 Special cases and particular values

8.751

1.

${P}_{\nu }^{m}\left(x\right)={\left(-1\right)}^{m}\frac{\Gamma \left(\nu +m+1\right){\left(1-{x}^{2}\right)}^{\frac{m}{2}}}{{2}^{m}\Gamma \left(\nu -m+1\right)m!}F\left(m-\nu ,m+\nu +1;m+1;\frac{1-x}{2}\right)$

MO 84

2.

${P}_{\nu }^{m}\left(z\right)=\frac{\Gamma \left(\nu +m+1\right){\left({z}^{2}-1\right)}^{\frac{m}{2}}}{{2}^{m}m!\Gamma \left(\nu -m+1\right)}F\left(m-\nu ,m+\nu +1;m+1;\frac{1-z}{2}\right)$

MO 84

3.8

${Q}_{n+\frac{1}{2}}^{\mu }\left(z\right)=\frac{{e}^{\mu \text{π}i}\Gamma \left(\mu +n+\frac{3}{2}\right)}{{2}^{n+\frac{3}{2}}\left(n+1\right)!}{\left({z}^{2}-1\right)}^{\frac{\mu }{2}}{\text{π}}^{1/2}{z}^{-n-\mu -3/2}F\left(\frac{\mu +n+\frac{5}{2}}{2},\frac{\mu +n+\frac{3}{2}}{2};n+2;\frac{1}{{z}^{2}}\right)$

MO 84

8.752

1.

${P}_{\nu }^{m}\left(x\right)={\left(-1\right)}^{m}{\left(1-{x}^{2}\right)}^{\frac{m}{2}}\frac{{d}^{m}}{d{x}^{m}}{P}_{\nu }\left(x\right)$

WH, MO 84, EH I ...

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