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21: 19.18 Derivatives and Differential Equations

… ►The next four differential equations apply to the complete case of $R_F$ and $R_G$ in the form $R_{-a}(\frac{1}{2},\frac{1}{2};z_1,z_2)$ (see (19.16.20) and (19.16.23)). …

22: 19.25 Relations to Other Functions

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§19.25(i) Legendre’s Integrals as Symmetric Integrals

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§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

… ► … ►

§19.25(vii) Hypergeometric Function

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23: 19.28 Integrals of Elliptic Integrals

§19.28 Integrals of Elliptic Integrals

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19.28.5 $${\int }_z^{\mathrm{\infty }}{R}_D(x,y,t)dt=6R_F(x,y,z),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E5

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Annotations for §19.28 and Ch.19

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19.28.6 $${\int }_0^1{R}_D(x,y,v^{2}z+(1-v^2)p)dv=R_J(x,y,z,p).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E6

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See also:

Annotations for §19.28 and Ch.19

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19.28.7 $${\int }_0^{\mathrm{\infty }}{R}_J(x,y,z,r^2)dr=\frac{3}{2}\pi R_F(xy,xz,yz),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E7

Encodings:

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See also:

Annotations for §19.28 and Ch.19

… ►

24: Mathematical Introduction

… ►All chapters went through several drafts (nine in some cases) before the authors, validators, and editors were fully satisfied. … ►Similarly in the case of confluent hypergeometric functions (§13.2(i)). ►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). … ►However, in many cases the coloring of the surface is chosen instead to indicate the quadrant of the plane to which the phase of the function belongs, thereby achieving a 4D effect. In these cases the phase colors that correspond to the 1st, 2nd, 3rd, and 4th quadrants are arranged in alphabetical order: blue, green, red, and yellow, respectively, and a “Quadrant Colors” icon appears alongside the figure. …

25: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.2 $${}_0{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-}{-};𝐓)=\mathrm{etr}\left(𝐓\right),$$ $𝐓\in 𝓢$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $𝓢$: space of real symmetric $m\times m$ matrices and $𝐓$: real symmetric $m\times m$ matrix

Permalink:

http://dlmf.nist.gov/35.8.E2

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Annotations for §35.8(ii), §35.8 and Ch.35

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35.8.4 $$A_{\nu }\left(𝐓\right)=\frac{1}{{\mathrm{\Gamma }}_m\left(\nu +\frac{1}{2}(m+1)\right)}{}_0{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-}{\nu +\frac{1}{2}(m+1)};-𝐓),$$ $𝐓\in 𝓢$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.8.E4

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See also:

Annotations for §35.8(ii), §35.8 and Ch.35

►

§35.8(iii) ${}_3{}^{}F_2^{}$ Case

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35.8.12 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{\gamma -\frac{1}{2}(m+1)}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};-𝐗)d𝐗={\mathrm{\Gamma }}_m\left(\gamma \right){\left|𝐓\right|}^{-\gamma }{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,\gamma }{b_1,\mathrm{\dots },b_q};-{𝐓}^{-1}),$$ $\mathrm{\Re }\left(\gamma \right)>\frac{1}{2}(m-1)$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: 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

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See also:

Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

… ►These multidimensional integrals reduce to the classical Mellin–Barnes integrals (§5.19(ii)) in the special case $m=1$. …

26: 18.39 Applications in the Physical Sciences

… ►Brief mention of non-unit normalized solutions in the case of mixed spectra appear, but as these solutions are not OP’s details appear elsewhere, as referenced. … ►defines the potential for a symmetric restoring force $-kx$ for displacements from $x=0$. … ►These cases correspond to the two distinct orthogonality conditions of (18.35.6) and (18.35.6_3). … ►In the attractive case (18.35.6_4) for the discrete parts of the weight function where with , are also simplified: …The weight functions for both the attractive and repulsive cases are now unit normalized, see Bank and Ismail (1985), and Ismail (2009). …

27: Bibliography S

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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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Parabolic Cylinder Functions and their Application in Symmetric Two-centre Shell Model. In Proceedings of the Conference on Mathematical Analysis and its Applications (Inst. Engrs., Mysore, 1977), Matscience Rep., Vol. 91, Aarhus, pp. P81–P89.

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External Links:

MathReview

Referenced by:

§12.17.

Encodings:

BibTeX

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Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 (1), pp. 76–115.

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External Links:

ISSN 0001-8708, Document, MathReview (Dennis White), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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Error bounds for asymptotic solutions of differential equations. I. The distinct eigenvalue case. J. Res. Nat. Bur. Standards Sect. B 70B, pp. 167–186.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

…

28: Bibliography N

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Bounds for symmetric elliptic integrals. J. Approx. Theory 122 (2), pp. 249–259.

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External Links:

ISSN 0021-9045, Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§19.24(ii), §19.24(i), §19.9(i).

Encodings:

BibTeX

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Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48 (1), pp. 3–167.

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External Links:

ISSN 0021-9045, Document, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.32.

Encodings:

BibTeX

…

29: Bibliography T

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Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.

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External Links:

ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Harmonic Analysis on Symmetric Spaces and Applications. II. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-96663-3, MathReview (Stephen Gelbart), ZentralBlatt

Referenced by:

§35.1, §35.5(ii), Notations K.

Encodings:

BibTeX

…

30: 19.31 Probability Distributions

§19.31 Probability Distributions

… ►§19.16(iii) shows that for $n=3$ the incomplete cases of $R_F$ and $R_G$ occur when $\mu =-1/2$ and $\mu =1/2$, respectively, while their complete cases occur when $n=2$. …