Ferrers graph

SearchAdvancedHelp

(0.002 seconds)

6 matching pages

1: 26.9 Integer Partitions: Restricted Number and Part Size

… ►

Table 26.9.1: Partitions $p_k\left(n\right)$.

► ►►

$n$	$k$
	…

►

►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. … ►

…

Figure 26.9.1: Ferrers graph of the partition $7+4+3+3+2+1$.

►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. …

2: 14.4 Graphics

… ►

§14.4(i) Ferrers Functions: 2D Graphs

…

3: 14.20 Conical (or Mehler) Functions

… ►Another real-valued solution ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ of (14.20.1) was introduced in Dunster (1991). …${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ exists except when $\mu =\frac{1}{2},\frac{3}{2},\mathrm{\dots }$ and $\tau =0$; compare §14.3(i). …provided that ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ exists. … ►Approximations for ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ and ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ can then be achieved via (14.9.7) and (14.20.3). … ►For zeros of ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ see Hobson (1931, §237). …

4: Bibliography C

… ►

Work Group of Computational Mathematics, University of Kassel, Germany.

ⓘ

Notes:

Computer Algebra and Orthogonal Polynomials: a Web service using Maple for online generation of formulas and graphs of orthogonal polynomials belonging to the Askey scheme. For further information see Swarttouw (1997) and Koepf (1999).

External Links:

CAOP

Referenced by:

1st item.

Encodings:

BibTeX

… ►

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

ⓘ

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

►

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

ⓘ

External Links:

Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(14.13.2), §14.3(iii), §14.3(iii).

Encodings:

BibTeX

… ►

Department of Computer Science, University of Victoria, Canada.

ⓘ

Notes:

Permutations, set and integer partitions, sequences, graphs, and trees.

External Links:

Combinatorial Object Server

Referenced by:

1st item.

Encodings:

BibTeX

…

5: 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). … ►Special functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs. …

6: Errata

… ►

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

… ►

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

… ►

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 ${𝖯}_{\nu }\left(x\right)$.

… ►

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

… ►

The captions for these figures have been corrected to read, in part, “as a function of $k^2=\mathrm{i}{\kappa }^2$” (instead of $k^2=\mathrm{i}\kappa$). Also, the resolution of the graph in Figure 22.3.22 was improved near $\kappa =3$.

Reported 2011-10-30 by Paul Abbott.

…