Cauchy determinant

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1: 1.3 Determinants, Linear Operators, and Spectral Expansions

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§1.3(i) Determinants: Elementary Properties

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Relationships Between Determinants

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§1.3(ii) Special Determinants

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Vandermonde Determinant or Vandermondian

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Cauchy Determinant

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4: 24.14 Sums

… ►These identities can be regarded as higher-order recurrences. Let

\det[a_{r+s}]

denote a Hankel (or persymmetric) determinant, that is, an

(n+1)\times(n+1)

determinant with element

a_{r+s}

in row

r

and column

s

for

r,s=0,1,\dots,n

. … ►

24.14.11

\det[B_{r+s}]=(-1)^{n(n+1)/2}\left(\prod_{k=1}^{n}k!\right)^{6}\Bigg{/}\left(% \prod_{k=1}^{2n+1}k!\right),

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\det$ : determinant, $!$ : factorial (as in $n!$ ), $k$ : integer, $n$ : integer, $r$ : row and $s$ : column
Referenced by:: §24.14(ii)
Permalink:: http://dlmf.nist.gov/24.14.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

►

24.14.12

\det[E_{r+s}]=(-1)^{n(n+1)/2}\left(\prod_{k=1}^{n}k!\right)^{2}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\det$ : determinant, $!$ : factorial (as in $n!$ ), $k$ : integer, $n$ : integer, $r$ : row and $s$ : column
Referenced by:: §24.14(ii)
Permalink:: http://dlmf.nist.gov/24.14.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

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5: 21.5 Modular Transformations

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21.5.3

\det\boldsymbol{{\Gamma}}=1,

ⓘ

Symbols:: $\det$ : determinant and Matrix $\boldsymbol{{\Gamma}}$
Permalink:: http://dlmf.nist.gov/21.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §21.5(i), §21.5 and Ch.21

… ►

21.5.4

\theta\left(\left[[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right]^{% \mathrm{T}}\mathbf{z}\middle|[\mathbf{A}\boldsymbol{{\Omega}}+\mathbf{B}][% \mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right)=\xi(\boldsymbol{{% \Gamma}})\sqrt{\det[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]}e^{\pi i% \mathbf{z}\cdot\left[[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\mathbf{% C}\right]\cdot\mathbf{z}}\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right).

ⓘ

Symbols:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $\boldsymbol{{\Omega}}$ : a Riemann matrix, Matrix $\boldsymbol{{\Gamma}}$ and $\xi(\boldsymbol{{\Gamma}})$ : eighth root of unity
Referenced by:: §21.5(i)
Permalink:: http://dlmf.nist.gov/21.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §21.5(i), §21.5 and Ch.21

… ►

\theta\left(\boldsymbol{{\Omega}}^{-1}\mathbf{z}\middle|-\boldsymbol{{\Omega}}% ^{-1}\right)=\sqrt{\det\left[-i\boldsymbol{{\Omega}}\right]}e^{\pi i\mathbf{z}% \cdot\boldsymbol{{\Omega}}^{-1}\cdot\mathbf{z}}\theta\left(\mathbf{z}\middle|% \boldsymbol{{\Omega}}\right),

… ►

21.5.9

\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{D}\boldsymbol{{\alpha}}-\mathbf{C}% \boldsymbol{{\beta}}+\tfrac{1}{2}\operatorname{diag}[\mathbf{C}\mathbf{D}^{% \mathrm{T}}]}{-\mathbf{B}\boldsymbol{{\alpha}}+\mathbf{A}\boldsymbol{{\beta}}+% \tfrac{1}{2}\operatorname{diag}[\mathbf{A}\mathbf{B}^{\mathrm{T}}]}\left(\left% [[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right]^{\mathrm{T}}\mathbf{% z}\middle|[\mathbf{A}\boldsymbol{{\Omega}}+\mathbf{B}][\mathbf{C}\boldsymbol{{% \Omega}}+\mathbf{D}]^{-1}\right)=\kappa(\boldsymbol{{\alpha}},\boldsymbol{{% \beta}},\boldsymbol{{\Gamma}})\sqrt{\det[\mathbf{C}\boldsymbol{{\Omega}}+% \mathbf{D}]}e^{\pi i\mathbf{z}\cdot\left[[\mathbf{C}\boldsymbol{{\Omega}}+% \mathbf{D}]^{-1}\mathbf{C}\right]\cdot\mathbf{z}}\theta\genfrac{[}{]}{0.0pt}{}% {\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle|% \boldsymbol{{\Omega}}\right),

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector, $\operatorname{diag}\mathbf{A}$ : Transpose of $[A_{11},A_{22},\ldots,A_{gg}]$ and Matrix $\boldsymbol{{\Gamma}}$
Permalink:: http://dlmf.nist.gov/21.5.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §21.5(ii), §21.5 and Ch.21

