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11: Staff

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Howard S. Cohl, Technical Editor, NIST

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William P. Reinhardt, University of Washington, Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23

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William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23

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12: How to Cite

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NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.0 of 2024-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

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13: 18.42 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3). …

14: Bibliography C

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Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).

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External Links:

Link, ISSN 2073-8994, Document, ZentralBlatt

Referenced by:

§14.6(ii), §14.6(ii), Erratum (V1.1.1) for Subsection 14.6(ii).

Encodings:

BibTeX

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Gauss hypergeometric representations of the Ferrers function of the second kind. SIGMA Symmetry Integrability Geom. Methods Appl. 17, pp. Paper 053, 33.

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External Links:

Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(14.13.2), §14.3(iii), §14.3(iii).

Encodings:

BibTeX

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Derivatives with respect to the degree and order of associated Legendre functions for $|z|>1$ using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Dmitriy Karp), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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On parameter differentiation for integral representations of associated Legendre functions. SIGMA Symmetry Integrability Geom. Methods Appl. 7, pp. Paper 050, 16.

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External Links:

ISSN 1815-0659, Document, FullText, MathReview (Michael Doschoris), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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On a generalization of the generating function for Gegenbauer polynomials. Integral Transforms Spec. Funct. 24 (10), pp. 807–816.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.28(ii), §14.28(ii).

Encodings:

BibTeX

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15: 14.6 Integer Order

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14.6.6 $${𝖯}_{\nu }^{-m}\left(x\right)={\left(1-x^2\right)}^{-m/2}{\int }_x^1\mathrm{\dots }{\int }_x^1{𝖯}_{\nu }\left(x\right){\left(dx\right)}^m.$$

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Symbols:

: Ferrers function of the first kind, $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.12(i), Erratum (V1.0.28) for Equation (14.6.6)

Permalink:

http://dlmf.nist.gov/14.6.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

The right-hand side has been corrected by replacing the Legendre function $P_{\nu }\left(x\right)$ with the Ferrers function ${𝖯}_{\nu }\left(x\right)$.

See also:

Annotations for §14.6(ii), §14.6 and Ch.14

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14.6.7 $$P_{\nu }^{-m}\left(x\right)={\left(x^2-1\right)}^{-m/2}{\int }_1^x\mathrm{\dots }{\int }_1^x{P}_{\nu }\left(x\right){\left(dx\right)}^m,$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$ , $\int$: integral, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

A&S Ref:

8.14.17

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.6(ii), §14.6 and Ch.14

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14.6.8 $$Q_{\nu }^{-m}\left(x\right)={(-1)}^m{\left(x^2-1\right)}^{-m/2}{\int }_x^{\mathrm{\infty }}\mathrm{\dots }{\int }_x^{\mathrm{\infty }}{Q}_{\nu }\left(x\right){\left(dx\right)}^m.$$

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Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $dx$: differential of $x$ , $\int$: integral, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

A&S Ref:

8.14.18

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §14.6(ii), §14.6 and Ch.14

… ►For generalizations see Cohl and Costas-Santos (2020). …

16: 14.28 Sums

… ►1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively. …

17: DLMF Project News

error generating summary

18: About the Project

… ► Cohl as Technical Editor, and Marje A. …

19: 14.11 Derivatives with Respect to Degree or Order

… ►See also Szmytkowski (2006, 2009, 2011, 2012), Cohl (2010, 2011) and Magnus et al. (1966, pp. 177–178).

20: 14.13 Trigonometric Expansions

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14.13.2 $${𝖰}_{\nu }^{\mu }\left(\cos \theta \right)={\pi }^{1/2}{2}^{\mu }{(\sin \theta )}^{\mu }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\nu +\mu +k+1\right)}{\mathrm{\Gamma }\left(\nu +k+\frac{3}{2}\right)}\frac{{\left(\mu +\frac{1}{2}\right)}_k}{k!}\cos \left((\nu +\mu +2k+1)\theta \right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : variable

Source:

Cohl et al. (2021, Theorem 6.2)

A&S Ref:

8.7.2

Referenced by:

§14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2)

Permalink:

http://dlmf.nist.gov/14.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

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