愛知東邦大学学士成绩单【购证 微kaa77788】54Z

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31: 5.7 Series Expansions

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32: 35.6 Confluent Hypergeometric Functions of Matrix Argument

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35.6.1 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a}{b};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a\right]}_{\kappa }}{{\left[b\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right).$$

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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, $!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $k$: nonnegative integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(i), §35.6 and Ch.35

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33: Bibliography V

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Some Wonderful Formulas $\mathrm{\dots }$ an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.

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External Links:

MathReview (F. Beukers), ZentralBlatt

Referenced by:

(25.6.10), (25.6.8), (25.6.9), §25.6(i), §25.6.

Encodings:

BibTeX

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

Encodings:

BibTeX

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34: 9.13 Generalized Airy Functions

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9.13.2 $$w=z^{1/2}{𝒵}_p\left(\zeta \right),$$

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Defines:

$w$: function (locally)

Symbols:

${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, p. 1401)

Permalink:

http://dlmf.nist.gov/9.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

… ►and ${𝒵}_p$ is any linear combination of the modified Bessel functions $I_p$ and ${\mathrm{e}}^{p\pi \mathrm{i}}{K}_p$ (§10.25(ii)). …

35: 4.24 Inverse Trigonometric Functions: Further Properties

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36: 19.12 Asymptotic Approximations

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37: 23.15 Definitions

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23.15.7 $$J\left(\tau \right)=\frac{{\left({\theta }_2^8(0,q)+{\theta }_3^8(0,q)+{\theta }_4^8(0,q)\right)}^3}{54{\left({\theta }_1^{\prime }(0,q)\right)}^8},$$

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Defines:

$J\left(\tau \right)$: Klein’s complete invariant

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

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38: Bibliography Z

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A proof of Andrews’ $q$-Dyson conjecture. Discrete Math. 54 (2), pp. 201–224.

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External Links:

ISSN 0012-365X, Document, MathReview (Ira Gessel), ZentralBlatt

Referenced by:

§17.14.

Encodings:

BibTeX

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39: 2.4 Contour Integrals

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$z$ ranges along a ray or over an annular sector ${\theta }_1\le \theta \le {\theta }_2$, $|z|\ge Z$, where $\theta =\mathrm{ph}z$, , and $Z>0$. $I(z)$ converges at $b$ absolutely and uniformly with respect to $z$.

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40: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.1 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a\right]}_{\kappa }{\left[b\right]}_{\kappa }}{{\left[c\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right),$$ $-c+\frac{1}{2}(j+1)\notin \mathbb{N}$, $1\le j\le m$; .

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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, $!$: factorial (as in $n!$), $\notin$: not an element of, $\mathbb{N}$: set of all positive integers, ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $\Vert 𝐓\Vert$: spectral norm of $𝐓$ , $c$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(i), §35.7 and Ch.35

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