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1: 26.15 Permutations: Matrix Notation

§26.15 Permutations: Matrix Notation

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2: 35.1 Special Notation

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$a,b$	complex variables.
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… ►Related notations for the Bessel functions are ${𝒥}_{\nu +\frac{1}{2}(m+1)}(𝐓)=A_{\nu }\left(𝐓\right)/A_{\nu }\left(𝟎\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_m(0,\mathrm{\dots },0,\nu |𝐒,𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Terras (1988, pp. 49–64)), and ${𝒦}_{\nu }(𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Faraut and Korányi (1994, pp. 357–358)).

3: 21.1 Special Notation

§21.1 Special Notation

►(For other notation see Notation for the Special Functions.) ► ►►►►

$g,h$	positive integers.
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$𝛀$	$g\times g$ complex, symmetric matrix with $\mathrm{\Im }𝛀$ strictly positive definite, i.e., a Riemann matrix.
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${\mathrm{𝟎}}_g$	$g\times g$ zero matrix.
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… ►Uppercase boldface letters are $g\times g$ real or complex matrices. …

4: 1.2 Elementary Algebra

… ►The full index form of an $m\times n$ matrix $𝐀$ is …

5: 32.8 Rational Solutions

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32.8.10 $${\tau }_n(z)=𝒲\left\{p_1(z),p_3(z),\mathrm{\dots },p_{2n-1}(z)\right\}.$$

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Symbols:

$𝒲$: Wronskian, $n$: integer, $z$: real, $p_m(z)$: polynomials and ${\tau }_n(z)$: determinant

Referenced by:

§32.8(ii), Erratum (V1.2.0) for Equations (32.8.10), (32.10.9)

Permalink:

http://dlmf.nist.gov/32.8.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.2.0):

The right-hand side of this equation, which was originally written as a matrix determinant, was rewritten using the Wronskian determinant notation.

See also:

Annotations for §32.8(ii), §32.8 and Ch.32

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6: 32.10 Special Function Solutions

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32.10.9 $${\tau }_n(z)=𝒲\left\{\varphi (z),{\varphi }^{\prime }(z),\mathrm{\dots },{\varphi }^{(n-1)}(z)\right\},$$

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Symbols:

$𝒲$: Wronskian, $n$: integer, $z$: real, $\varphi (z)$: function and ${\tau }_n(z)$: determinant

Referenced by:

§32.10(ii), Erratum (V1.2.0) for Equations (32.8.10), (32.10.9)

Permalink:

http://dlmf.nist.gov/32.10.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.2.0):

The right-hand side of this equation, which was originally written as a matrix determinant, was rewritten using the Wronskian determinant notation.

See also:

Annotations for §32.10(ii), §32.10 and Ch.32

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7: Bibliography J

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Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 (2), pp. 475–501.

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External Links:

FullText, MathReview (I. Olkin), ZentralBlatt

Referenced by:

§35.11, §35.4(i), §35.4(ii), §35.8(i), §35.8(ii), §35.8(iv), §35.9, Ch.35.

Encodings:

BibTeX

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Simple-periodic and Non-periodic Lamé Functions. Mathematical Centre Tracts, No. 72, Mathematisch Centrum, Amsterdam.

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External Links:

ISBN 90-6196-130-0, MathReview (V. P. Madan)

Referenced by:

§29.1, §29.1, §29.18(i), §29.19(i), §29.20(i), §29.20(i), §29.22(ii), §29.3(v), §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), Ch.29, Notations L, Notations L, Notations L, Notations L.

Encodings:

BibTeX

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Methods of Mathematical Physics. 3rd edition, Cambridge University Press, Cambridge.

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Notes:

Reprinted in 1999.

External Links:

MathReview, ZentralBlatt

Referenced by:

§10.1, §12.17, Ch.14, Ch.6, §7.18(iv), Ch.9, Notations H, Notations H, Notations K.

Encodings:

BibTeX

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Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. D 1 (1), pp. 80–158.

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External Links:

ISSN 0167-2789, Document, MathReview, ZentralBlatt

Referenced by:

§32.16.

Encodings:

BibTeX

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Calculus of Finite Differences. Hungarian Agent Eggenberger Book-Shop, Budapest.

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Notes:

Third edition published in 1965 by Chelsea Publishing Co., New York, and American Mathematical Society, AMS Chelsea Book Series, Providence, RI

External Links:

MathReview (W. E. Milne), ZentralBlatt

Referenced by:

§26.1, §26.1, §26.8(vii), §5.16, Notations S.

Encodings:

BibTeX

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8: 1.3 Determinants, Linear Operators, and Spectral Expansions

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1.3.5 $$det\left({𝐀}^{\mathrm{T}}\right)=det(𝐀),$$

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Symbols:

$det$: determinant, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $j$: integer and $k$: integer

Referenced by:

Erratum (V1.2.0) for Equations (1.3.5), (1.3.6), (1.3.7)

Permalink:

http://dlmf.nist.gov/1.3.E5

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

Previously we did use the notation $[a_{jk}]$ for $𝐀$.

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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1.3.6 $$det\left({𝐀}^{-1}\right)=\frac{1}{det(𝐀)},$$

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Symbols:

$det$: determinant, ${𝐀}^{-1}$: matrix inverse, $j$: integer and $k$: integer

Referenced by:

Erratum (V1.2.0) for Equations (1.3.5), (1.3.6), (1.3.7)

Permalink:

http://dlmf.nist.gov/1.3.E6

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

Previously we did use the notation $[a_{jk}]$ for $𝐀$.

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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9: Errata

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1.3.5 $$det\left({𝐀}^{\mathrm{T}}\right)=det(𝐀)$$

1.3.6 $$det\left({𝐀}^{-1}\right)=\frac{1}{det(𝐀)}$$

1.3.7 $$det(𝐀𝐁)=det(𝐀)det(𝐁)$$

Previously we used the notation $[a_{jk}]$, $[b_{jk}]$, for $𝐀$, $𝐁$ respectively.

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32.8.10 $${\tau }_n(z)=𝒲\left\{p_1(z),p_3(z),\mathrm{\dots },p_{2n-1}(z)\right\}$$

32.10.9 $${\tau }_n(z)=𝒲\left\{\varphi (z),{\varphi }^{\prime }(z),\mathrm{\dots },{\varphi }^{(n-1)}(z)\right\}$$

The right-hand side of these equation, which was originally written as a matrix determinant, was rewritten using the Wronskian determinant notation. Also, in each preceding sentence, the word ‘determinant’ was replaced with ‘Wronskian determinant’.

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10: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►Let $\alpha =-n$, $n=0,1,2,\mathrm{\dots }$, and $q_{n,m}$, $m=0,1,\mathrm{\dots },n$, be the eigenvalues of the tridiagonal matrix … ►

31.5.2 $${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

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Defines:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.5 and Ch.31

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