generalizations

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21: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

§10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

►The function

\phi\left(\rho,\beta;z\right)

is defined by ►

10.46.1

\phi\left(\rho,\beta;z\right)=\sum_{k=0}^{\infty}\frac{z^{k}}{k!\Gamma\left(% \rho k+\beta\right)},

\rho>-1

ⓘ

Defines:: $\phi\left(\NVar{\rho},\NVar{\beta};\NVar{z}\right)$ : generalized Bessel function
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.46.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.46 and Ch.10

… ►For asymptotic expansions of

\phi\left(\rho,\beta;z\right)

z\to\infty

in various sectors of the complex

z

-plane for fixed real values of

\rho

and fixed real or complex values of

\beta

, see Wright (1935) when

\rho>0

, and Wright (1940b) when

-1<\rho<0

. … ►The Laplace transform of

\phi\left(\rho,\beta;z\right)

can be expressed in terms of the Mittag-Leffler function: …

22: 8.1 Special Notation

… ►Unless otherwise indicated, primes denote derivatives with respect to the argument. ►The functions treated in this chapter are the incomplete gamma functions

\gamma\left(a,z\right)

\Gamma\left(a,z\right)

\gamma^{*}\left(a,z\right)

P\left(a,z\right)

, and

Q\left(a,z\right)

; the incomplete beta functions

\mathrm{B}_{x}\left(a,b\right)

and

I_{x}\left(a,b\right)

; the generalized exponential integral

E_{p}\left(z\right)

; the generalized sine and cosine integrals

\operatorname{si}\left(a,z\right)

\operatorname{ci}\left(a,z\right)

\operatorname{Si}\left(a,z\right)

, and

\operatorname{Ci}\left(a,z\right)

. ►Alternative notations include: Prym’s functions

P_{z}(a)=\gamma\left(a,z\right)

Q_{z}(a)=\Gamma\left(a,z\right)

, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63);

(a,z)!=\gamma\left(a+1,z\right)

[a,z]!=\Gamma\left(a+1,z\right)

, Dingle (1973);

B(a,b,x)=\mathrm{B}_{x}\left(a,b\right)

I(a,b,x)=I_{x}\left(a,b\right)

, Magnus et al. (1966);

\operatorname{Si}\left(a,x\right)\to\operatorname{Si}\left(1-a,x\right)

\operatorname{Ci}\left(a,x\right)\to\operatorname{Ci}\left(1-a,x\right)

, Luke (1975).

23: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

…

24: 16.25 Methods of Computation

§16.25 Methods of Computation

…

25: 16 Generalized Hypergeometric Functions & Meijer G-Function

Chapter 16 Generalized Hypergeometric Functions and Meijer $G$ -Function

…

26: 35.10 Methods of Computation

§35.10 Methods of Computation

… ►See Yan (1992) for the

{{}_{1}F_{1}}

and

{{}_{2}F_{1}}

functions of matrix argument in the case

m=2

, and Bingham et al. (1992) for Monte Carlo simulation on

\mathbf{O}(m)

applied to a generalization of the integral (35.5.8). …

27: 16.6 Transformations of Variable

§16.6 Transformations of Variable

►

Quadratic

►

16.6.1

{{}_{3}F_{2}}\left({a,b,c\atop a-b+1,a-c+1};z\right)=(1-z)^{-a}{{}_{3}F_{2}}% \left({a-b-c+1,\frac{1}{2}a,\frac{1}{2}(a+1)\atop a-b+1,a-c+1};\frac{-4z}{(1-z% )^{2}}\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.6
Permalink:: http://dlmf.nist.gov/16.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.6, §16.6 and Ch.16

►

Cubic

►

16.6.2

{{}_{3}F_{2}}\left({a,2b-a-1,2-2b+a\atop b,a-b+\frac{3}{2}};\frac{z}{4}\right)% =(1-z)^{-a}{{}_{3}F_{2}}\left({\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{% 3}a+\frac{2}{3}\atop b,a-b+\frac{3}{2}};\frac{-27z}{4(1-z)^{3}}\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.6
Permalink:: http://dlmf.nist.gov/16.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.6, §16.6 and Ch.16

…

28: 30.12 Generalized and Coulomb Spheroidal Functions

§30.12 Generalized and Coulomb Spheroidal Functions

►Generalized spheroidal wave functions and Coulomb spheroidal functions are solutions of the differential equation … ►Another generalization is provided by the differential equation …

29: 6 Exponential, Logarithmic, Sine, and
Cosine Integrals

…

30: 7 Error Functions, Dawson’s and Fresnel Integrals

…