# derivatives with respect to order

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##### 4: 14.11 Derivatives with Respect to Degree or Order
###### §14.11 Derivatives with Respectto Degree or Order
14.11.1 $\frac{\partial}{\partial\nu}\mathsf{P}^{\mu}_{\nu}\left(x\right)=\pi\cot\left(% \nu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x\right)-\frac{1}{\pi}\mathsf{A}_{\nu% }^{\mu}(x),$
14.11.2 $\frac{\partial}{\partial\nu}\mathsf{Q}^{\mu}_{\nu}\left(x\right)=-\tfrac{1}{2}% \pi^{2}\mathsf{P}^{\mu}_{\nu}\left(x\right)+\frac{\pi\sin\left(\mu\pi\right)}{% \sin\left(\nu\pi\right)\sin\left((\nu+\mu)\pi\right)}\mathsf{Q}^{\mu}_{\nu}% \left(x\right)-\tfrac{1}{2}\cot\left((\nu+\mu)\pi\right)\mathsf{A}_{\nu}^{\mu}% (x)+\tfrac{1}{2}\csc\left((\nu+\mu)\pi\right)\mathsf{A}_{\nu}^{\mu}(-x),$
14.11.4 $\left.\frac{\partial}{\partial\mu}\mathsf{P}^{\mu}_{\nu}\left(x\right)\right|_% {\mu=0}=\left(\psi\left(-\nu\right)-\pi\cot\left(\nu\pi\right)\right)\mathsf{P% }_{\nu}\left(x\right)+\mathsf{Q}_{\nu}\left(x\right),$
14.11.5 $\left.\frac{\partial}{\partial\mu}\mathsf{Q}^{\mu}_{\nu}\left(x\right)\right|_% {\mu=0}=-\tfrac{1}{4}\pi^{2}\mathsf{P}_{\nu}\left(x\right)+\left(\psi\left(-% \nu\right)-\pi\cot\left(\nu\pi\right)\right)\mathsf{Q}_{\nu}\left(x\right).$
##### 5: 11.4 Basic Properties
11.4.27 $\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{\nu}\mathbf{H}_{\nu}\left(z\right)% \right)=z^{\nu}\mathbf{H}_{\nu-1}\left(z\right),$
11.4.28 $\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{-\nu}\mathbf{H}_{\nu}\left(z\right)% \right)=\frac{2^{-\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)}-z^{-\nu% }\mathbf{H}_{\nu+1}\left(z\right),$
11.4.31 ${\cal H}_{\nu-m}(z)=z^{m-\nu}\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}% \right)^{m}(z^{\nu}{\cal H}_{\nu}(z)),$ $m=1,2,3,\dots$,
###### §11.4(vi) Derivatives with RespecttoOrder
For derivatives with respect to the order $\nu$, see Apelblat (1989) and Brychkov and Geddes (2005). …
##### 7: 14.29 Generalizations
14.29.1 $\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{% \mathrm{d}w}{\mathrm{d}z}+{\left(\nu(\nu+1)-\frac{\mu_{1}^{2}}{2(1-z)}-\frac{% \mu_{2}^{2}}{2(1+z)}\right)w}=0$
##### 8: Bibliography
• A. Apelblat (1989) Derivatives and integrals with respect to the order of the Struve functions $\mathbf{H}_{\nu}(x)$ and $\mathbf{L}_{\nu}(x)$ . J. Math. Anal. Appl. 137 (1), pp. 17–36.
• A. Apelblat (1991) Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42 (5), pp. 708–714.
##### 10: 14.10 Recurrence Relations and Derivatives
14.10.4 $\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{% \mathrm{d}x}={(\mu-\nu-1)\mathsf{P}^{\mu}_{\nu+1}\left(x\right)+(\nu+1)x% \mathsf{P}^{\mu}_{\nu}\left(x\right)},$