general properties

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1: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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Convergence Properties

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§35.8(iv) General Properties

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2: 16.2 Definition and Analytic Properties

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§16.2(ii) Case $p\le q$

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§16.2(iii) Case $p=q+1$

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§16.2(iv) Case $p>q+1$

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§16.2(v) Behavior with Respect to Parameters

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3: Bibliography R

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

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Total positivity properties of generalized hypergeometric functions of matrix argument. J. Statist. Phys. 116 (1-4), pp. 907–922.

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External Links:

ISSN 0022-4715, Document, MathReview (Stamatis Koumandos), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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4: 16.5 Integral Representations and Integrals

… ►where the contour of integration separates the poles of $\mathrm{\Gamma }\left(a_k+s\right)$, $k=1,\mathrm{\dots },p$, from those of $\mathrm{\Gamma }\left(-s\right)$. …

5: 1.15 Summability Methods

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General Cesàro Summability

… ►and satisfies the property …

6: 25.16 Mathematical Applications

… ►For further properties of $H(s,z)$ see Apostol and Vu (1984). …

7: 9.13 Generalized Airy Functions

… ►Properties of $A_n\left(z\right)$ and $B_n\left(z\right)$ follow from the corresponding properties of the modified Bessel functions. … ►The distribution in $ℂ$ and asymptotic properties of the zeros of $A_n\left(z\right)$, $A_n^{\prime }\left(z\right)$, $B_n\left(z\right)$, and $B_n^{\prime }\left(z\right)$ are investigated in Swanson and Headley (1967) and Headley and Barwell (1975). …

8: 18.5 Explicit Representations

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18.5.5 $$p_n(x)=\frac{1}{{\kappa }_{n}w(x)}\frac{d^n}{{dx}^n}\left(w(x){(F(x))}^n\right).$$

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Symbols:

$\frac{df}{dx}$: derivative, $w(x)$: weight function, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable

A&S Ref:

22.1.6 (incorrectly stated as property of general OP’s)

Referenced by:

§18.17(v), §18.17(vi), §18.17(vii), Table 18.5.1, §18.9(iii)

Permalink:

http://dlmf.nist.gov/18.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(ii), §18.5 and Ch.18

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9: Bibliography G

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Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.

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Notes:

Translated from the Russian by Eugene Saletan. A paperbound reprinting by the same publisher appeared in 1977

External Links:

MathReview, ZentralBlatt

Referenced by:

§2.6(i).

Encodings:

BibTeX

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Generalized Fermi-Dirac functions and derivatives: Properties and evaluation. Comput. Phys. Comm. 136 (3), pp. 294–309.

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Notes:

Double-precision Fortran, maximum accuracy 20D

External Links:

Document, Code, ZentralBlatt

Referenced by:

7th item.

Encodings:

BibTeX

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10: 14.29 Generalizations

§14.29 Generalizations

… ►are called Generalized Associated Legendre Functions. …For properties see Virchenko and Fedotova (2001) and Braaksma and Meulenbeld (1967). ►For inhomogeneous versions of the associated Legendre equation, and properties of their solutions, see Babister (1967, pp. 252–264).