determinant formula

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2: 1.3 Determinants, Linear Operators, and Spectral Expansions

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Krattenthaler’s Formula

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3: Bibliography M

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T. Masuda, Y. Ohta, and K. Kajiwara (2002) A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.

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External Links:: ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(v).
Encodings:: BibTeX

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T. Masuda (2003) On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46 (1), pp. 121–171.

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External Links:: ISSN 0532-8721, FullText, MathReview (Andrew Pickering), ZentralBlatt
Referenced by:: §32.9(iii).
Encodings:: BibTeX

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5: Bibliography B

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W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.

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External Links:: ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.

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External Links:: Document, MathReview (T. M. Apostol), ZentralBlatt
Referenced by:: §24.16(ii).
Encodings:: BibTeX

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B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §24.16(iii), §24.8(ii).
Encodings:: BibTeX

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M. V. Berry and J. P. Keating (1992) A new asymptotic representation for

\zeta(\frac{1}{2}+it)

and quantum spectral determinants. Proc. Roy. Soc. London Ser. A 437, pp. 151–173.

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External Links:: ISSN 0962-8444, MathReview (Jack Quine), ZentralBlatt
Referenced by:: §25.9.
Encodings:: BibTeX

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R. Bo and R. Wong (1999) A uniform asymptotic formula for orthogonal polynomials associated with

\exp(-x^{4})

. J. Approx. Theory 98, pp. 146–166.

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External Links:: ISSN 0021-9045, Document, MathReview (V. L. Deshpande), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

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6: 23.10 Addition Theorems and Other Identities

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23.10.5

\begin{vmatrix}1&\wp\left(u\right)&\wp'\left(u\right)\\ 1&\wp\left(v\right)&\wp'\left(v\right)\\ 1&\wp\left(w\right)&\wp'\left(w\right)\end{vmatrix}=0,

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Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\det$ : determinant, $\mathbb{L}$ : lattice and $w=-u-v$
Referenced by:: §22.8(iii)
Permalink:: http://dlmf.nist.gov/23.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(i), §23.10 and Ch.23

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§23.10(ii) Duplication Formulas

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§23.10(iii) $n$ -Tuple Formulas

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7: Bibliography V

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A. J. van der Poorten (1980) Some Wonderful Formulas

\ldots

an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.

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External Links:: MathReview (F. Beukers), ZentralBlatt
Referenced by:: (25.6.10), (25.6.8), (25.6.9), §25.6(i), §25.6.
Encodings:: BibTeX

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R. Vein and P. Dale (1999) Determinants and Their Applications in Mathematical Physics. Applied Mathematical Sciences, Vol. 134, Springer-Verlag, New York.

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External Links:: ISBN 0-387-98558-1, MathReview (Jorma Kaarlo Merikoski), ZentralBlatt
Referenced by:: §1.3(ii).
Encodings:: BibTeX

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A. Verma and V. K. Jain (1983) Certain summation formulae for

q

-series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).

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External Links:: ISSN 0019-5839, MathReview Entry, ZentralBlatt
Referenced by:: (17.6.4_5), §17.6(i), (17.8.8), §17.8.
Encodings:: BibTeX

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8: Errata

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Chapter 1 Additions

The following additions were made in Chapter 1:

§1.2

New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
§1.3

The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
§1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
§1.8

In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
§1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
§1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
§1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
§1.16

We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
§1.17

Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
§1.18

An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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9: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.4

\lim_{c\to\infty}{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{I}-c\mathbf{T}^{-1}% \right)=\left|\mathbf{T}\right|^{b}\Psi\left(b;b-a+\tfrac{1}{2}(m+1);\mathbf{T% }\right).

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ : confluent hypergeometric function of matrix argument (second kind), $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $b$ : complex variable, $c$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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35.7.5

{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left% (a,c-a\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|}^{% c-a-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{T}\mathbf{X}\right|}^{-b}\,% \mathrm{d}{\mathbf{X}},

\Re\left(a\right),\Re\left(c-a\right)>\frac{1}{2}(m-1)

\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\int$ : integral, $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, $b$ : complex variable, $<$ : less-than for matrices, $c$ : complex variable, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Referenced by:: §35.7(iv)
Permalink:: http://dlmf.nist.gov/35.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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35.7.6

{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\left|\mathbf{I}-\mathbf{T}% \right|^{c-a-b}{{}_{2}F_{1}}\left({c-a,c-b\atop c};\mathbf{T}\right)=\left|% \mathbf{I}-\mathbf{T}\right|^{-a}{{}_{2}F_{1}}\left({a,c-b\atop c};-\mathbf{T}% (\mathbf{I}-\mathbf{T})^{-1}\right)=\left|\mathbf{I}-\mathbf{T}\right|^{-b}{{}% _{2}F_{1}}\left({c-a,b\atop c};-\mathbf{T}(\mathbf{I}-\mathbf{T})^{-1}\right).

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $b$ : complex variable and $c$ : complex variable
Permalink:: http://dlmf.nist.gov/35.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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Gauss Formula

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Reflection Formula

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10: 37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$

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§37.17(iv) Mehler Formula

►The Poisson kernel (37.13.6) of

\mathcal{V}_{n}({\mathbb{R}}^{d})

is given explicitly by the Mehler formula … ►

37.17.14

\langle{f},{g}\rangle_{\mathbf{A}}={\pi}^{-\frac{1}{2}d}\det(\mathbf{A})^{% \frac{1}{2}}\int_{{\mathbb{R}}^{d}}f(\mathbf{x})g(\mathbf{x}){\mathrm{e}}^{-% \left\langle\mathbf{A}\mathbf{x},\mathbf{x}\right\rangle}\,\mathrm{d}\mathbf{x}.

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Defines:: $\langle{f},{g}\rangle_{\mathbf{A}}$ : inner product for $d$ -dimensional Hermite polynomials with a positive definite matrix (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\int$ : integral, $\mathbb{R}$ : real line, $d$ : positive integer, usually $\geq 2$ , ${\mathbb{R}}^{d}$ : $d$ -dimensional Euclidean space and $\mathbf{x}$ : vector in $d$ -dimensional Euclidean space
Permalink:: http://dlmf.nist.gov/37.17.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §37.17(vi), §37.17, Part 37.III and Ch.37

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determinant formula

1—10 of 14 matching pages

1: 1.12 Continued Fractions

Determinant Formula

2: 1.3 Determinants, Linear Operators, and Spectral Expansions

Krattenthaler’s Formula

3: Bibliography M

4: 1.2 Elementary Algebra

5: Bibliography B

6: 23.10 Addition Theorems and Other Identities

§23.10(ii) Duplication Formulas

§23.10(iii) $n$ -Tuple Formulas

7: Bibliography V

8: Errata

9: 35.7 Gaussian Hypergeometric Function of Matrix Argument

Gauss Formula

Reflection Formula

10: 37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$

§37.17(iv) Mehler Formula

determinant formula

1—10 of 14 matching pages

1: 1.12 Continued Fractions

Determinant Formula

2: 1.3 Determinants, Linear Operators, and Spectral Expansions

Krattenthaler’s Formula

3: Bibliography M

4: 1.2 Elementary Algebra

5: Bibliography B

6: 23.10 Addition Theorems and Other Identities

§23.10(ii) Duplication Formulas

§23.10(iii) n -Tuple Formulas

7: Bibliography V

8: Errata

9: 35.7 Gaussian Hypergeometric Function of Matrix Argument

Gauss Formula

Reflection Formula

10: 37.17 Hermite Polynomials on ℝ d

§37.17(iv) Mehler Formula

§23.10(iii) $n$ -Tuple Formulas

10: 37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$