representation by squares

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11: Errata

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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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12: Bibliography K

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S. H. Khamis (1965) Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution. Justus von Liebig Verlag, Darmstadt (German, English).

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Notes:: In cooperation with W. Rudert
External Links:: MathReview, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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A. I. Kheyfits (2004) Closed-form representations of the Lambert

W

function. Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.

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External Links:: ISSN 1311-0454, MathReview (Lewis H. Ryder), ZentralBlatt
Referenced by:: §4.13.
Encodings:: BibTeX

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T. H. Koornwinder (1994) Compact quantum groups and

q

-special functions. In Representations of Lie Groups and Quantum Groups, Pitman Res. Notes Math. Ser., Vol. 311, pp. 46–128.

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Notes:: Sections 3 and 4 also available as arXiv:math/9403216
External Links:: MathReview (Eric M. Opdam)
Referenced by:: (18.27.14_3).
Encodings:: BibTeX

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13: 10.9 Integral Representations

§10.9 Integral Representations

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§10.9(ii) Contour Integrals

… ► … ►where the square root has its principal value. … ► …

14: 4.13 Lambert $W$ -Function

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W_{0}\left(z\right)

is a single-valued analytic function on

\mathbb{C}\setminus(-\infty,-{\mathrm{e}}^{-1}]

, real-valued when

z>-{\mathrm{e}}^{-1}

, and has a square root branch point at

z=-{\mathrm{e}}^{-1}

. …The other branches

W_{k}\left(z\right)

are single-valued analytic functions on

\mathbb{C}\setminus(-\infty,0]

, have a logarithmic branch point at

z=0

, and, in the case

k=\pm 1

, have a square root branch point at

z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}

respectively. … ►Explicit representations for the

p_{n}(x)

are given in Kalugin and Jeffrey (2011). …See Jeffrey and Murdoch (2017) for an explicit representation for the

c_{n}

in terms of associated Stirling numbers. … ►For these and other integral representations of the Lambert

W

-function see Kheyfits (2004), Kalugin et al. (2012) and Mező (2020). …

15: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.3

{{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{1}&0\\ 0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{% \left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c% \right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}% \left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $a$ : complex variable, $t_{j}$ : eigenvalues of $\mathbf{T}$ , $b$ : complex variable, $c$ : complex variable and $k$ : nonnegative integer
Referenced by:: Erratum (V1.0.25) for Equation (35.7.3)
Permalink:: http://dlmf.nist.gov/35.7.E3
Encodings:: pMML, png, TeX
Notational change (effective with 1.0.25):: Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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Integral Representation

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16: 18.18 Sums

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Expansion of $L^{2}$ functions

►In all three cases of Jacobi, Laguerre and Hermite, if

f(x)

L^{2}

on the corresponding interval with respect to the corresponding weight function and if

a_{n},b_{n},d_{n}

are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in

L^{2}

sense. … ►For integral representations for products implied by (18.18.8) and (18.18.9) see (18.17.5) and (18.17.6), respectively. …

17: Bibliography S

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B. E. Sagan (2001) The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. 2nd edition, Graduate Texts in Mathematics, Vol. 203, Springer-Verlag, New York.

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Notes:: First edition published by Wadsworth & Brooks/Cole Advanced Books & Software, 1991.
External Links:: ISBN 0-387-95067-2, MathReview, ZentralBlatt
Referenced by:: §26.19.
Encodings:: BibTeX

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B. I. Schneider, X. Guan, and K. Bartschat (2016) Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation. Adv. Quantum Chem. 72, pp. 95–127.

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Referenced by:: §18.38(i).
Encodings:: BibTeX

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R. Shail (1980) On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.

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External Links:: ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt
Referenced by:: §29.17(i), §29.8.
Encodings:: BibTeX

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H. Shanker (1940a) On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

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B. L. Shea (1988) Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.

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Notes:: Double-precision Fortran.
External Links:: FullText, Code
Referenced by:: 5th item.
Encodings:: BibTeX

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18: 14.30 Spherical and Spheroidal Harmonics

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Explicit Representation

… ►For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii). … ►Here, in spherical coordinates,

\mathrm{L}^{2}

is the squared angular momentum operator: …

19: 21.7 Riemann Surfaces

… ►If a local coordinate

z

is chosen on the Riemann surface, then the local coordinate representation of these holomorphic differentials is given by … ►Either branch of the square roots may be chosen, as long as the branch is consistent across

\Gamma

. …