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

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

## §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 $|\mathop{\mathrm{ph}\/}\nolimits\!\left(1-z\right)|\leq\pi$, is the principal branch (or principal value) of $\mathop{{{}_{q+1}F_{q}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$; 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(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 $\mathop{{{}_{p+1}F_{q}}\/}\nolimits\!\left({-m,\mathbf{a}\atop\mathbf{b}};z% \right)=\frac{{\left(\mathbf{a}\right)_{m}}(-z)^{m}}{{\left(\mathbf{b}\right)_% {m}}}\mathop{{{}_{q+1}F_{p}}\/}\nolimits\!\left({-m,1-m-\mathbf{b}\atop 1-m-% \mathbf{a}};\frac{(-1)^{p+q}}{z}\right)$

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

### Non-Polynomials

See §16.5 for the definition of $\mathop{{{}_{p}F_{q}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$ 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 $\mathop{{{}_{p}{\mathbf{F}}_{q}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z% \right)=\ifrac{\mathop{{{}_{p}F_{q}}\/}\nolimits\!\left({a_{1},\dots,a_{p}% \atop b_{1},\dots,b_{q}};z\right)}{\left(\mathop{\Gamma\/}\nolimits\!\left(b_{% 1}\right)\cdots\mathop{\Gamma\/}\nolimits\!\left(b_{q}\right)\right)}=\sum_{k=% 0}^{\infty}\frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_{k}}}{% \mathop{\Gamma\/}\nolimits\!\left(b_{1}+k\right)\cdots\mathop{\Gamma\/}% \nolimits\!\left(b_{q}+k\right)}\frac{z^{k}}{k!};$ Defines: $\mathop{{{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\/}\nolimits\!\left(\NVar{% \mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or $\mathop{{{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\/}\nolimits\!\left({\NVar{% \mathbf{a}}\atop\NVar{\mathbf{b}}};\NVar{z}\right)$: scaled (or Olver’s) generalized hypergeometric function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{{{}_{\NVar{p}}F_{\NVar{q}}}\/}\nolimits\!\left(\NVar{a_{1},\dots,a_{p}% };\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)$ or $\mathop{{{}_{\NVar{p}}F_{\NVar{q}}}\/}\nolimits\!\left({\NVar{a_{1},\dots,a_{p% }}\atop\NVar{b_{1},\dots,b_{q}}};\NVar{z}\right)$: alternatively $\mathop{{{}_{\NVar{p}}F_{\NVar{q}}}\/}\nolimits\!\left(\NVar{\mathbf{a}};\NVar% {\mathbf{b}};\NVar{z}\right)$ or $\mathop{{{}_{\NVar{p}}F_{\NVar{q}}}\/}\nolimits\!\left({\NVar{\mathbf{a}}\atop% \NVar{\mathbf{b}}};\NVar{z}\right)$ 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)

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 $\mathop{{{}_{p}{\mathbf{F}}_{q}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$ is an entire function of each of the parameters $a_{1},\dots,a_{p},b_{1},\dots,b_{q}$.