Ferrers board

(0.001 seconds)

11—20 of 62 matching pages

12: 14.10 Recurrence Relations and Derivatives

§14.10 Recurrence Relations and Derivatives

►

14.10.1

{\mathsf{P}^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}% \mathsf{P}^{\mu+1}_{\nu}\left(x\right)}+(\nu-\mu)(\nu+\mu+1)\mathsf{P}^{\mu}_{% \nu}\left(x\right)=0,

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.10
Permalink:: http://dlmf.nist.gov/14.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

►

14.10.2

{\left(1-x^{2}\right)^{1/2}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)% \mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)x\mathsf{P}^{\mu}_{\nu}% \left(x\right)=0,

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

►

14.10.3

{(\nu-\mu+2)\mathsf{P}^{\mu}_{\nu+2}\left(x\right)-(2\nu+3)x\mathsf{P}^{\mu}_{% \nu+1}\left(x\right)}+(\nu+\mu+1)\mathsf{P}^{\mu}_{\nu}\left(x\right)=0,

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.5.3 (modified)
Referenced by:: §14.10, §14.14, §14.21(iii)
Permalink:: http://dlmf.nist.gov/14.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

… ►

\mathsf{Q}^{\mu}_{\nu}\left(x\right)

also satisfies (14.10.1)–(14.10.5). …

13: 14.11 Derivatives with Respect to Degree or Order

§14.11 Derivatives with Respect to Degree or Order

… ►

14.11.4

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

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.11, §14.11
Permalink:: http://dlmf.nist.gov/14.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §14.11 and Ch.14

►

14.11.5

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

ⓘ

Symbols:: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.11
Permalink:: http://dlmf.nist.gov/14.11.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §14.11 and Ch.14

►(14.11.1) holds if

\mathsf{P}^{\mu}_{\nu}\left(x\right)

is replaced by

P^{\mu}_{\nu}\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

\mathsf{P}^{\mu}_{\nu}\left(x\right)

\mathsf{P}_{\nu}\left(x\right)

, and

\mathsf{Q}_{\nu}\left(x\right)

are replaced by

P^{\mu}_{\nu}\left(x\right)

P_{\nu}\left(x\right)

, and

Q_{\nu}\left(x\right)

, respectively. …

14: 14.9 Connection Formulas

… ►

§14.9(i) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{P}^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\mathsf{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right)$

… ►

\mathsf{P}^{\mu}_{-\nu-1}\left(x\right)=\mathsf{P}^{\mu}_{\nu}\left(x\right),

►

\mathsf{P}^{-\mu}_{-\nu-1}\left(x\right)=\mathsf{P}^{-\mu}_{\nu}\left(x\right),

►

14.9.6

\pi\cos\left(\nu\pi\right)\cos\left(\mu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x% \right)=\sin\left((\nu+\mu)\pi\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\sin% \left((\nu-\mu)\pi\right)\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right).

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §14.9(i), §14.9 and Ch.14

►

§14.9(ii) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{-\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$

…

16: 14.7 Integer Degree and Order

§14.7 Integer Degree and Order

… ►

§14.7(ii) Rodrigues-Type Formulas

… ►

§14.7(iii) Reflection Formulas

… ►

§14.7(iv) Generating Functions

… ►

Abramowitz and Stegun (1964, Chapter 8) tabulates $\mathsf{P}_{n}\left(x\right)$ for $n=0(1)3,9,10$ , $x=0(.01)1$ , 5–8D; $\mathsf{P}_{n}'\left(x\right)$ for $n=1(1)4,9,10$ , $x=0(.01)1$ , 5–7D; $\mathsf{Q}_{n}\left(x\right)$ and $\mathsf{Q}_{n}'\left(x\right)$ for $n=0(1)3,9,10$ , $x=0(.01)1$ , 6–8D; $P_{n}\left(x\right)$ and $P_{n}'\left(x\right)$ for $n=0(1)5,9,10$ , $x=1(.2)10$ , 6S; $Q_{n}\left(x\right)$ and $Q_{n}'\left(x\right)$ for $n=0(1)3,9,10$ , $x=1(.2)10$ , 6S. (Here primes denote derivatives with respect to $x$ .)

►

•

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 $m=0(1)4$ , $\theta=10^{\circ},30^{\circ},150^{\circ}$ .

►

•

Belousov (1962) tabulates $\mathsf{P}^{m}_{n}\left(\cos\theta\right)$ (normalized) for $m=0(1)36$ , $n-m=0(1)56$ , $\theta=0(2.5^{\circ})90^{\circ}$ , 6D.

►

•

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 $-1<x<1$ .

…

18: 14.18 Sums

§14.18 Sums

… ►

§14.18(ii) Addition Theorems

… ►

14.18.3

\mathsf{Q}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2% }\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{Q}_{\nu}% \left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}% \left(\cos\theta_{1}\right)\mathsf{Q}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos% \left(m\phi\right).

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\cos\NVar{z}$ : cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\nu$ : general degree, $\theta_{1}$ , $\theta_{2}$ : real and $\phi$ : real
Referenced by:: §14.18(ii), Ch.14
Permalink:: http://dlmf.nist.gov/14.18.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §14.18(ii), §14.18 and Ch.14

… ►The formulas are also valid with the Ferrers functions as in §14.3(i) with

\mu=0

. … ►

19: Nico M. Temme

… ►He has served on the editorial boards of the SIAM Journal on Mathematical Analysis, Mathematics of Computation, ZAMP, and Integral Transforms and Special Functions. …

20: Roderick S. C. Wong

… ►Wong has published numerous papers in international journals and is serving on the Editorial Boards of several journals. …

Ferrers board

11—20 of 62 matching pages

11: 14.17 Integrals

§14.17(i) Indefinite Integrals

§14.17(ii) Barnes’ Integral

§14.17(iii) Orthogonality Properties

§14.17(v) Laplace Transforms

12: 14.10 Recurrence Relations and Derivatives

§14.10 Recurrence Relations and Derivatives

13: 14.11 Derivatives with Respect to Degree or Order

§14.11 Derivatives with Respect to Degree or Order