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

Abstract

In our recent works [R. Szmytkowski, J. Phys. A 39 (2006) 15147; corrigendum: 40 (2007) 7819; addendum: 40 (2007) 14887], we have investigated the derivative of the Legendre function of the first kind, $P_{\nu }(z)$, with respect to its degree $\nu$. In the present work, we extend these studies and construct several representations of the derivative of the associated Legendre function of 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-integral representations of $\partial P_{\nu }^{\pm m}(z)/\partial \nu$. They are then used to derive Rodrigues-type formulas for $[\partial P_{\nu }^{\pm m}(z)/\partial \nu {]}_{\nu =n}$ with $n\in \mathbb{N}$. Next, some closed-form expressions for $[\partial P_{\nu }^{\pm m}(z)/\partial \nu {]}_{\nu =n}$ are obtained. These results are applied to find several representations, both explicit and of the Rodrigues type, for the associated Legendre function of the second kind of integer degree and order, $Q_n^{\pm m}(z)$; the explicit representations are suitable for use for numerical purposes in various regions of the complex $z$-plane. Finally, the derivatives $[{\partial }^2{P}_{\nu }^m(z)/\partial {\nu }^2{]}_{\nu =n}$, $[\partial Q_{\nu }^m(z)/\partial \nu {]}_{\nu =n}$ and $[\partial Q_{\nu }^m(z)/\partial \nu {]}_{\nu =-n-1}$, all with $m>n$, are evaluated in terms of $[\partial P_{\nu }^{-m}(\pm z)/\partial \nu {]}_{\nu =n}$.

arXiv commentLateX, 40 pages, 1 figure, extensive referencing

References in corpus (4)

• A note on parameter derivatives of classical orthogonal polynomials
• Green's function for the wavized Maxwell fish-eye problem
• Some differentiation formulas for Legendre polynomials
• On harmonic binomial series

Cited by in corpus (8)

• A note on parameter derivatives of classical orthogonal polynomials
• On Parameter Differentiation for Integral Representations of Associated Legendre Functions
• The scaling properties and the multiple derivative of Legendre polynomials
• On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)
• On the derivatives ${\partial }^2{P}_{\nu }(z)/\partial {\nu }^2$ and $\partial Q_{\nu }(z)/\partial \nu$ of the Legendre functions with respect to their degrees
• Representations for the parameter derivatives of some Koornwinder polynomials
• Ferrers functions of arbitrary degree and order and related functions
• Generating Functions for Lacunary Legendre and Legendre-like Polynomials