§16.2 Definition and Analytic Properties

§16.2(i) Generalized Hypergeometric Series

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 ${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\sum_{k% =0}^{\infty}\frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}}{{% \left(b_{1}\right)_{k}}\cdots{\left(b_{q}\right)_{k}}}\frac{z^{k}}{k!}.$

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 $p\leq q$

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 $p=q+1$

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 $p>q+1$

Polynomials

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 ${{}_{p+1}F_{q}}\left({-m,\mathbf{a}\atop\mathbf{b}};z\right)=\frac{{\left(% \mathbf{a}\right)_{m}}(-z)^{m}}{{\left(\mathbf{b}\right)_{m}}}{{}_{q+1}F_{p}}% \left({-m,1-m-\mathbf{b}\atop 1-m-\mathbf{a}};\frac{(-1)^{p+q}}{z}\right)$

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 $\sum_{k=0}^{m}\frac{{\left(\mathbf{a}\right)_{k}}}{{\left(\mathbf{b}\right)_{k% }}}\frac{z^{k}}{k!}=\frac{{\left(\mathbf{a}\right)_{m}}z^{m}}{{\left(\mathbf{b% }\right)_{m}}m!}{{}_{q+2}F_{p}}\left({-m,1,1-m-\mathbf{b}\atop 1-m-\mathbf{a}}% ;\frac{(-1)^{p+q+1}}{z}\right).$

Non-Polynomials

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

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 $a_{1},\dots,a_{p},b_{1},\dots,b_{q}$.