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21: 12.14 The Function $W(a,x)$

… ►In other cases the general theory of (12.2.2) is available. …

22: Bibliography K

… ►

Some special cases of the generalized hypergeometric function ${}_{q+1}{}^{}F_q^{}$ . J. Comput. Appl. Math. 78 (1), pp. 79–95.

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External Links:

ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§16.7.

Encodings:

BibTeX

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23: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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§35.8(iii) ${}_3{}^{}F_2^{}$ Case

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24: 18.18 Sums

… ►See (18.2.41) for the Poisson kernel in case of general OP’s. …

25: 10.13 Other Differential Equations

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10.13.4 $$w^{\prime \prime }+\frac{1\mp 2\nu }{z}{w}^{\prime }+{\lambda }^{2}w=0,$$ $w=z^{\pm \nu }{𝒞}_{\nu }\left(\lambda z\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.52

Referenced by:

Erratum (V1.0.8) for Equation (10.13.4)

Permalink:

http://dlmf.nist.gov/10.13.E4

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.8):

This equation has been generalized to allow an additional case. Originally only the case $w=z^{\nu }{𝒞}_{\nu }\left(\lambda z\right)$ was covered.

Suggested 2014-01-27 by Tom Koornwinder

See also:

Annotations for §10.13 and Ch.10

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26: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to $\mathit{3}j$ symbols, for which see Edmonds (1974, Chapter 4). …

27: 10.38 Derivatives with Respect to Order

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10.38.1 $$\frac{\partial I_{\pm \nu }\left(z\right)}{\partial \nu }=\pm I_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.42

Referenced by:

§10.38, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.38.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial I_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.38.2).

See also:

Annotations for §10.38 and Ch.10

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28: 21.7 Riemann Surfaces

… ►Fay derives (21.7.10) as a special case of a more general class of addition theorems for Riemann theta functions on Riemann surfaces. …

29: 10.15 Derivatives with Respect to Order

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10.15.1 $$\frac{\partial J_{\pm \nu }\left(z\right)}{\partial \nu }=\pm J_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.64

Referenced by:

§10.15, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.15.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial J_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.15.2).

See also:

Annotations for §10.15, §10.15 and Ch.10

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30: 18.2 General Orthogonal Polynomials

… ►See Ismail (2009, §3.4) for another expression of the discriminant in the case of a general OP. …