16 Generalized Hypergeometric Functions & Meijer G-Function Generalized Hypergeometric Functions 16.1 Special Notation 16.3 Derivatives and Contiguous Functions

§16.2 Definition and Analytic Properties

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Defines:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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 and ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function
Referenced by:: §18.20(ii), §18.23, §18.26(i), §18.5(iii), §34.2, §34.4, §5.16
Permalink:: http://dlmf.nist.gov/16.2
See also:: Annotations for Ch.16

§16.2(i) Generalized Hypergeometric Series
§16.2(ii) Case $p\leq q$
§16.2(iii) Case $p=q+1$
§16.2(iv) Case $p>q+1$
§16.2(v) Behavior with Respect to Parameters

§16.2(i) Generalized Hypergeometric Series

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Keywords:: definition, definitions, generalized hypergeometric function ${{}_{0}F_{2}}$ , generalized hypergeometric functions, generalized hypergeometric series, notation
Referenced by:: §13.1, §13.31(iii), §17.4(i), §2.10(iii), §7.11
Permalink:: http://dlmf.nist.gov/16.2.i
See also:: Annotations for §16.2 and Ch.16

Throughout this chapter it is assumed that none of the bottom parameters $b_{1}$ , $b_{2}$ , $\dots$ , $b_{q}$ is a nonpositive integer, unless stated otherwise. Then formally

16.2.1		${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\sum_{k% =0}^{\infty}\frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}}{{% \left(b_{1}\right)_{k}}\cdots{\left(b_{q}\right)_{k}}}\frac{z^{k}}{k!}.$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : 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: §10.16, §10.16, §10.39, §16.2(iv), §16.2(ii), §16.2(iii), §16.2(iii), §16.2(iii), §16.5, §18.17(v), §18.17(vi), §18.17(vii), §18.30 Permalink: http://dlmf.nist.gov/16.2.E1 Encodings: TeX, pMML, png See also: Annotations for §16.2(i), §16.2 and Ch.16

Equivalently, the function is denoted by ${{}_{p}F_{q}}\left({\mathbf{a}\atop\mathbf{b}};z\right)$ or ${{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ , and sometimes, for brevity, by ${{}_{p}F_{q}}\left(z\right)$ .

§16.2(ii) Case $p\leq q$

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Keywords:: analytic properties, generalized hypergeometric functions
Referenced by:: §18.34(i), §2.10(iii), §7.11
Permalink:: http://dlmf.nist.gov/16.2.ii
See also:: Annotations for §16.2 and Ch.16

When $p\leq q$ the series (16.2.1) converges for all finite values of $z$ and defines an entire function.

§16.2(iii) Case $p=q+1$

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Keywords:: analytic properties, generalized hypergeometric functions, principal branch (value), terminating
Notes:: See Luke (1975, §5.2.1) and Slater (1966, §2.2).
Referenced by:: §16.5
Permalink:: http://dlmf.nist.gov/16.2.iii
See also:: Annotations for §16.2 and Ch.16

Suppose first one or more of the top parameters $a_{j}$ is a nonpositive integer. Then the series (16.2.1) terminates and the generalized hypergeometric function is a polynomial in $z$ .

If none of the $a_{j}$ is a nonpositive integer, then the radius of convergence of the series (16.2.1) is $1$ , and outside the open disk $|z|<1$ the generalized hypergeometric function is defined by analytic continuation with respect to $z$ . The branch obtained by introducing a cut from $1$ to $+\infty$ on the real axis, that is, the branch in the sector $|\operatorname{ph}\left(1-z\right)|\leq\pi$ , is the principal branch (or principal value) of ${{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ ; compare §4.2(i). Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at $z=0,1$ , and $\infty$ . Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values.

