Krattenthaler formula for determinants

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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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Cauchy Determinant

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Krattenthaler’s Formula

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3: Bibliography K

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K. Kajiwara and Y. Ohta (1996) Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37 (9), pp. 4693–4704.

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External Links:: ISSN 0022-2488, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(ii).
Encodings:: BibTeX

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K. Kajiwara and Y. Ohta (1998) Determinant structure of the rational solutions for the Painlevé IV equation. J. Phys. A 31 (10), pp. 2431–2446.

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External Links:: ISSN 0305-4470, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

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R. P. Kelisky (1957) On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §24.15(iv).
Encodings:: BibTeX

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T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

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External Links:: Document, MathReview (M. Mikolas), ZentralBlatt
Referenced by:: (18.14.8), §18.14(i), §18.17(ii), §18.18(ii).
Encodings:: BibTeX

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C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively

q

-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:: Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.
External Links:: Code, Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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5: 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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6: 18.2 General Orthogonal Polynomials

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§18.2(v) Christoffel–Darboux Formula

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Confluent Form

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§18.2(ix) Moments

… ►It is to be noted that, although formally correct, the results of (18.2.30) are of little utility for numerical work, as Hankel determinants are notoriously ill-conditioned. … ►

Degree lowering and raising differentiation formulas and structure relations

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7: Bibliography S

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N. J. A. Sloane (2003) The On-Line Encyclopedia of Integer Sequences. Notices Amer. Math. Soc. 50 (8), pp. 912–915.

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External Links:: ISSN 0002-9920, FullText, Electronic Database, MathReview (Christian Krattenthaler), ZentralBlatt
Referenced by:: (19.20.2), (19.20.22), (19.30.13), (25.11.40), (3.12.1), (3.12.3), (3.12.4), §3.12, (4.2.11), (4.2.17), (4.2.18), (4.4.19), (5.17.6), (5.2.3), (5.4.10), (5.4.6), (5.4.7), (5.4.8), (5.4.9), (6.13.1).
Encodings:: BibTeX

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R. Spira (1971) Calculation of the gamma function by Stirling’s formula. Math. Comp. 25 (114), pp. 317–322.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §5.11(i), §5.11(ii).
Encodings:: BibTeX

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A. H. Stroud and D. Secrest (1966) Gaussian Quadrature Formulas. Prentice-Hall Inc., Englewood Cliffs, N.J..

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External Links:: MathReview (P. C. Hammer), ZentralBlatt
Referenced by:: §3.5(v).
Encodings:: BibTeX

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8: Bibliography V

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

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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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9: 1.12 Continued Fractions

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Determinant Formula

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10: 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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Krattenthaler formula for determinants

1—10 of 264 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

Cauchy Determinant

Krattenthaler’s Formula

2: 17.19 Software

3: Bibliography K

4: 1.2 Elementary Algebra

The Determinant

Relation of Eigenvalues to the Determinant and Trace

5: 23.10 Addition Theorems and Other Identities

§23.10(ii) Duplication Formulas

§23.10(iii) $n$ -Tuple Formulas

6: 18.2 General Orthogonal Polynomials

§18.2(v) Christoffel–Darboux Formula

Confluent Form

§18.2(ix) Moments

Degree lowering and raising differentiation formulas and structure relations

7: Bibliography S

8: Bibliography V

9: 1.12 Continued Fractions

Determinant Formula

10: 35.7 Gaussian Hypergeometric Function of Matrix Argument

Gauss Formula

Reflection Formula

Krattenthaler formula for determinants

1—10 of 264 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

Cauchy Determinant

Krattenthaler’s Formula

2: 17.19 Software

3: Bibliography K

4: 1.2 Elementary Algebra

The Determinant

Relation of Eigenvalues to the Determinant and Trace

5: 23.10 Addition Theorems and Other Identities

§23.10(ii) Duplication Formulas

§23.10(iii) n -Tuple Formulas

6: 18.2 General Orthogonal Polynomials

§18.2(v) Christoffel–Darboux Formula

Confluent Form

§18.2(ix) Moments

Degree lowering and raising differentiation formulas and structure relations

7: Bibliography S

8: Bibliography V

9: 1.12 Continued Fractions

Determinant Formula

10: 35.7 Gaussian Hypergeometric Function of Matrix Argument

Gauss Formula

Reflection Formula

§23.10(iii) $n$ -Tuple Formulas