functions Fℓ(η,ρ),Gℓ(η,ρ),H±ℓ(η,ρ)

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11—20 of 236 matching pages

11: Bille C. Carlson

… ►In his paper Lauricella’s hypergeometric function

F_{D}

(1963), he defined the

R

-function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. …

12: 33.13 Complex Variable and Parameters

… ►The functions

F_{\ell}\left(\eta,\rho\right)

G_{\ell}\left(\eta,\rho\right)

, and

{H^{\pm}_{\ell}}\left(\eta,\rho\right)

may be extended to noninteger values of

\ell

by generalizing

(2\ell+1)!=\Gamma\left(2\ell+2\right)

, and supplementing (33.6.5) by a formula derived from (33.2.8) with

U\left(a,b,z\right)

expanded via (13.2.42). …

13: 33.2 Definitions and Basic Properties

… ►

§33.2(ii) Regular Solution $F_{\ell}\left(\eta,\rho\right)$

►The function

F_{\ell}\left(\eta,\rho\right)

is recessive (§2.7(iii)) at

\rho=0

, and is defined by … ►

F_{\ell}\left(\eta,\rho\right)

is a real and analytic function of

\rho

on the open interval

0<\rho<\infty

, and also an analytic function of

\eta

when

-\infty<\eta<\infty

. … ►

§33.2(iii) Irregular Solutions $G_{\ell}\left(\eta,\rho\right),{H^{\pm}_{\ell}}\left(\eta,\rho\right)$

… ►As in the case of

F_{\ell}\left(\eta,\rho\right)

, the solutions

{H^{\pm}_{\ell}}\left(\eta,\rho\right)

and

G_{\ell}\left(\eta,\rho\right)

are analytic functions of

\rho

when

0<\rho<\infty

. …

14: 16.12 Products

… ►

16.12.1

{{}_{0}F_{1}}\left(-;a;z\right){{}_{0}F_{1}}\left(-;b;z\right)={{}_{2}F_{3}}% \left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop a,b,a+b-1};4z\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:: (10.8.3), §10.8, §10.8, §16.12, Erratum (V1.0.17) for Section 10.8
Permalink:: http://dlmf.nist.gov/16.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

… ►

16.12.2

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

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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:: §16.12, §17.9(iii), Erratum (V1.1.12) for Subsection 17.9(iii)
Permalink:: http://dlmf.nist.gov/16.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

… ►

16.12.3

\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)\right)^{2}=\sum_{k=0}^{\infty}% \frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}}{\left(c-\frac{1}{2}\right)_{k% }}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}k!}{{}_{4}F_{3}}\left({-\frac{1% }{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}\atop a+\frac{1}{2},b+% \frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k},

|z|<1

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $z$ : 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:: §16.12
Permalink:: http://dlmf.nist.gov/16.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.12 and Ch.16

…

15: 16.5 Integral Representations and Integrals

… ►

16.5.1

\left({\textstyle\ifrac{\prod\limits_{k=1}^{p}\Gamma\left(a_{k}\right)}{\prod% \limits_{k=1}^{q}\Gamma\left(b_{k}\right)}}\right){{}_{p}F_{q}}\left({a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac{1}{2\pi\mathrm{i}}\int_{L}% \left({\textstyle\ifrac{\prod\limits_{k=1}^{p}\Gamma\left(a_{k}+s\right)}{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}+s\right)}}\right)\Gamma\left(-s\right)% (-z)^{s}\,\mathrm{d}s,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : 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:: §13.6(vi), §16.11(i), §16.5, §16.5
Permalink:: http://dlmf.nist.gov/16.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

… ►In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as

z\to 0

in the sector

|\operatorname{ph}\left(-z\right)|\leq(p+1-q-\delta)\pi/2

, where

\delta

is an arbitrary small positive constant. … ►

16.5.2

{{}_{p+1}F_{q+1}}\left({a_{0},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=% \frac{\Gamma\left(b_{0}\right)}{\Gamma\left(a_{0}\right)\Gamma\left(b_{0}-a_{0% }\right)}\int_{0}^{1}t^{a_{0}-1}(1-t)^{b_{0}-a_{0}-1}{{}_{p}F_{q}}\left({a_{1}% ,\dots,a_{p}\atop b_{1},\dots,b_{q}};zt\right)\,\mathrm{d}t,

