# Ferrers functions

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## 11—20 of 47 matching pages

##### 11: 14.2 Differential Equations
###### §14.2(ii) Associated Legendre Equation
Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations $\mathsf{P}^{0}_{\nu}\left(x\right)=\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}^{0}_{\nu}\left(x\right)=\mathsf{Q}_{\nu}\left(x\right)$, $P^{0}_{\nu}\left(x\right)=P_{\nu}\left(x\right)$, $Q^{0}_{\nu}\left(x\right)=Q_{\nu}\left(x\right)$, $\boldsymbol{Q}^{0}_{\nu}\left(x\right)=\boldsymbol{Q}_{\nu}\left(x\right)=Q_{% \nu}\left(x\right)/\Gamma\left(\nu+1\right)$. … Unless stated otherwise in §§14.214.20 it is assumed that the arguments of the functions $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ and $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ lie in the interval $(-1,1)$, and the arguments of the functions $P^{\mu}_{\nu}\left(x\right)$, $Q^{\mu}_{\nu}\left(x\right)$, and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ lie in the interval $(1,\infty)$. …
###### §14.2(iv) Wronskians and Cross-Products
14.2.5 $\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)-% \mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)=% \frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+2\right)},$
##### 12: 14.9 Connection Formulas
###### §14.9(i) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(x\right)$, $\mathsf{P}^{\pm\mu}_{-\nu-1}\left(x\right)$, $\mathsf{Q}^{\pm\mu}_{\nu}\left(x\right)$, $\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right)$
14.9.4 $\mathsf{Q}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{% \Gamma\left(\nu+m+1\right)}\mathsf{Q}^{m}_{\nu}\left(x\right),$ $\nu\neq m-1,m-2,\dots$.
###### §14.9(ii) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(\pm x\right)$, $\mathsf{Q}^{-\mu}_{\nu}\left(\pm x\right)$, $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$
14.9.9 $\frac{2}{\Gamma\left(\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\mathsf{Q}^{% \mu}_{\nu}\left(x\right)=-\cos\left(\nu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(% x\right)+\cos\left(\mu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(-x\right),$
##### 14: 30.8 Expansions in Series of Ferrers Functions
###### §30.8 Expansions in Series of FerrersFunctions
where $\mathsf{P}^{m}_{n+2k}\left(x\right)$ is the Ferrers function of the first kind (§14.3(i)), $R=\left\lfloor\frac{1}{2}(n-m)\right\rfloor$, and the coefficients $a^{m}_{n,k}(\gamma^{2})$ are given by …
30.8.9 $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)=\sum_{k=-\infty}^{-N-1}(-1)^{k}{a% ^{\prime}}^{m}_{n,k}(\gamma^{2})\mathsf{P}^{m}_{n+2k}\left(x\right)+\sum_{k=-N% }^{\infty}(-1)^{k}a^{m}_{n,k}(\gamma^{2})\mathsf{Q}^{m}_{n+2k}\left(x\right),$
where $\mathsf{P}^{m}_{n}$ and $\mathsf{Q}^{m}_{n}$ are again the Ferrers functions and $N=\left\lfloor\frac{1}{2}(n+m)\right\rfloor$. …
##### 15: 14.33 Tables
###### §14.33 Tables
• Zhang and Jin (1996, Chapter 4) tabulates $\mathsf{P}_{n}\left(x\right)$ for $n=2(1)5,10$, $x=0(.1)1$, 7D; $\mathsf{P}_{n}\left(\cos\theta\right)$ for $n=1(1)4,10$, $\theta=0(5^{\circ})90^{\circ}$, 8D; $\mathsf{Q}_{n}\left(x\right)$ for $n=0(1)2,10$, $x=0(.1)0.9$, 8S; $\mathsf{Q}_{n}\left(\cos\theta\right)$ for $n=0(1)3,10$, $\theta=0(5^{\circ})90^{\circ}$, 8D; $\mathsf{P}^{m}_{n}\left(x\right)$ for $m=1(1)4$, $n-m=0(1)2$, $n=10$, $x=0,0.5$, 8S; $\mathsf{Q}^{m}_{n}\left(x\right)$ for $m=1(1)4$, $n=0(1)2,10$, 8S; $\mathsf{P}^{m}_{\nu}\left(\cos\theta\right)$ for $m=0(1)3$, $\nu=0(.25)5$, $\theta=0(15^{\circ})90^{\circ}$, 5D; $P_{n}\left(x\right)$ for $n=2(1)5,10$, $x=1(1)10$, 7S; $Q_{n}\left(x\right)$ for $n=0(1)2,10$, $x=2(1)10$, 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 $\nu$-zeros of $\mathsf{P}^{m}_{\nu}\left(\cos\theta\right)$ and of its derivative for $m=0(1)4$, $\theta=10^{\circ},30^{\circ},150^{\circ}$.

