Ferrers functions

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31: 30.5 Functions of the Second Kind

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30.5.3 $${\mathrm{𝖰𝗌}}_n^m(x,0)={𝖰}_n^m\left(x\right);$$

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Symbols:

: Ferrers function of the second kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer and $n\ge m$: integer degree

Permalink:

http://dlmf.nist.gov/30.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

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32: 30.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ and the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$, and $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$. …Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathrm{𝖯𝗌}$, $\mathrm{𝖰𝗌}$, $\mathrm{𝑃𝑠}$, $\mathrm{𝑄𝑠}$, respectively. ►

Other Notations

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33: Bibliography V

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Fourier series representation of Ferrers function $𝖯$ .

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External Links:

FullText

Referenced by:

(14.13.1).

Encodings:

BibTeX

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34: 10.43 Integrals

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10.43.22

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Keywords:

Laplace transform, Mellin transform

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.43.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

… ►For the Ferrers function $𝖯$ and the associated Legendre function $P$, see §§14.3(i) and 14.21(i). …

35: 30.16 Methods of Computation

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30.16.9 $${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)=\underset{d\to \mathrm{\infty }}{lim}\sum _{j=1}^d{(-1)}^{j-p}{e}_{j,d}{𝖯}_{n+2(j-p)}^m\left(x\right).$$

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Symbols:

: Ferrers function of the first kind, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.16.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.16(ii), §30.16 and Ch.30

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36: 18.3 Definitions

… ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …

37: Mathematical Introduction

… ►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). …

38: 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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39: 14.34 Software

… ►A more complete list of available software for computing these functions is found in the Software Index. For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C9). ►

§14.34(ii) Legendre Functions: Real Argument and Parameters

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§14.34(iii) Legendre Functions: Complex Argument and/or Parameters

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§14.34(iv) Conical (Mehler) and/or Toroidal Functions

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40: 18.17 Integrals

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18.17.7 $${\left(P_n\left(x\right)\right)}^2+4{\pi }^{-2}{\left({𝖰}_n\left(x\right)\right)}^2=4{\pi }^{-2}{\int }_1^{\mathrm{\infty }}{Q}_n\left(x^2+(1-x^2)t\right){(t^2-1)}^{-\frac{1}{2}}dt,$$ .

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $t$: real variable, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(iii), §18.17(iii), §18.17 and Ch.18

►For the Ferrers function ${𝖰}_n\left(x\right)$ and Legendre function $Q_n\left(x\right)$ see §§14.3(i) and 14.3(ii), with $\mu =0$ and $\nu =n$. …