\Re b_{0}>\Re a_{0}>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Keywords:: Mellin transform
Referenced by:: §16.5, §16.5
Permalink:: http://dlmf.nist.gov/16.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

►

16.5.3

{{}_{p+1}F_{q}}\left({a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac% {1}{\Gamma\left(a_{0}\right)}\int_{0}^{\infty}{\mathrm{e}}^{-t}t^{a_{0}-1}{{}_% {p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\cdots,b_{q}};zt\right)\,\mathrm{% d}t,

\Re z<1

\Re a_{0}>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Keywords:: Laplace transform, Mellin transform
Referenced by:: §16.5
Permalink:: http://dlmf.nist.gov/16.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

►

16.5.4

{{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=\frac% {\Gamma\left(b_{0}\right)}{2\pi\mathrm{i}}\int_{c-\mathrm{i}\infty}^{c+\mathrm% {i}\infty}{\mathrm{e}}^{t}t^{-b_{0}}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b% _{1},\dots,b_{q}};\frac{z}{t}\right)\,\mathrm{d}t,

c>0

\Re b_{0}>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Keywords:: Mellin transform
Referenced by:: §16.5, §16.5
Permalink:: http://dlmf.nist.gov/16.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

…

16: 15.6 Integral Representations

… ►The function

\mathbf{F}\left(a,b;c;z\right)

(not

F\left(a,b;c;z\right)

) has the following integral representations: ►

15.6.1

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re b>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
Permalink:: http://dlmf.nist.gov/15.6.E1
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

►

15.6.2

\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1+b-c\right)}{2\pi\mathrm{i}% \Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}\,% \mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

c-b\neq 1,2,3,\dots

\Re b>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.6
Permalink:: http://dlmf.nist.gov/15.6.E2
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

… ►

15.6.6

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)% \Gamma\left(b\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma% \left(a+t\right)\Gamma\left(b+t\right)\Gamma\left(-t\right)}{\Gamma\left(c+t% \right)}(-z)^{t}\,\mathrm{d}t,

|\operatorname{ph}\left(-z\right)|<\pi

;

a,b\neq 0,-1,-2,\dots

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
A&S Ref:: 15.3.2 (modified)
Referenced by:: §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
Permalink:: http://dlmf.nist.gov/15.6.E6
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

… ►

15.6.8

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(c-d\right)}\int_{0}^{1}% \mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-1}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re d>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Proof sketch:: Follows from (15.6.9) with $\lambda=a+b$ .
Referenced by:: (15.6.9), §15.6, §15.6, §15.6, §18.17(iv), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)
Permalink:: http://dlmf.nist.gov/15.6.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

…

17: 16.18 Special Cases

… ►The

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

and

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

functions introduced in Chapters 13 and 15, as well as the more general

{{}_{p}F_{q}}

functions introduced in the present chapter, are all special cases of the Meijer

G

-function. … ►

16.18.1

{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\left({% \textstyle\ifrac{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_% {k=1}^{p}\Gamma\left(a_{k}\right)}}\right){G^{1,p}_{p,q+1}}\left(-z;{1-a_{1},% \dots,1-a_{p}\atop 0,1-b_{1},\dots,1-b_{q}}\right)=\left({\textstyle\ifrac{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}{\prod\limits_{k=1}^{p}\Gamma% \left(a_{k}\right)}}\right){G^{p,1}_{q+1,p}}\left(-\frac{1}{z};{1,b_{1},\dots,% b_{q}\atop a_{1},\dots,a_{p}}\right).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.18.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.18 and Ch.16

►As a corollary, special cases of the

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

and

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

functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer

G

-function. …

18: 15.4 Special Cases

… ►

15.4.20

F\left(a,b;c;1\right)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{% \Gamma\left(c-a\right)\Gamma\left(c-b\right)}.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
A&S Ref:: 15.1.20
Referenced by:: §13.23(i), §15.4(ii), §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.4.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §15.4(ii), §15.4 and Ch.15

… ►

15.4.21

\lim_{z\to 1-}\frac{F\left(a,b;a+b;z\right)}{-\ln\left(1-z\right)}=\frac{% \Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\ln\NVar{z}$ : principal branch of logarithm function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter
Referenced by:: §15.4(ii)
Permalink:: http://dlmf.nist.gov/15.4.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §15.4(ii), §15.4 and Ch.15

