# determinants

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## 1—10 of 31 matching pages

##### 2: 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 $r,s=0,1,\dots,n$. …
##### 3: 21.5 Modular Transformations
21.5.3 $\det\boldsymbol{{\Gamma}}=1,$
21.5.4 $\theta\left(\left[[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right]^{% \mathrm{T}}\mathbf{z}\middle|[\mathbf{A}\boldsymbol{{\Omega}}+\mathbf{B}][% \mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right)=\xi(\boldsymbol{{% \Gamma}})\sqrt{\det[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]}e^{\pi i% \mathbf{z}\cdot\left[[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\mathbf{% C}\right]\cdot\mathbf{z}}\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right).$
$\theta\left(\boldsymbol{{\Omega}}^{-1}\mathbf{z}\middle|-\boldsymbol{{\Omega}}% ^{-1}\right)=\sqrt{\det\left[-i\boldsymbol{{\Omega}}\right]}e^{\pi i\mathbf{z}% \cdot\boldsymbol{{\Omega}}^{-1}\cdot\mathbf{z}}\theta\left(\mathbf{z}\middle|% \boldsymbol{{\Omega}}\right),$
21.5.9 $\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{D}\boldsymbol{{\alpha}}-\mathbf{C}% \boldsymbol{{\beta}}+\tfrac{1}{2}\operatorname{diag}[\mathbf{C}\mathbf{D}^{% \mathrm{T}}]}{-\mathbf{B}\boldsymbol{{\alpha}}+\mathbf{A}\boldsymbol{{\beta}}+% \tfrac{1}{2}\operatorname{diag}[\mathbf{A}\mathbf{B}^{\mathrm{T}}]}\left(\left% [[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right]^{\mathrm{T}}\mathbf{% z}\middle|[\mathbf{A}\boldsymbol{{\Omega}}+\mathbf{B}][\mathbf{C}\boldsymbol{{% \Omega}}+\mathbf{D}]^{-1}\right)=\kappa(\boldsymbol{{\alpha}},\boldsymbol{{% \beta}},\boldsymbol{{\Gamma}})\sqrt{\det[\mathbf{C}\boldsymbol{{\Omega}}+% \mathbf{D}]}e^{\pi i\mathbf{z}\cdot\left[[\mathbf{C}\boldsymbol{{\Omega}}+% \mathbf{D}]^{-1}\mathbf{C}\right]\cdot\mathbf{z}}\theta\genfrac{[}{]}{0.0pt}{}% {\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle|% \boldsymbol{{\Omega}}\right),$
##### 4: 19.31 Probability Distributions
19.31.2 $\int_{{\mathbb{R}}^{n}}(\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x})^{\mu}\exp% \left(-\mathbf{x}^{\mathrm{T}}\mathbf{B}\mathbf{x}\right)\mathrm{d}x_{1}\cdots% \mathrm{d}x_{n}=\frac{\pi^{n/2}\Gamma\left(\mu+\tfrac{1}{2}n\right)}{\sqrt{% \det\mathbf{B}}\Gamma\left(\tfrac{1}{2}n\right)}R_{\mu}\left(\tfrac{1}{2},% \dots,\tfrac{1}{2};\lambda_{1},\dots,\lambda_{n}\right),$ $\mu>-\tfrac{1}{2}n$.
##### 5: 3.9 Acceleration of Convergence
3.9.9 $t_{n,2k}=\frac{H_{k+1}(s_{n})}{H_{k}(\Delta^{2}s_{n})},$ $n=0,1,2,\dots$,
where $H_{m}$ is the Hankel determinant
3.9.10 $H_{m}(u_{n})=\begin{vmatrix}u_{n}&u_{n+1}&\cdots&u_{n+m-1}\\ u_{n+1}&u_{n+2}&\cdots&u_{n+m}\\ \vdots&\vdots&\ddots&\vdots\\ u_{n+m-1}&u_{n+m}&\cdots&u_{n+2m-2}\end{vmatrix}.$
The ratio of the Hankel determinants in (3.9.9) can be computed recursively by Wynn’s epsilon algorithm: …
##### 6: 32.8 Rational Solutions
$\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 $n\times n$ determinantFor 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). …
##### 7: 1.11 Zeros of Polynomials
$D_{2}=\begin{vmatrix}a_{1}&a_{3}\\ a_{0}&a_{2}\end{vmatrix},$
$D_{3}=\begin{vmatrix}a_{1}&a_{3}&a_{5}\\ a_{0}&a_{2}&a_{4}\\ 0&a_{1}&a_{3}\end{vmatrix},$
1.11.26 $D_{k}=\det[h_{k}^{(1)},h_{k}^{(3)},\dots,h_{k}^{(2k-1)}],$
##### 8: 32.10 Special Function Solutions
where $\tau_{n}(z)$ is the $n\times n$ determinantFor determinantal representations see Forrester and Witte (2002) and Okamoto (1987c). … For determinantal representations see Forrester and Witte (2001) and Okamoto (1986). … For determinantal representations see Forrester and Witte (2002), Masuda (2004), and Okamoto (1987b). … For determinantal representations see Forrester and Witte (2004) and Masuda (2004). …
##### 9: 35.1 Special Notation
 $a,b$ complex variables. … determinant of $\mathbf{X}$ (except when $m=1$ where it means either determinant or absolute value, depending on the context). …
##### 10: 35.3 Multivariate Gamma and Beta Functions
35.3.1 $\Gamma_{m}\left(a\right)=\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{% X}\right)\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right)>\frac{1}{2}(m-1)$.
35.3.2 $\Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}\mathrm{etr% }\left(-\mathbf{X}\right)\left|\mathbf{X}\right|^{s_{m}-\frac{1}{2}(m+1)}\prod% _{j=1}^{m-1}|(\mathbf{X})_{j}|^{s_{j}-s_{j+1}}\mathrm{d}{\mathbf{X}},$ $s_{j}\in\mathbb{C}$, $\Re\left(s_{j}\right)>\frac{1}{2}(j-1)$, $j=1,\dots,m$.
35.3.3 $\mathrm{B}_{m}\left(a,b\right)=\int\limits_{\boldsymbol{{0}}<\mathbf{X}<% \mathbf{I}}\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}-% \mathbf{X}\right|^{b-\frac{1}{2}(m+1)}\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.
35.3.8 $\mathrm{B}_{m}\left(a,b\right)=\int_{\boldsymbol{\Omega}}\left|\mathbf{X}% \right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}+\mathbf{X}\right|^{-(a+b)}\mathrm% {d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.