English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In our recent works [R. Szmytkowski, J. Phys. A 39 (2006) 15147; corrigendum:40 (2007) 7819; addendum: 40 (2007) 14887], we have investigated the derivativeof the Legendre function of the first kind, $P_{\nu}(z)$, with respect to itsdegree $\nu$. In the present work, we extend these studies and constructseveral representations of the derivative of the associated Legendre functionof the first kind, $P_{\nu}^{\pm m}(z)$, with respect to the degree $\nu$, for$m\in\mathbb{N}$. At first, we establish several contour-integralrepresentations of $\partial P_{\nu}^{\pm m}(z)/\partial\nu$. They are thenused to derive Rodrigues-type formulas for $[\partial P_{\nu}^{\pmm}(z)/\partial\nu]_{\nu=n}$ with $n\in\mathbb{N}$. Next, some closed-formexpressions for $[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}$ areobtained. These results are applied to find several representations, bothexplicit and of the Rodrigues type, for the associated Legendre function of thesecond kind of integer degree and order, $Q_{n}^{\pm m}(z)$; the explicitrepresentations are suitable for use for numerical purposes in various regionsof the complex $z$-plane. Finally, the derivatives$[\partial^{2}P_{\nu}^{m}(z)/\partial\nu^{2}]_{\nu=n}$, $[\partialQ_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$ and $[\partialQ_{\nu}^{m}(z)/\partial\nu]_{\nu=-n-1}$, all with $m>n$, are evaluated in termsof $[\partial P_{\nu}^{-m}(\pm z)/\partial\nu]_{\nu=n}$.

On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree