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1: Bille C. Carlson

… ►Also, the homogeneity of the $R$-function has led to a new type of mean value for several variables, accompanied by various inequalities. …

2: 1.16 Distributions

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§1.16(vi) Distributions of Several Variables

… ►For a multi-index $𝜶=({\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_n)$, define ►

1.16.31 $$P(𝐱)=\sum _{𝜶}{c}_{𝜶}{𝐱}^{𝜶}=\sum _{𝜶}{c}_{𝜶}{x}_1^{{\alpha }_1}\mathrm{\cdots }{x}_n^{{\alpha }_n},$$

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Symbols:

$n$: nonnegative integer and $P$: polynomial of several variables

Permalink:

http://dlmf.nist.gov/1.16.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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1.16.36 $$⟨ℱ\left(P(𝐃)u\right),\varphi ⟩=⟨P_{-}ℱ\left(u\right),\varphi ⟩=⟨ℱ\left(u\right),P_{-}\varphi ⟩,$$

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Symbols:

$⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $ℱ\left(u\right)$: Fourier transform of a tempered distribution, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

item (iv), §1.16(vii), §1.16(viii), item (iv)

Permalink:

http://dlmf.nist.gov/1.16.E36

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

Previously this equation was written as

$$ℱ(P(𝐷)u)=P(-𝐱)ℱ(u).$$

It has been rewritten using the notation of (1.16.32) and (1.16.35).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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3: 17.12 Bailey Pairs

… ►A sequence of pairs of rational functions of several variables $({\alpha }_n,{\beta }_n)$, $n=0,1,2,\mathrm{\dots }$, is called a Bailey pair provided that for each $n\geqq 0$ …

4: 18.37 Classical OP’s in Two or More Variables

… ►Orthogonal polynomials associated with root systems are certain systems of trigonometric polynomials in several variables, symmetric under a certain finite group (Weyl group), and orthogonal on a torus. …In several variables they occur, for $q=1$, as Jack polynomials and also as Jacobi polynomials associated with root systems; see Macdonald (1995, Chapter VI, §10), Stanley (1989), Kuznetsov and Sahi (2006, Part 1), Heckman (1991). …

5: Bibliography O

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Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.

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Notes:

Reprinted in 2000 by the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania

External Links:

MathReview (J. F. Traub), ZentralBlatt (Nelli Dimitrova (Sofia))

Referenced by:

§3.8(vii).

Encodings:

BibTeX

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6: Bibliography D

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Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-80043-9, MathReview (Marcel de Jeu), ZentralBlatt

Referenced by:

§18.37(ii), §18.37(i).

Encodings:

BibTeX

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7: Bibliography S

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Topics in Complex Function Theory. Vol. III: Abelian Functions and Modular Functions of Several Variables. Interscience Tracts in Pure and Applied Mathematics, No. 25, Wiley-Interscience, [John Wiley & Sons, Inc], New York-London-Sydney.

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Notes:

Translated from the original German by E. Gottschling and M. Tretkoff. Reprinted by John Wiley & Sons, New York, 1989.

External Links:

ISBN 0-471-79090-7, MathReview, ZentralBlatt

Referenced by:

Ch.21.

Encodings:

BibTeX

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8: Bibliography K

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Orthogonal Polynomials on $n$-spheres: Gegenbauer, Jacobi and Heun. In Topics in Polynomials of One and Several Variables and their Applications, pp. 299–322.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§31.16(ii).

Encodings:

BibTeX

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9: Bibliography H

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Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.

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Notes:

Reprinted in 1979.

External Links:

ISBN 0-8218-1556-3, MathReview, ZentralBlatt

Referenced by:

§35.4(i).

Encodings:

BibTeX

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10: 19.28 Integrals of Elliptic Integrals

… ►To replace a single component of $𝐳$ in $R_{-a}(𝐛;𝐳)$ by several different variables (as in (19.28.6)), see Carlson (1963, (7.9)). …