• Žurina and Karmazina (1964, 1965) tabulate the conical functions $\mathsf{P}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)50$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)50$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7D. Auxiliary tables are included to facilitate computation for larger values of $\tau$ when $-1.

• Žurina and Karmazina (1963) tabulates the conical functions $\mathsf{P}^{1}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)25$, $x=-0.9(.1)0.9$, 7S; $P^{1}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)25$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7S. Auxiliary tables are included to assist computation for larger values of $\tau$ when $-1.

• ##### 16: 14.12 Integral Representations
###### §14.12(i) $-1
14.12.1 $\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{2^{1/2}(\sin\theta)^{\mu}}% {\pi^{1/2}\Gamma\left(\frac{1}{2}-\mu\right)}\int_{0}^{\theta}\frac{\cos\left(% \left(\nu+\frac{1}{2}\right)t\right)}{(\cos t-\cos\theta)^{\mu+(1/2)}}\mathrm{% d}t,$ $0<\theta<\pi$, $\Re\mu<\tfrac{1}{2}$.
14.12.2 $\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{\left(1-x^{2}\right)^{-\mu/2}}{% \Gamma\left(\mu\right)}\int_{x}^{1}\mathsf{P}_{\nu}\left(t\right)(t-x)^{\mu-1}% \mathrm{d}t,$ $\Re\mu>0$;
14.12.3 $\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{\pi^{1/2}\Gamma\left(\nu+% \mu+1\right)(\sin\theta)^{\mu}}{2^{\mu+1}\Gamma\left(\mu+\frac{1}{2}\right)% \Gamma\left(\nu-\mu+1\right)}\*\left(\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{% (\cos\theta+i\sin\theta\cosh t)^{\nu+\mu+1}}\mathrm{d}t+\int_{0}^{\infty}\frac% {(\sinh t)^{2\mu}}{(\cos\theta-i\sin\theta\cosh t)^{\nu+\mu+1}}\mathrm{d}t% \right),$ $0<\theta<\pi$, $\Re\mu>-\tfrac{1}{2}$, $\Re\nu\pm\mu>-1$.
##### 17: 14.13 Trigonometric Expansions
###### §14.13 Trigonometric Expansions
14.13.1 $\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{2^{\mu+1}(\sin\theta)^{\mu% }}{\pi^{1/2}}\*\sum_{k=0}^{\infty}\frac{\Gamma\left(\nu+\mu+k+1\right)}{\Gamma% \left(\nu+k+\frac{3}{2}\right)}\frac{{\left(\mu+\frac{1}{2}\right)_{k}}}{k!}\*% \sin\left((\nu+\mu+2k+1)\theta\right),$
14.13.2 $\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^{1/2}2^{\mu}(\sin\theta)^{% \mu}\*\sum_{k=0}^{\infty}\frac{\Gamma\left(\nu+\mu+k+1\right)}{\Gamma\left(\nu% +k+\frac{3}{2}\right)}\frac{{\left(\mu+\frac{1}{2}\right)_{k}}}{k!}\*\cos\left% ((\nu+\mu+2k+1)\theta\right).$
