§14.6 Integer Order

§14.6(i) Nonnegative Integer Orders

For $m=0,1,2,\dots$,

 14.6.1 $\displaystyle\mathop{\mathsf{P}^{m}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathop{% \mathsf{P}_{\nu}\/}\nolimits\!\left(x\right)}{{\mathrm{d}x}^{m}},$ 14.6.2 $\displaystyle\mathop{\mathsf{Q}^{m}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathop{% \mathsf{Q}_{\nu}\/}\nolimits\!\left(x\right)}{{\mathrm{d}x}^{m}}.$
 14.6.3 $\displaystyle\mathop{P^{m}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathop{P_{\nu}% \/}\nolimits\!\left(x\right)}{{\mathrm{d}x}^{m}},$ 14.6.4 $\displaystyle\mathop{Q^{m}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathop{Q_{\nu}% \/}\nolimits\!\left(x\right)}{{\mathrm{d}x}^{m}},$
 14.6.5 ${\left(\nu+1\right)_{m}}\mathop{\boldsymbol{Q}^{m}_{\nu}\/}\nolimits\!\left(x% \right)=(-1)^{m}\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathop{% \boldsymbol{Q}_{\nu}\/}\nolimits\!\left(x\right)}{{\mathrm{d}x}^{m}}.$

§14.6(ii) Negative Integer Orders

For $m=1,2,3,\dots$,

 14.6.6 $\displaystyle\mathop{\mathsf{P}^{-m}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\left(1-x^{2}\right)^{-m/2}\int_{x}^{1}\!\dots\!\int_{x}^{1}% \mathop{P_{\nu}\/}\nolimits\!\left(x\right)\left(\!\mathrm{d}x\right)^{m}.$ 14.6.7 $\displaystyle\mathop{P^{-m}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\left(x^{2}-1\right)^{-m/2}\int_{1}^{x}\!\dots\!\int_{1}^{x}% \mathop{P_{\nu}\/}\nolimits\!\left(x\right)\left(\!\mathrm{d}x\right)^{m},$ 14.6.8 $\displaystyle\mathop{Q^{-m}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=(-1)^{m}\left(x^{2}-1\right)^{-m/2}\*\int_{x}^{\infty}\!\dots\!% \int_{x}^{\infty}\mathop{Q_{\nu}\/}\nolimits\!\left(x\right)\left(\!\mathrm{d}% x\right)^{m}.$

For connections between positive and negative integer orders see (14.9.3), (14.9.4), and (14.9.13).