determinant formula

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10 matching pages

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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4: 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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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: 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(i), §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
Referenced by:: (17.6.4_5), §17.6(i), (17.8.8), §17.8.
Encodings:: BibTeX

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7: 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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8: 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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A. A. Kapaev and A. V. Kitaev (1993) Connection formulae for the first Painlevé transcendent in the complex domain. Lett. Math. Phys. 27 (4), pp. 243–252.

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External Links:: ISSN 0377-9017, Document, MathReview (B. L. J. Braaksma), ZentralBlatt
Referenced by:: §32.11(i), §32.12(i).
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(i), §18.17(ii).
Encodings:: BibTeX

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9: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.3

{{}_{2}F_{1}}\left({a,b\atop b};\mathbf{T}\right)={{}_{1}F_{0}}\left({a\atop-}% ;\mathbf{T}\right)=\left|\mathbf{I}-\mathbf{T}\right|^{-a},

\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, $a$ : 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, $<$ : less-than for matrices and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(ii), §35.8 and Ch.35

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Pfaff–Saalschütz Formula

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35.8.12

{\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right% )\left|\mathbf{X}\right|^{\gamma-\frac{1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}};-\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}% }=\Gamma_{m}\left(\gamma\right)\left|\mathbf{T}\right|^{-\gamma}{{}_{p+1}F_{q}% }\left({a_{1},\dots,a_{p},\gamma\atop b_{1},\dots,b_{q}};-\mathbf{T}^{-1}% \right),

\Re\left(\gamma\right)>\frac{1}{2}(m-1)

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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, $\operatorname{etr}\left(\NVar{\mathbf{X}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $a$ : complex variable, $\mathbf{T}$ : 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, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $b$ : complex variable, $p$ : nonnegative integer, $q$ : nonnegative integer and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.8.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

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35.8.13

\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|\mathbf{X}\right|^{a% _{1}-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{X}\right|}^{b_{1}-a_{1}-\frac{% 1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{2},\dots,a_{p+1}\atop b_{2},\dots,b_{q+1}}% ;\mathbf{T}\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}=\frac{1}{\mathrm{B}_{m}% \left(b_{1}-a_{1},a_{1}\right)}{{}_{p+1}F_{q+1}}\left({a_{1},\dots,a_{p+1}% \atop b_{1},\dots,b_{q+1}};\mathbf{T}\right),

\Re\left(b_{1}-a_{1}\right),\Re\left(a_{1}\right)>\frac{1}{2}(m-1)

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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, $p$ : nonnegative integer, $q$ : nonnegative integer, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.8.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

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10: Errata

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Paragraph Inversion Formula (in §35.2)

The wording was changed to make the integration variable more apparent.

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Section 1.13

In Equation (1.13.4), the determinant form of the two-argument Wronskian

1.13.4

\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z)

was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the $n$ -argument Wronskian is given by $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$ , where $1\leq j,k\leq n$ . Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$ -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for $n$ th-order differential equations. A reference to Ince (1926, §5.2) was added.

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Usability

Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

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Subsection 14.18(iii)

This subsection now identifies Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

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Table 18.3.1

Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

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determinant formula

10 matching pages

1: 1.12 Continued Fractions

Determinant Formula

2: 1.3 Determinants

Krattenthaler’s Formula

3: Bibliography M

4: Bibliography B

5: 23.10 Addition Theorems and Other Identities

§23.10(ii) Duplication Formulas

§23.10(iii) $n$ -Tuple Formulas

6: Bibliography V

7: 35.7 Gaussian Hypergeometric Function of Matrix Argument

Gauss Formula

Reflection Formula

8: Bibliography K

9: 35.8 Generalized Hypergeometric Functions of Matrix Argument

Pfaff–Saalschütz Formula

10: Errata

determinant formula

10 matching pages

1: 1.12 Continued Fractions

Determinant Formula

2: 1.3 Determinants

Krattenthaler’s Formula

3: Bibliography M

4: Bibliography B

5: 23.10 Addition Theorems and Other Identities

§23.10(ii) Duplication Formulas

§23.10(iii) n -Tuple Formulas

6: Bibliography V

7: 35.7 Gaussian Hypergeometric Function of Matrix Argument

Gauss Formula

Reflection Formula

8: Bibliography K

9: 35.8 Generalized Hypergeometric Functions of Matrix Argument

Pfaff–Saalschütz Formula

10: Errata

§23.10(iii) $n$ -Tuple Formulas