# associated Legendre equation

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##### 2: 14.29 Generalizations
For inhomogeneous versions of the associated Legendre equation, and properties of their solutions, see Babister (1967, pp. 252–264).
##### 3: 14.2 Differential Equations
###### §14.2(ii) AssociatedLegendreEquation
Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations $\mathsf{P}^{0}_{\nu}\left(x\right)=\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}^{0}_{\nu}\left(x\right)=\mathsf{Q}_{\nu}\left(x\right)$, $P^{0}_{\nu}\left(x\right)=P_{\nu}\left(x\right)$, $Q^{0}_{\nu}\left(x\right)=Q_{\nu}\left(x\right)$, $\boldsymbol{Q}^{0}_{\nu}\left(x\right)=\boldsymbol{Q}_{\nu}\left(x\right)=Q_{% \nu}\left(x\right)/\Gamma\left(\nu+1\right)$. …
###### §14.2(iii) Numerically Satisfactory Solutions
14.2.7 $\mathscr{W}\left\{P^{-\mu}_{\nu}\left(x\right),P^{\mu}_{\nu}\left(x\right)% \right\}=\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x\right),\mathsf{P}^{% \mu}_{\nu}\left(x\right)\right\}=\frac{2\sin\left(\mu\pi\right)}{\pi\left(1-x^% {2}\right)},$
##### 4: 14.31 Other Applications
###### §14.31(iii) Miscellaneous
Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …
##### 5: 30.2 Differential Equations
If $\gamma=0$, Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). …
##### 6: Howard S. Cohl
Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …
##### 7: 14.3 Definitions and Hypergeometric Representations
14.3.8 $P^{m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(% \nu-m+1\right)}\left(x^{2}-1\right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;% \tfrac{1}{2}-\tfrac{1}{2}x\right).$
##### 8: Errata
• Equation (14.8.9)
14.8.9 $\boldsymbol{Q}_{\nu}\left(x\right)=-\frac{\ln\left(x-1\right)}{2\Gamma\left(% \nu+1\right)}+\frac{\frac{1}{2}\ln 2-\gamma-\psi\left(\nu+1\right)}{\Gamma% \left(\nu+1\right)}+O\left(\left(x-1\right)\ln\left(x-1\right)\right),$ $\nu\neq-1,-2,-3,\dots$

The symbol $O\left(x-1\right)$ has been corrected to be $O\left(\left(x-1\right)\ln\left(x-1\right)\right)$.

Reported by Mark Ashbaugh on 2022-02-08

• Subsection 14.2(iii)

Previously the exponents of the associated Legendre differential equation (14.2.2) at infinity were given incorrectly by $\left\{-\nu-1,\nu\right\}$. These were replaced by $\left\{\nu+1,-\nu\right\}$.

Reported by Hans Volkmer on 2019-01-30

• Equation (10.22.72)
10.22.72 $\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\nu}\left(bt\right)J_{\nu}\left(ct% \right)t^{1-\mu}\,\mathrm{d}t=\frac{(bc)^{\mu-1}\sin\left((\mu-\nu)\pi\right)(% \sinh\chi)^{\mu-\frac{1}{2}}}{(\frac{1}{2}{\pi}^{3})^{\frac{1}{2}}a^{\mu}}{% \mathrm{e}}^{(\mu-\frac{1}{2})\mathrm{i}\pi}Q^{\frac{1}{2}-\mu}_{\nu-\frac{1}{% 2}}\left(\cosh\chi\right),$ $\Re\mu>-\tfrac{1}{2},\Re\nu>-1,a>b+c,\cosh\chi=(a^{2}-b^{2}-c^{2})/(2bc)$

Originally, the factor on the right-hand side was written as $\frac{(bc)^{\mu-1}\cos\left(\nu\pi\right)(\sinh\chi)^{\mu-\frac{1}{2}}}{(\frac% {1}{2}{\pi}^{3})^{\frac{1}{2}}a^{\mu}}$, which was taken directly from Watson (1944, p. 412, (13.46.5)), who uses a different normalization for the associated Legendre function of the second kind $Q^{\mu}_{\nu}$. Watson’s $Q_{\nu}^{\mu}$ equals $\frac{\sin\left((\nu+\mu)\pi\right)}{\sin\left(\nu\pi\right)}{\mathrm{e}}^{-% \mu\pi\mathrm{i}}Q^{\mu}_{\nu}$ in the DLMF.

Reported by Arun Ravishankar on 2018-10-22

• Equation (14.2.7)

The Wronskian was generalized to include both associated Legendre and Ferrers functions.

• ##### 9: Bibliography D
• T. M. Dunster (2004) Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3), pp. 245–270.
• ##### 10: 10.19 Asymptotic Expansions for Large Order
10.19.9 $\rselection{{H^{(1)}_{\nu}}\left(\nu+a\nu^{\frac{1}{3}}\right)\\ {H^{(2)}_{\nu}}\left(\nu+a\nu^{\frac{1}{3}}\right)}\sim\frac{2^{\frac{4}{3}}}{% \nu^{\frac{1}{3}}}e^{\mp\pi i/3}\operatorname{Ai}\left(e^{\mp\pi i/3}2^{\frac{% 1}{3}}a\right)\sum_{k=0}^{\infty}\frac{P_{k}(a)}{\nu^{2k/3}}+\frac{2^{\frac{5}% {3}}}{\nu}e^{\pm\pi i/3}\operatorname{Ai}'\left(e^{\mp\pi i/3}2^{\frac{1}{3}}a% \right)\sum_{k=0}^{\infty}\frac{Q_{k}(a)}{\nu^{2k/3}},$