Krattenthaler formula for determinants

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11: 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, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : 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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12: 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

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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),

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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

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24.14.12

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

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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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13: 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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S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:: Document, ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

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S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

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External Links:: ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

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D. S. Moak (1984) The

q

-analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.

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External Links:: ISSN 0035-7596, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §5.18(ii).
Encodings:: BibTeX

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14: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function

p\left(n\right)

for

n<N

. … ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function

\tau\left(n\right)

, and the values can be checked by the congruence (27.14.20). …

15: 21.5 Modular Transformations

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21.5.3

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

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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

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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).

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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

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\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),

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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),

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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

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M. Abramowitz and I. A. Stegun (Eds.) (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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Notes:: Corrections appeared in later printings up to the 10th Printing, December, 1972. Reproductions by other publishers, in whole or in part, have been available since 1965.
External Links:: FullText, MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §10.1, §10.21(ix), §10.47(i), §10.68(iv), §10.70, 4th item, 2nd item, 4th item, 5th item, 1st item, 1st item, 4th item, 1st item, 1st item, Ch.10, §11.10(vi), 1st item, 1st item, Ch.11, 1st item, Ch.12, 3rd item, Ch.13, 1st item, Ch.14, Ch.15, §18.14(i), §18.3, §18.41(i), §18.41(ii), §18.5(iv), §18.8, §19.1, §19.14(i), §19.14(ii), §19.25(v), §19.36(iii), §19.37(ii), §19.37(ii), §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.38, Ch.19, §20.15, Ch.20, §22.1, §22.15(ii), Ch.22, §23.1, §23.20(i), §23.23, §23.23, §23.9, §23.9, §23.9, Ch.23, §24.2(iv), §24.20, Ch.24, 1st item, §26.1, §26.1, §26.2, §26.21, §26.3(i), §26.4(i), §26.8(i), §27.2(ii), §27.21, §28.1, Ch.28, §30.1, Ch.30, 1st item, Ch.33, §4.44, §4.46, §5.11(i), §5.22(ii), §5.22(iii), §5.7(i), Ch.5, 1st item, 1st item, 1st item, Ch.6, 1st item, 2nd item, 1st item, Ch.7, (8.17.24), §8.22(ii), 1st item, Ch.8, 1st item, 1st item, 1st item, §9.8(i), Ch.9, Profile Frank W. J. Olver, About the Project, Preface, Possible Errors in DLMF, Notations E, Notations E, Notations F, Notations F, Notations H, Notations H, Notations J, Notations K, Notations P, Notations P, Notations S, Notations Y, H. E. Fettis and J. C. Caslin (1973).
Encodings:: BibTeX

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W. A. Al-Salam and L. Carlitz (1959) Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

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External Links:: FullText, MathReview (D. P. Banerjee), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

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K. Aomoto (1987) Special value of the hypergeometric function

{}_{3}F_{2}

and connection formulae among asymptotic expansions. J. Indian Math. Soc. (N.S.) 51, pp. 161–221.

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External Links:: ISSN 0019-5839, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

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T. M. Apostol (1985a) Formulas for higher derivatives of the Riemann zeta function. Math. Comp. 44 (169), pp. 223–232.

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External Links:: ISSN 0025-5718, FullText, MathReview (Aleksandar Ivić), ZentralBlatt
Referenced by:: (25.11.19), (25.11.20), (25.11.6), §25.11(vi), §25.18(i), (25.4.5), (25.4.6), §25.4, (25.6.11), (25.6.12), (25.6.13), (25.6.14), §25.6(ii), §25.6.
Encodings:: BibTeX

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T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:: §24.4(iii).
Encodings:: BibTeX

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17: 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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18: Errata

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Equations (1.3.5), (1.3.6), (1.3.7)

1.3.5

\det\left(\mathbf{A}^{\mathrm{T}}\right)=\det(\mathbf{A})

1.3.6

\det\left({\mathbf{A}}^{-1}\right)=\frac{1}{\det(\mathbf{A})}

1.3.7

\det(\mathbf{A}\mathbf{B})=\det(\mathbf{A})\det(\mathbf{B})

Previously we used the notation $[a_{jk}]$ , $[b_{jk}]$ , for $\mathbf{A}$ , $\mathbf{B}$ respectively.

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Equations (32.8.10), (32.10.9)

32.8.10

\tau_{n}(z)=\mathscr{W}\left\{p_{1}(z),p_{3}(z),\ldots,p_{2n-1}(z)\right\}

32.10.9

\tau_{n}(z)=\mathscr{W}\left\{\phi(z),\phi^{\prime}(z),\ldots,\phi^{(n-1)}(z)\right\}

The right-hand side of these equation, which was originally written as a matrix determinant, was rewritten using the Wronskian determinant notation. Also, in each preceding sentence, the word ‘determinant’ was replaced with ‘Wronskian determinant’.

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Chapter 1 Additions

The following additions were made in Chapter 1:

Section 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).
Section 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).
Section 1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
Section 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).
Section 1.10

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

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

Another form of Parseval’s formula, (1.14.7_5).
Section 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.
Section 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.
Section 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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Subsection 17.9(iii)

The title of the paragraph which was previously “Gasper’s $q$ -Analog of Clausen’s Formula” has been changed to “Gasper’s $q$ -Analog of Clausen’s Formula (16.12.2)”.

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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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19: 1.1 Special Notation

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

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20: Howard S. Cohl

… ►Howard is the project leader for the NIST Digital Repository of Mathematical Formulae seeding and development projects. In this regard, he has been exploring mathematical knowledge management and the digital expression of mostly unambiguous context-free full semantic information for mathematical formulae.