… ►

15.4.23

\lim_{z\to 1-}\frac{F\left(a,b;c;z\right)}{(1-z)^{c-a-b}}=\frac{\Gamma\left(c% \right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.4(ii)
Permalink:: http://dlmf.nist.gov/15.4.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §15.4(ii), §15.4 and Ch.15

… ►

15.4.26

F\left(a,b;a-b+1;-1\right)=\frac{\Gamma\left(a-b+1\right)\Gamma\left(\tfrac{1}% {2}a+1\right)}{\Gamma\left(a+1\right)\Gamma\left(\tfrac{1}{2}a-b+1\right)}.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $a$ : real or complex parameter and $b$ : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.4.E26
Encodings:: pMML, png, TeX
See also:: Annotations for §15.4(iii), §15.4 and Ch.15

… ►

15.4.31

F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2}-2a;-\tfrac{1}{3}\right)=\left(\frac{8}{9}% \right)^{-2a}\frac{\Gamma\left(\tfrac{4}{3}\right)\Gamma\left(\tfrac{3}{2}-2a% \right)}{\Gamma\left(\tfrac{3}{2}\right)\Gamma\left(\tfrac{4}{3}-2a\right)}.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function and $a$ : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.4.E31
Encodings:: pMML, png, TeX
See also:: Annotations for §15.4(iii), §15.4 and Ch.15

…

19: 33.6 Power-Series Expansions in $\rho$

… ►

33.6.1

F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\sum_{k=\ell+1}^{% \infty}A_{k}^{\ell}(\eta)\rho^{k},

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $A_{\ell+1}^{\ell}$ : coefficient
A&S Ref:: 14.1.4, 14.1.5
Referenced by:: §33.6
Permalink:: http://dlmf.nist.gov/33.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §33.6 and Ch.33

►

33.6.2

F_{\ell}'\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\sum_{k=\ell+1}^{% \infty}kA_{k}^{\ell}(\eta)\rho^{k-1},

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $A_{\ell+1}^{\ell}$ : coefficient
A&S Ref:: 14.1.4, 14.1.5
Referenced by:: §33.6
Permalink:: http://dlmf.nist.gov/33.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §33.6 and Ch.33

… ►

33.6.4

A_{k}^{\ell}(\eta)=\dfrac{(-\mathrm{i})^{k-\ell-1}}{(k-\ell-1)!}\*{{}_{2}F_{1}% }\left(\ell+1-k,\ell+1-\mathrm{i}\eta;2\ell+2;2\right).

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $A_{\ell+1}^{\ell}$ : coefficient
Referenced by:: §33.6
Permalink:: http://dlmf.nist.gov/33.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §33.6 and Ch.33

…

20: 15.2 Definitions and Analytical Properties

… ►The hypergeometric function

F\left(a,b;c;z\right)

is defined by the Gauss series … ►The principal branch of

\mathbf{F}\left(a,b;c;z\right)

is an entire function of

a

b

, and

c

. …As a multivalued function of

z

\mathbf{F}\left(a,b;c;z\right)

is analytic everywhere except for possible branch points at

z=0

1

, and

\infty

. The same properties hold for

F\left(a,b;c;z\right)

, except that as a function of

c

F\left(a,b;c;z\right)

in general has poles at

c=0,-1,-2,\dots

. … ►For example, when

a=-m

m=0,1,2,\dots

, and

c\neq 0,-1,-2,\dots

F\left(a,b;c;z\right)

is a polynomial: …

functions Fℓ(η,ρ),Gℓ(η,ρ),H±ℓ(η,ρ)

11—20 of 236 matching pages

11: Bille C. Carlson

12: 33.13 Complex Variable and Parameters

13: 33.2 Definitions and Basic Properties

§33.2(ii) Regular Solution F ℓ ⁡ ( η , ρ )

§33.2(iii) Irregular Solutions G ℓ ⁡ ( η , ρ ) , H ℓ ± ⁡ ( η , ρ )

14: 16.12 Products

15: 16.5 Integral Representations and Integrals

16: 15.6 Integral Representations

17: 16.18 Special Cases

18: 15.4 Special Cases

19: 33.6 Power-Series Expansions in ρ

20: 15.2 Definitions and Analytical Properties

§33.2(ii) Regular Solution $F_{\ell}\left(\eta,\rho\right)$

§33.2(iii) Irregular Solutions $G_{\ell}\left(\eta,\rho\right),{H^{\pm}_{\ell}}\left(\eta,\rho\right)$

19: 33.6 Power-Series Expansions in $\rho$