Ferrers functions

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41: 10.22 Integrals

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10.22.71 $${\int }_0^{\mathrm{\infty }}{J}_{\mu }\left(at\right)J_{\nu }\left(bt\right)J_{\nu }\left(ct\right)t^{1-\mu }dt=\frac{{(bc)}^{\mu -1}{(\sin \varphi )}^{\mu -\frac{1}{2}}}{{(2\pi )}^{\frac{1}{2}}{a}^{\mu }}{𝖯}_{\nu -\frac{1}{2}}^{\frac{1}{2}-\mu }\left(\cos \varphi \right),$$ .

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

§10.22(iv), §10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E71

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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

42: Errata

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Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, ${𝖯}_n,{𝖰}_n,P_n,Q_n,{𝑸}_n$ and the Laguerre polynomial, $L_n$, were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

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The constraints in (14.5.3), (14.5.4) on $\nu +\mu$ have been corrected to exclude all negative integers since the Ferrers function of the second kind is not defined for these values.

Reported by Hans Volkmer on 2021-06-02

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

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

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

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A new Subsection Addendum to §14.5(ii) $\mu =0$, $\nu =2$ , containing the values of Legendre and Ferrers functions for degree $\nu =2$ has been added.

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43: 14.21 Definitions and Basic Properties

… ►The generating function expansions (14.7.19) (with $𝖯$ replaced by $P$) and (14.7.22) apply when ; (14.7.21) (with $𝖯$ replaced by $P$) applies when $|h|>\max \left|z\pm {\left(z^2-1\right)}^{1/2}\right|$.

44: Software Index

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	Open Source													With Book						Commercial
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14.34(ii) ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(x\right)$, $Q_{\nu }\left(x\right)$, $x,\nu \in ℝ$	✓				✓	✓					✓	a	✓	✓		✓	✓		✓		✓	✓	✓	✓
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

45: 14.14 Continued Fractions

§14.14 Continued Fractions

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14.14.1 $$\frac{1}{2}{\left(x^2-1\right)}^{1/2}\frac{P_{\nu }^{\mu }\left(x\right)}{P_{\nu }^{\mu -1}\left(x\right)}=\frac{x_0}{y_0+}\frac{x_1}{y_1+}\frac{x_2}{y_2+}\mathrm{\cdots },$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.14 and Ch.14

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14.14.3 $$(\nu -\mu )\frac{Q_{\nu }^{\mu }\left(x\right)}{Q_{\nu -1}^{\mu }\left(x\right)}=\frac{x_0}{y_0-}\frac{x_1}{y_1-}\frac{x_2}{y_2-}\mathrm{\cdots },$$ $\nu \ne \mu$,

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Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.14 and Ch.14

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46: 26.9 Integer Partitions: Restricted Number and Part Size

… ► $p_k\left(n\right)$ denotes the number of partitions of $n$ into at most $k$ parts. … … ►A useful representation for a partition is the Ferrers graph in which the integers in the partition are each represented by a row of dots. … ►The conjugate partition is obtained by reflecting the Ferrers graph across the main diagonal or, equivalently, by representing each integer by a column of dots. … ►

§26.9(ii) Generating Functions

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47: 18.41 Tables

… ►For $P_n\left(x\right)$ ($={𝖯}_n\left(x\right)$) see §14.33. …

48: 26.15 Permutations: Matrix Notation

… ►The rook polynomial is the generating function for $r_j(B)$: … ► $N(x,B)$ is the generating function: … ►

Example 2

►The Ferrers board of shape $(b_1,b_2,\mathrm{\dots },b_n)$, $0\le b_1\le b_2\le \mathrm{\cdots }\le b_n$, is the set $B=\{(j,k)|\mathrm{\hspace{0.25em}1}\le j\le n,1\le k\le b_j\}$. …If $B$ is the Ferrers board of shape $(0,1,2,\mathrm{\dots },n-1)$, then …