# §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 exponential function, $p$: 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)
 16.11.2 $H_{p,q}(z)=\sum_{m=1}^{p}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\mathop{\Gamma% \/}\nolimits\!\left(a_{m}+k\right)\left({\textstyle\ifrac{\prod\limits_{% \begin{subarray}{c}\ell=1\\ \ell\neq m\end{subarray}}^{p}\mathop{\Gamma\/}\nolimits\!\left(a_{\ell}-a_{m}-% k\right)}{\prod\limits_{\ell=1}^{q}\mathop{\Gamma\/}\nolimits\!\left(b_{\ell}-% a_{m}-k\right)}}\right)z^{-a_{m}-k}.$ Defines: $H_{p,q}(z)$: formal infinite series (locally) Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: 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)

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)

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)

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)

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

compare (16.8.8).

### Case $p=q$

As $z\to\infty$ in $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi$,

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

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 $\mathop{\mathrm{ph}\/}\nolimits z\mp\pi$, respectively. (Either sign may be used when $\mathop{\mathrm{ph}\/}\nolimits z=0$ 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 $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi$,

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

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

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

As $z\to\infty$ in $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi$,

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

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

## §16.11(iii) Expansions for Large Parameters

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

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

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

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

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

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