14.13.3 $\mathsf{P}_{n}\left(\cos\theta\right)=\frac{2^{2n+2}(n!)^{2}}{\pi(2n+1)!}\*% \sum_{k=0}^{\infty}\frac{1\cdot 3\cdots(2k-1)}{k!}\*\frac{(n+1)(n+2)\cdots(n+k% )}{(2n+3)(2n+5)\cdots(2n+2k+1)}\*\sin\left((n+2k+1)\theta\right),$
14.13.4 $\mathsf{Q}_{n}\left(\cos\theta\right)=\frac{2^{2n+1}(n!)^{2}}{(2n+1)!}\*\sum_{% k=0}^{\infty}\frac{1\cdot 3\cdots(2k-1)}{k!}\*\frac{(n+1)(n+2)\cdots(n+k)}{(2n% +3)(2n+5)\cdots(2n+2k+1)}\*\cos\left((n+2k+1)\theta\right),$
##### 18: 14.8 Behavior at Singularities
###### §14.8 Behavior at Singularities
14.8.1 $\mathsf{P}^{\mu}_{\nu}\left(x\right)\sim\frac{1}{\Gamma\left(1-\mu\right)}% \left(\frac{2}{1-x}\right)^{\mu/2},$ $\mu\neq 1,2,3,\dots$,
14.8.3 $\mathsf{Q}_{\nu}\left(x\right)=\frac{1}{2}\ln\left(\frac{2}{1-x}\right)-\gamma% -\psi\left(\nu+1\right)+O\left(1-x\right),$ $\nu\neq-1,-2,-3,\dots$,
14.8.4 $\mathsf{Q}^{\mu}_{\nu}\left(x\right)\sim\frac{1}{2}\cos\left(\mu\pi\right)% \Gamma\left(\mu\right)\left(\frac{2}{1-x}\right)^{\mu/2},$ $\mu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\dots$,
14.8.6 $\mathsf{Q}^{-\mu}_{\nu}\left(x\right)\sim\frac{\Gamma\left(\mu\right)\Gamma% \left(\nu-\mu+1\right)}{2\Gamma\left(\nu+\mu+1\right)}\left(\frac{2}{1-x}% \right)^{\mu/2},$ $\nu\pm\mu\neq-1,-2,-3,\dots$.
##### 19: 14.23 Values on the Cut
14.23.1 $P^{\mu}_{\nu}\left(x\pm i0\right)=e^{\mp\mu\pi i/2}\mathsf{P}^{\mu}_{\nu}\left% (x\right),$
14.23.2 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)=\frac{e^{\pm\mu\pi i/2}}{\Gamma% \left(\nu+\mu+1\right)}\left(\mathsf{Q}^{\mu}_{\nu}\left(x\right)\mp\tfrac{1}{% 2}\pi i\mathsf{P}^{\mu}_{\nu}\left(x\right)\right).$
14.23.4 $\mathsf{P}^{\mu}_{\nu}\left(x\right)=e^{\pm\mu\pi i/2}P^{\mu}_{\nu}\left(x\pm i% 0\right),$
14.23.5 $\mathsf{Q}^{\mu}_{\nu}\left(x\right)=\tfrac{1}{2}\Gamma\left(\nu+\mu+1\right)% \left(e^{-\mu\pi i/2}\boldsymbol{Q}^{\mu}_{\nu}\left(x+i0\right)+e^{\mu\pi i/2% }\boldsymbol{Q}^{\mu}_{\nu}\left(x-i0\right)\right),$
14.23.6 $\mathsf{Q}^{\mu}_{\nu}\left(x\right)=e^{\mp\mu\pi i/2}\Gamma\left(\nu+\mu+1% \right)\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)\pm\tfrac{1}{2}\pi ie^{% \pm\mu\pi i/2}P^{\mu}_{\nu}\left(x\pm i0\right).$
##### 20: 18.11 Relations to Other Functions
###### Ultraspherical
18.11.1 $\mathsf{P}^{m}_{n}\left(x\right)={\left(\tfrac{1}{2}\right)_{m}}(-2)^{m}(1-x^{% 2})^{\frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x\right)={\left(n+1\right)_{m% }}(-2)^{-m}(1-x^{2})^{\frac{1}{2}m}P^{(m,m)}_{n-m}\left(x\right),$ $0\leq m\leq n$.
For the Ferrers function $\mathsf{P}^{m}_{n}\left(x\right)$, see §14.3(i). …