…

$x, y$	real variables.
…
$\det(\mathbf{A})$	determinant of the square matrix $\mathbf{A}$
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7: 19.31 Probability Distributions

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19.31.2

\int_{{\mathbb{R}}^{n}}(\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x})^{\mu}\exp% \left(-\mathbf{x}^{\mathrm{T}}\mathbf{B}\mathbf{x}\right)\,\mathrm{d}x_{1}% \cdots\,\mathrm{d}x_{n}=\frac{\pi^{n/2}\Gamma\left(\mu+\tfrac{1}{2}n\right)}{% \sqrt{\det\mathbf{B}}\Gamma\left(\tfrac{1}{2}n\right)}R_{\mu}\left(\tfrac{1}{2% },\dots,\tfrac{1}{2};\lambda_{1},\dots,\lambda_{n}\right),

\mu>-\tfrac{1}{2}n

ⓘ

Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\mathbb{R}$ : real line, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $n$ : nonnegative integer and $\lambda_{n}$ : eigenvalues
Referenced by:: §19.31
Permalink:: http://dlmf.nist.gov/19.31.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.31 and Ch.19

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8: 3.9 Acceleration of Convergence

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3.9.9

t_{n,2k}=\frac{H_{k+1}(s_{n})}{H_{k}(\Delta^{2}s_{n})},

n=0,1,2,\dots

ⓘ

Symbols:: $s_{n}$ : sequence, $t_{n}$ : sequence and $H_{m}(u)$ : Hankel determinant
Referenced by:: §3.9(iv)
Permalink:: http://dlmf.nist.gov/3.9.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §3.9(iv), §3.9 and Ch.3

►where

H_{m}

is the Hankel determinant ►

3.9.10

H_{m}(u_{n})=\begin{vmatrix}u_{n}&u_{n+1}&\cdots&u_{n+m-1}\\ u_{n+1}&u_{n+2}&\cdots&u_{n+m}\\ \vdots&\vdots&\ddots&\vdots\\ u_{n+m-1}&u_{n+m}&\cdots&u_{n+2m-2}\end{vmatrix}.

ⓘ

Defines:: $H_{m}(u)$ : Hankel determinant (locally)
Symbols:: $\det$ : determinant
Permalink:: http://dlmf.nist.gov/3.9.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §3.9(iv), §3.9 and Ch.3

►The ratio of the Hankel determinants in (3.9.9) can be computed recursively by Wynn’s epsilon algorithm: …

9: 17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions

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Cauchy’s Sum

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10: 32.8 Rational Solutions

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\mbox{P}_{\mbox{\scriptsize II}}

–

\mbox{P}_{\mbox{\scriptsize VI}}

possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. … ►where

\tau_{n}(z)

is the

n\times n

Wronskian determinant … ►For determinantal representations see Kajiwara and Masuda (1999). … ►For determinantal representations see Kajiwara and Ohta (1998) and Noumi and Yamada (1999). … ►For determinantal representations see Masuda et al. (2002). …

Cauchy determinant

1—10 of 68 matching pages

1: 1.3 Determinants, Linear Operators, and Spectral Expansions

§1.3(i) Determinants: Elementary Properties

Relationships Between Determinants

§1.3(ii) Special Determinants

Vandermonde Determinant or Vandermondian

Cauchy Determinant

2: 1.2 Elementary Algebra

The Determinant

Relation of Eigenvalues to the Determinant and Trace

3: 1.7 Inequalities

Cauchy–Schwarz Inequality

Cauchy–Schwarz Inequality

4: 24.14 Sums

5: 21.5 Modular Transformations

6: 1.1 Special Notation

7: 19.31 Probability Distributions

8: 3.9 Acceleration of Convergence

9: 17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions

Cauchy’s Sum

10: 32.8 Rational Solutions

Cauchy determinant

1—10 of 68 matching pages

1: 1.3 Determinants, Linear Operators, and Spectral Expansions

§1.3(i) Determinants: Elementary Properties

Relationships Between Determinants

§1.3(ii) Special Determinants

Vandermonde Determinant or Vandermondian

Cauchy Determinant

2: 1.2 Elementary Algebra

The Determinant

Relation of Eigenvalues to the Determinant and Trace

3: 1.7 Inequalities

Cauchy–Schwarz Inequality

Cauchy–Schwarz Inequality

4: 24.14 Sums

5: 21.5 Modular Transformations

6: 1.1 Special Notation

7: 19.31 Probability Distributions

8: 3.9 Acceleration of Convergence

9: 17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions

Cauchy’s Sum

10: 32.8 Rational Solutions

9: 17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions