# §16.11 Asymptotic Expansions

## §16.11(i) Formal Series

For subsequent use we define two formal infinite series, $E_{p,q}(z)$ and $H_{p,q}(z)$, as follows:

 16.11.1 $E_{p,q}(z)=(2\pi)^{\ifrac{(p-q)}{2}}\kappa^{-\nu-(\ifrac{1}{2})}{\mathrm{e}}^{% \kappa z^{\ifrac{1}{\kappa}}}\sum_{k=0}^{\infty}c_{k}\left(\kappa z^{\ifrac{1}% {\kappa}}\right)^{\nu-k},$ $p, ⓘ Defines: $E_{p,q}(z)$: formal infinite series (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $\kappa$ and $\nu$ Referenced by: §16.11(i), §16.11(ii) Permalink: http://dlmf.nist.gov/16.11.E1 Encodings: TeX, pMML, png See also: Annotations for §16.11(i), §16.11 and Ch.16
 16.11.2 $H_{p,q}(z)=\sum_{m=1}^{p}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\Gamma\left(a_{% m}+k\right)\left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}\ell=1\\ \ell\neq m\end{subarray}}^{p}\Gamma\left(a_{\ell}-a_{m}-k\right)}{\prod\limits% _{\ell=1}^{q}\Gamma\left(b_{\ell}-a_{m}-k\right)}}\right)z^{-a_{m}-k}.$ ⓘ Defines: $H_{p,q}(z)$: formal infinite series (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $!$: 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: §16.11(ii), §16.11(ii) Permalink: http://dlmf.nist.gov/16.11.E2 Encodings: TeX, pMML, png See also: Annotations for §16.11(i), §16.11 and Ch.16

In (16.11.1)

 16.11.3 $\displaystyle\kappa$ $\displaystyle=q-p+1,$ $\displaystyle\nu$ $\displaystyle=a_{1}+\dots+a_{p}-b_{1}-\dots-b_{q}+\tfrac{1}{2}(q-p),$ ⓘ Defines: $\kappa$ (locally) and $\nu$ (locally) Symbols: $p$: nonnegative integer, $q$: 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/16.11.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §16.11(i), §16.11 and Ch.16

and

 16.11.4 $\displaystyle c_{0}$ $\displaystyle=1,$ $\displaystyle c_{k}$ $\displaystyle=-\frac{1}{k\kappa^{\kappa}}\sum_{m=0}^{k-1}c_{m}e_{k,m},$ $k\geq 1$, ⓘ Symbols: $\kappa$ and $e_{k,m}$ Permalink: http://dlmf.nist.gov/16.11.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §16.11(i), §16.11 and Ch.16

where

 16.11.5 $e_{k,m}=\sum_{j=1}^{q+1}{\left(1-\nu-\kappa b_{j}+m\right)_{\kappa+k-m}}\left(% {\textstyle\ifrac{\prod\limits_{\ell=1}^{p}(a_{\ell}-b_{j})}{\prod\limits_{% \begin{subarray}{c}\ell=1\\ \ell\neq j\end{subarray}}^{q+1}(b_{\ell}-b_{j})}}\right),$ ⓘ Defines: $e_{k,m}$ (locally) Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $p$: nonnegative integer, $q$: nonnegative integer, $a,a_{1},\ldots,a_{p}$: real or complex parameters, $b,b_{1},\ldots,b_{q}$: real or complex parameters, $\kappa$ and $\nu$ Permalink: http://dlmf.nist.gov/16.11.E5 Encodings: TeX, pMML, png See also: Annotations for §16.11(i), §16.11 and Ch.16

and $b_{q+1}=1$.

It may be observed that $H_{p,q}(z)$ represents the sum of the residues of the poles of the integrand in (16.5.1) at $s=-a_{j},-a_{j}-1,\dots$, $j=1,\dots,p$, provided that these poles are all simple, that is, no two of the $a_{j}$ differ by an integer. (If this condition is violated, then the definition of $H_{p,q}(z)$ has to be modified so that the residues are those associated with the multiple poles. In consequence, logarithmic terms may appear. See (15.8.8) for an example.)

## §16.11(ii) Expansions for Large Variable

In this subsection we assume that none of $a_{1},a_{2},\dots,a_{p}$ is a nonpositive integer.

