# §14.6 Integer Order

## §14.6(i) Nonnegative Integer Orders

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

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

## §14.6(ii) Negative Integer Orders

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

 14.6.6 $\displaystyle\mathsf{P}^{-m}_{\nu}\left(x\right)$ $\displaystyle=\left(1-x^{2}\right)^{-m/2}\int_{x}^{1}\!\dots\!\int_{x}^{1}% \mathsf{P}_{\nu}\left(x\right)\left(\!\,\mathrm{d}x\right)^{m}.$ ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the first kind, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$: Ferrers function of the first kind, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$: Legendre function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree Referenced by: §14.12(i), Erratum (V1.0.28) for Equation (14.6.6) Permalink: http://dlmf.nist.gov/14.6.E6 Encodings: TeX, pMML, png Clarification (effective with 1.0.28): The right-hand side has been corrected by replacing the Legendre function $P_{\nu}\left(x\right)$ with the Ferrers function $\mathsf{P}_{\nu}\left(x\right)$. See also: Annotations for §14.6(ii), §14.6 and Ch.14 14.6.7 $\displaystyle P^{-m}_{\nu}\left(x\right)$ $\displaystyle=\left(x^{2}-1\right)^{-m/2}\int_{1}^{x}\!\dots\!\int_{1}^{x}P_{% \nu}\left(x\right)\left(\!\,\mathrm{d}x\right)^{m},$ 14.6.8 $\displaystyle Q^{-m}_{\nu}\left(x\right)$ $\displaystyle=(-1)^{m}\left(x^{2}-1\right)^{-m/2}\*\int_{x}^{\infty}\!\dots\!% \int_{x}^{\infty}Q_{\nu}\left(x\right)\left(\!\,\mathrm{d}x\right)^{m}.$

For connections between positive and negative integer orders see (14.9.3), (14.9.4), and (14.9.13). For generalizations see Cohl and Costas-Santos (2020).