On the circle $|z|=1$ the series (16.2.1) is absolutely convergent if $\Re\gamma_{q}>0$ , convergent except at $z=1$ if $-1<\Re\gamma_{q}\leq 0$ , and divergent if $\Re\gamma_{q}\leq-1$ , where

16.2.2		$\gamma_{q}=(b_{1}+\dots+b_{q})-(a_{1}+\dots+a_{q+1}).$
ⓘ Symbols: $q$ : nonnegative integer, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters Permalink: http://dlmf.nist.gov/16.2.E2 Encodings: TeX, pMML, png See also: Annotations for §16.2(iii), §16.2 and Ch.16

§16.2(iv) Case $p>q+1$

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Keywords:: analytic properties, generalized hypergeometric functions
Notes:: See Luke (1975, §5.2.1).
Referenced by:: §13.6(vi), §18.34(i)
Permalink:: http://dlmf.nist.gov/16.2.iv
See also:: Annotations for §16.2 and Ch.16

Polynomials

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Keywords:: generalized hypergeometric functions, polynomial cases, terminating
See also:: Annotations for §16.2(iv), §16.2 and Ch.16

In general the series (16.2.1) diverges for all nonzero values of $z$ . However, when one or more of the top parameters $a_{j}$ is a nonpositive integer the series terminates and the generalized hypergeometric function is a polynomial in $z$ . Note that if $-m$ is the value of the numerically largest $a_{j}$ that is a nonpositive integer, then the identity

16.2.3		${{}_{p+1}F_{q}}\left({-m,\mathbf{a}\atop\mathbf{b}};z\right)=\frac{{\left(% \mathbf{a}\right)_{m}}(-z)^{m}}{{\left(\mathbf{b}\right)_{m}}}{{}_{q+1}F_{p}}% \left({-m,1-m-\mathbf{b}\atop 1-m-\mathbf{a}};\frac{(-1)^{p+q}}{z}\right)$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $p$ : nonnegative integer, $q$ : nonnegative integer and $z$ : complex variable Referenced by: §13.6(v) Permalink: http://dlmf.nist.gov/16.2.E3 Encodings: TeX, pMML, png See also: Annotations for §16.2(iv), §16.2(iv), §16.2 and Ch.16

can be used to interchange $p$ and $q$ .

Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via

16.2.4		$\sum_{k=0}^{m}\frac{{\left(\mathbf{a}\right)_{k}}}{{\left(\mathbf{b}\right)_{k% }}}\frac{z^{k}}{k!}=\frac{{\left(\mathbf{a}\right)_{m}}z^{m}}{{\left(\mathbf{b% }\right)_{m}}m!}{{}_{q+2}F_{p}}\left({-m,1,1-m-\mathbf{b}\atop 1-m-\mathbf{a}}% ;\frac{(-1)^{p+q+1}}{z}\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/16.2.E4 Encodings: TeX, pMML, png See also: Annotations for §16.2(iv), §16.2(iv), §16.2 and Ch.16

Non-Polynomials

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See also:: Annotations for §16.2(iv), §16.2 and Ch.16

See §16.5 for the definition of ${{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ as a contour integral when $p>q+1$ and none of the $a_{k}$ is a nonpositive integer. (However, except where indicated otherwise in the DLMF we assume that when $p>q+1$ at least one of the $a_{k}$ is a nonpositive integer.)

§16.2(v) Behavior with Respect to Parameters

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Keywords:: analytic properties, as functions of parameters, generalized hypergeometric functions
Notes:: The statement that follows (16.2.5) follows from the uniform convergence of (16.2.5) when $p\leq q$ , and also when $p=q+1$ provided that $|z|<1$ . For other values of $z$ , apply the straightforward generalization (to higher-order differential equations) of Theorem 3.2 in Olver (1997b, Chapter 5).
Permalink:: http://dlmf.nist.gov/16.2.v
See also:: Annotations for §16.2 and Ch.16

Let

16.2.5		${{}_{p}{\mathbf{F}}_{q}}\left(\mathbf{a};\mathbf{b};z\right)=\ifrac{{{}_{p}F_{% q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)}{\left(\Gamma% \left(b_{1}\right)\cdots\Gamma\left(b_{q}\right)\right)}=\sum_{k=0}^{\infty}% \frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}}{\Gamma\left(b_{1% }+k\right)\cdots\Gamma\left(b_{q}+k\right)}\frac{z^{k}}{k!};$
ⓘ Defines: ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : 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: §16.2(v) Permalink: http://dlmf.nist.gov/16.2.E5 Encodings: TeX, pMML, png See also: Annotations for §16.2(v), §16.2 and Ch.16

compare (15.2.2) in the case $p=2$ , $q=1$ . When $p\leq q+1$ and $z$ is fixed and not a branch point, any branch of ${{}_{p}{\mathbf{F}}_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ is an entire function of each of the parameters $a_{1},\dots,a_{p},b_{1},\dots,b_{q}$ .

§16.2 Definition and Analytic Properties

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