### Case $p=q+1$

The formal series (16.11.2) for $H_{q+1,q}(z)$ converges if $\left|z\right|>1$, and

 16.11.6 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q+1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q+1}F_{q}}% \left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=H_{q+1,q}(-z),$ $|\operatorname{ph}\left(-z\right)|\leq\pi$;

compare (16.8.8).

### Case $p=q$

As $z\to\infty$ in $|\operatorname{ph}z|\leq\pi$,

 16.11.7 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q}F_{q}}% \left({a_{1},\dots,a_{q}\atop b_{1},\dots,b_{q}};z\right)\sim H_{q,q}(z{% \mathrm{e}}^{\mp\pi\mathrm{i}})+E_{q,q}(z).$

Here the upper or lower signs are chosen according as $z$ lies in the upper or lower half-plane; in consequence, in the fractional powers (§4.2(iv)) of $ze^{\mp\pi i}$ its phases are $\operatorname{ph}z\mp\pi$, respectively. (Either sign may be used when $\operatorname{ph}z=0$ since the first term on the right-hand side becomes exponentially small compared with the second term.)

Explicit representations for the coefficients $c_{k}$ are given in Volkmer and Wood (2014). The special case $a_{1}=1$, $p=q=2$ is discussed in Kim (1972).

### Case $p=q-1$

As $z\to\infty$ in $|\operatorname{ph}z|\leq\pi$,

 16.11.8 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q-1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q-1}F_{q}}% \left({a_{1},\dots,a_{q-1}\atop b_{1},\dots,b_{q}};-z\right)\sim H_{q-1,q}(z)+% E_{q-1,q}(ze^{-\pi\mathrm{i}})+E_{q-1,q}(ze^{\pi\mathrm{i}}),$

with the same conventions on the phases of $ze^{\mp\pi i}$.

### Case $p\leq q-2$

As $z\to\infty$ in $|\operatorname{ph}z|\leq\pi$,

 16.11.9 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{p}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{p}F_{q}}% \left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};-z\right)\sim E_{p,q}(ze^{-% \pi\mathrm{i}})+E_{p,q}(ze^{\pi\mathrm{i}}),$

with the same conventions on the phases of $ze^{\mp\pi i}$.

## §16.11(iii) Expansions for Large Parameters

If $z$ is fixed and $|\operatorname{ph}\left(1-z\right)|<\pi$, then for each nonnegative integer $m$

 16.11.10 ${{}_{p+1}F_{p}}\left({a_{1}+r,\dots,a_{k-1}+r,a_{k},\dots,a_{p+1}\atop b_{1}+r% ,\dots,b_{k}+r,b_{k+1},\dots,b_{p}};z\right)=\sum_{n=0}^{m-1}\frac{{\left(a_{1% }+r\right)_{n}}\cdots{\left(a_{k-1}+r\right)_{n}}{\left(a_{k}\right)_{n}}% \cdots{\left(a_{p+1}\right)_{n}}}{{\left(b_{1}+r\right)_{n}}\cdots{\left(b_{k}% +r\right)_{n}}{\left(b_{k+1}\right)_{n}}\cdots{\left(b_{p}\right)_{n}}}\frac{z% ^{n}}{n!}+O\left(\frac{1}{r^{m}}\right),$

as $r\to+\infty$. Here $k$ can have any integer value from $1$ to $p$. Also if $p, then

 16.11.11 ${{}_{p}F_{q}}\left({a_{1}+r,\dots,a_{p}+r\atop b_{1}+r,\dots,b_{q}+r};z\right)% =\sum_{n=0}^{m-1}\frac{{\left(a_{1}+r\right)_{n}}\cdots{\left(a_{p}+r\right)_{% n}}}{{\left(b_{1}+r\right)_{n}}\cdots{\left(b_{q}+r\right)_{n}}}\frac{z^{n}}{n% !}+O\left(\frac{1}{r^{(q-p)m}}\right),$

again as $r\to+\infty$. For these and other results see Knottnerus (1960). See also Luke (1969a, §7.3).

Asymptotic expansions for the polynomials ${{}_{p+2}F_{q}}\left(-r,r+a_{0},\mathbf{a};\mathbf{b};z\right)$ as $r\to\infty$ through integer values are given in Fields and Luke (1963b, a) and Fields (1965).