symmetric forms

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21: 21.1 Special Notation

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$g,h$	positive integers.
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$𝛀$	$g\times g$ complex, symmetric matrix with $\mathrm{\Im }𝛀$ strictly positive definite, i.e., a Riemann matrix.
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$S_1{S}_2$	set of all elements of the form “$\text{element of }{S}_1\times \text{element of }{S}_2$”.
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22: 2.6 Distributional Methods

… ►This leads to integrals of the form … ►The distribution method outlined here can be extended readily to functions $f(t)$ having an asymptotic expansion of the form … ►The replacement of $f(t)$ by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form … ►It is easily seen that $K^{+}$ forms a commutative, associative linear algebra. … ►On inserting this identity into (2.6.54), we immediately encounter divergent integrals of the form …

23: 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.

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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.

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

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

Referenced by:

§28.8(iv).

Encodings:

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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).

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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.

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24: 21.7 Riemann Surfaces

… ►Then the prime form on the corresponding compact Riemann surface $\mathrm{\Gamma }$ is defined by … ►These are Riemann surfaces that may be obtained from algebraic curves of the form … ►

21.7.12 $$T_1\ominus T_2=(T_1\cup T_2)\setminus (T_1\cap T_2).$$

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Symbols:

$\cap$: intersection, $\setminus$: set subtraction, $\cup$: union and $T$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E12

Encodings:

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See also:

Annotations for §21.7(iii), §21.7 and Ch.21

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21.7.14 $$𝜼(T_1\ominus T_2)=𝜼(T_1)+𝜼(T_2),$$

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Symbols:

$T$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E14

Encodings:

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See also:

Annotations for §21.7(iii), §21.7 and Ch.21

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21.7.15 $$4{𝜼}^1(T)\cdot {𝜼}^2(T)=\frac{1}{2}\left(|T\ominus U|-g-1\right)(mod2),$$

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Symbols:

$g$: positive integer, $U$: subset of $B$ and $T$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E15

Encodings:

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See also:

Annotations for §21.7(iii), §21.7 and Ch.21

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25: 18.2 General Orthogonal Polynomials

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First Form

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Second Form

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Monic and Orthonormal Forms

… ►the monic recurrence relations (18.2.8) and (18.2.10) take the form … ►

Confluent Form

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26: Bibliography M

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Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.

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External Links:

Document, MathReview (K.-B. Gundlach), ZentralBlatt

Referenced by:

§35.4(ii).

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Symmetric Functions and Hall Polynomials. 2nd edition, The Clarendon Press, Oxford University Press, New York-Oxford.

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Notes:

With contributions by A. Zelevinsky. Paperback version published by Oxford University Press, July 1999.

External Links:

ISBN 0-19-853489-2; 0-19-850450-0, MathReview (John R. Stembridge), ZentralBlatt

Referenced by:

§18.37(iii), §26.19, §35.4(i), §35.4(ii).

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Symmetric Functions and Orthogonal Polynomials. University Lecture Series, Vol. 12, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0770-6, MathReview (Angèle M. Hamel), ZentralBlatt

Referenced by:

§17.14.

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Crossing symmetric expansions of physical scattering amplitudes: The $O(2,1)$ group and Lamé functions. J. Mathematical Phys. 12, pp. 281–293.

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External Links:

Document, MathReview

Referenced by:

§29.19(ii).

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An integral representation for the Bessel form. J. Comput. Appl. Math. 57 (1-2), pp. 251–260.

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External Links:

ISSN 0377-0427, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt

Referenced by:

§18.34(ii).

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27: 26.12 Plane Partitions

… ►The number of symmetric plane partitions in $B(r,r,t)$ is … ►The plane partition in Figure 26.12.1 is an example of a cyclically symmetric plane partition. The number of cyclically symmetric plane partitions in $B(r,r,r)$ is … ►A plane partition is totally symmetric if it is both symmetric and cyclically symmetric. … ►

§26.12(iv) Limiting Form

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28: 28.29 Definitions and Basic Properties

… ►It has the form … ►In the symmetric case $Q(z)=Q(-z)$, $w_{\text{I}}(z,\lambda )$ is an even solution and $w_{\text{II}}(z,\lambda )$ is an odd solution; compare §28.2(ii). …

29: 36.5 Stokes Sets

… ►The Stokes set takes different forms for $z=0$, , and $z>0$. … ►One of the sheets is symmetrical under reflection in the plane $y=0$, and is given by …

30: Bibliography C

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Asymptotic approximations for symmetric elliptic integrals. SIAM J. Math. Anal. 25 (2), pp. 288–303.

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External Links:

ISSN 0036-1410, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.12, §19.27(ii), §19.27(iii), §19.27(iv), §19.27(v), §19.27(vi).

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Inequalities for a symmetric elliptic integral. Proc. Amer. Math. Soc. 25 (3), pp. 698–703.

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External Links:

FullText, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.24(ii), §19.9(ii).

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Intégrandes à deux formes quadratiques. C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).

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External Links:

MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§19.31.

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Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.16.

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Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.

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External Links:

ISBN 0-387-94609-8; 0-387-98998-6, MathReview (M. Ram Murty), ZentralBlatt

Referenced by:

§23.20(v).

Encodings:

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