hypergeometric functions of matrix argument

SearchAdvancedHelp

(0.009 seconds)

11—20 of 25 matching pages

11: Bibliography B

… ►

Laplace approximations for hypergeometric functions with matrix argument. Ann. Statist. 30 (4), pp. 1155–1177.

ⓘ

External Links:

ISSN 0090-5364, Document, MathReview, ZentralBlatt

Referenced by:

§35.6(iv), §35.7(iv), §35.7(iv), §35.9.

Encodings:

BibTeX

…

12: Errata

… ►

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.

… ►

The generalized hypergeometric function of matrix argument ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$, was linked inadvertently as its single variable counterpart ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$. Furthermore, the Jacobi function of matrix argument $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and the Laguerre function of matrix argument $L_{\nu }^{(\gamma )}\left(𝐓\right)$, were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and $L_{\nu }^{(\gamma )}\left(𝐓\right)$. In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

…

13: Donald St. P. Richards

… ►Richards has published numerous papers on special functions of matrix argument, harmonic analysis, multivariate statistical analysis, probability inequalities, and applied probability. He is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis: A Conference in Honor of R. A. Kunze (with T. … ►

35 Functions of Matrix Argument

…

14: Bibliography G

… ►

Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Trans. Amer. Math. Soc. 301 (2), pp. 781–811.

ⓘ

External Links:

ISSN 0002-9947, FullText, MathReview (Wayne J. Holman, III), ZentralBlatt

Referenced by:

§35.8(i), §35.8(ii), §35.8(iv), Ch.35.

Encodings:

BibTeX

…

15: Bibliography O

… ►

Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

ⓘ

External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

►

Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.

ⓘ

External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

ⓘ

External Links:

ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

… ►

A class of integral identities with matrix argument. Duke Math. J. 26 (2), pp. 207–213.

ⓘ

External Links:

Document, MathReview (R. Bellman), ZentralBlatt

Referenced by:

§35.3(ii).

Encodings:

BibTeX

… ►

Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Hussain S. Nur), ZentralBlatt

Referenced by:

§10.17(v), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).

Encodings:

BibTeX

…

16: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$, or $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). ►For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function $W_{-\eta ,\mathrm{\ell }+\frac{1}{2}}\left(2\rho \right)$. … … ►The Coulomb functions given in this chapter are most commonly evaluated for real values of $\rho$, $r$, $\eta$, $ϵ$ and nonnegative integer values of $\mathrm{\ell }$, but they may be continued analytically to complex arguments and order $\mathrm{\ell }$ as indicated in §33.13. … ►

Searches for resonances as poles of the $S$-matrix in the complex half-plane . See for example Csótó and Hale (1997).

…

17: Bibliography Y

… ►

Tables of Airy Functions and Their Derivatives. Izdat. Nauka, Moscow (Russian).

ⓘ

Referenced by:

4th item.

Encodings:

BibTeX

►

$J$-matrix method: Extensions to arbitrary angular momentum and to Coulomb scattering. J. Math. Phys. 16, pp. 410–420.

ⓘ

Referenced by:

§18.39(iv).

Encodings:

BibTeX

… ►

Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.

ⓘ

External Links:

MathReview (B. M. Agrawal), ZentralBlatt

Referenced by:

§35.10.

Encodings:

BibTeX

… ►

Computation of Kummer functions $U(a,b,x)$ for large argument $x$ by using the $\tau$-method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).

ⓘ

Notes:

Japanese with English summary. Double-precision Fortran.

External Links:

ISSN 0387-5806, FullText, MathReview

Referenced by:

§13.32(ii).

Encodings:

BibTeX

… ►

Bessel Functions. Part IV: Kelvin Functions. Royal Society Mathematical Tables, Volume 10, Cambridge University Press, Cambridge-New York.

ⓘ

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§10.65(ii), §10.68(iii), §10.68(iv), §10.71(i), 1st item.

Encodings:

BibTeX

…

18: Software Index

… ► ►►►

	Open Source													With Book						Commercial
…
35 Functions of Matrix Argument
…

►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

19: 18.38 Mathematical Applications

… ►The Askey–Gasper inequality …For the generalized hypergeometric function ${}_3{}^{}F_2^{}$ see (16.2.1). … ►

Random Matrix Theory

… ►The $3j$ symbol (34.2.6), with an alternative expression as a terminating ${}_3{}^{}F_2^{}$ of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. … ►The $6j$ symbol (34.4.3), with an alternative expression as a terminating balanced ${}_4{}^{}F_3^{}$ of unit argument, can be expressend in terms of Racah polynomials (18.26.3). …

20: Bibliography H

… ►

Certain integrals that contain a probability function and degenerate hypergeometric functions. Bul. Akad. S̆tiince RSS Moldoven 1969 (2), pp. 40–47 (Russian).

ⓘ

External Links:

MathReview (Wazir Hasan Abdi)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

►

Some integrals that contain a probability function and hypergeometric functions. Bul. Akad. Štiince RSS Moldoven 1970 (1), pp. 49–62 (Russian).

ⓘ

External Links:

MathReview (R. G. Langebartel)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

… ►

Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

ⓘ

External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

… ►

Bessel functions of matrix argument. Ann. of Math. (2) 61 (3), pp. 474–523.

ⓘ

External Links:

FullText, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§35.1, §35.2, §35.3(ii), §35.5(i), §35.5(ii), §35.5(iii), §35.6(i), §35.6(ii), §35.6(iii), §35.6(iv), §35.7(i), §35.7(ii), §35.7(iv), §35.8(i), §35.8(ii), §35.8(iv), Ch.35, Notations P.

Encodings:

BibTeX

… ►

Development of a Gaussian hypergeometric function code in complex domains. Internat. J. Modern Phys. C 4 (4), pp. 805–840.

ⓘ

External Links:

ISSN 0129-1831, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§15.20(iii).

Encodings:

BibTeX

…