14 Legendre and Related Functions Real Arguments 14.10 Recurrence Relations and Derivatives 14.12 Integral Representations

§14.11 Derivatives with Respect to Degree or Order

Notes:: (14.11.1) may be derived from (14.3.1). (14.11.2) may be derived from (14.9.10) and (14.11.1). (14.11.4) may be derived from (14.3.1) and the hypergeometric expansion for $\mathop{\mathsf{Q}_{{\nu}}\/}\nolimits\!\left(x\right)$ (Hobson (1931, §132)). (14.11.5) may be derived from (14.9.8), (14.9.10), and (14.11.4).
Keywords:: Ferrers functions, associated Legendre functions, derivatives
Referenced by:: Other Changes
Permalink:: http://dlmf.nist.gov/14.11
Addition (effective with 1.0.5):: The references to Szmytkowski (2006, 2009, 2011, 2012) have been added at the end of this section.
Reported 2012-03-27

14.11.1 $\frac{\partial}{\partial\nu}\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!% \left(x\right)=\pi\mathop{\cot\/}\nolimits\!\left(\nu\pi\right)\mathop{\mathsf% {P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)-\frac{1}{\pi}\mathsf{A}_{{\nu}% }^{{\mu}}(x),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real variable, $\mu$ : general order, $\nu$ : general degree and $\mathsf{A}_{{\nu}}^{{\mu}}(x)$
Referenced by:: §14.11, §14.11
Permalink:: http://dlmf.nist.gov/14.11.E1
Encodings:: TeX, pMML, png

14.11.2 $\frac{\partial}{\partial\nu}\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!% \left(x\right)=-\tfrac{1}{2}\pi^{2}\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}% \nolimits\!\left(x\right)+\frac{\pi\mathop{\sin\/}\nolimits\!\left(\mu\pi% \right)}{\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{\sin\/}\nolimits% \!\left((\nu+\mu)\pi\right)}\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!% \left(x\right)-\tfrac{1}{2}\mathop{\cot\/}\nolimits\!\left((\nu+\mu)\pi\right)% \mathsf{A}_{{\nu}}^{{\mu}}(x)+\tfrac{1}{2}\mathop{\csc\/}\nolimits\!\left((\nu% +\mu)\pi\right)\mathsf{A}_{{\nu}}^{{\mu}}(-x),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\mathop{\sin\/}\nolimits z$ : sine function, : real variable, $\mu$ : general order, $\nu$ : general degree and $\mathsf{A}_{{\nu}}^{{\mu}}(x)$
Referenced by:: §14.11
Permalink:: http://dlmf.nist.gov/14.11.E2
Encodings:: TeX, pMML, png

where

14.11.3 $\mathsf{A}_{{\nu}}^{{\mu}}(x)=\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)% \left(\frac{1+x}{1-x}\right)^{{\mu/2}}\*\sum_{{k=0}}^{{\infty}}\frac{\left(% \frac{1}{2}-\frac{1}{2}x\right)^{k}\mathop{\Gamma\/}\nolimits\!\left(k-\nu% \right)\mathop{\Gamma\/}\nolimits\!\left(k+\nu+1\right)}{k!\mathop{\Gamma\/}% \nolimits\!\left(k-\mu+1\right)}\*\left(\mathop{\psi\/}\nolimits\!\left(k+\nu+% 1\right)-\mathop{\psi\/}\nolimits\!\left(k-\nu\right)\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable, $\mu$ : general order, $\nu$ : general degree and $\mathsf{A}_{{\nu}}^{{\mu}}(x)$
Referenced by:: §14.11
Permalink:: http://dlmf.nist.gov/14.11.E3
Encodings:: TeX, pMML, png

14.11.4 $\left.\frac{\partial}{\partial\mu}\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}% \nolimits\!\left(x\right)\right|_{{\mu=0}}=\left(\mathop{\psi\/}\nolimits\!% \left(-\nu\right)-\pi\mathop{\cot\/}\nolimits\!\left(\nu\pi\right)\right)% \mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(x\right)+\mathop{\mathsf{Q}_{{% \nu}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.11, §14.11
Permalink:: http://dlmf.nist.gov/14.11.E4
Encodings:: TeX, pMML, png

14.11.5 $\left.\frac{\partial}{\partial\mu}\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}% \nolimits\!\left(x\right)\right|_{{\mu=0}}=-\tfrac{1}{4}\pi^{2}\mathop{\mathsf% {P}_{{\nu}}\/}\nolimits\!\left(x\right)+\left(\mathop{\psi\/}\nolimits\!\left(% -\nu\right)-\pi\mathop{\cot\/}\nolimits\!\left(\nu\pi\right)\right)\mathop{% \mathsf{Q}_{{\nu}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.11
Permalink:: http://dlmf.nist.gov/14.11.E5
Encodings:: TeX, pMML, png

(14.11.1) holds if $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ is replaced by $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ , provided that the factor $(\ifrac{(1+x)}{(1-x)})^{{\mu/2}}$ in (14.11.3) is replaced by $(\ifrac{(x+1)}{(x-1)})^{{\mu/2}}$ . (14.11.4) holds if $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(x\right)$ , and $\mathop{\mathsf{Q}_{{\nu}}\/}\nolimits\!\left(x\right)$ are replaced by $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ , $\mathop{P_{{\nu}}\/}\nolimits\!\left(x\right)$ , and $\mathop{Q_{{\nu}}\/}\nolimits\!\left(x\right)$ , respectively.

See also Szmytkowski (2006, 2009, 2011, 2012) and Magnus et al. (1966, pp. 177–178).

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