§16.11 Asymptotic Expansions

Referenced by:: Ch.16
Permalink:: http://dlmf.nist.gov/16.11

§16.11(i) Formal Series
§16.11(ii) Expansions for Large Variable
§16.11(iii) Expansions for Large Parameters

§16.11(i) Formal Series

Notes:: See Paris and Kaminski (2001, §2.3).
Keywords:: generalized hypergeometric functions
Referenced by:: ¶ ‣ §2.10(iii)
Permalink:: http://dlmf.nist.gov/16.11.i

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

16.11.1 $E_{{p,q}}(z)=(2\pi)^{{\ifrac{(p-q)}{2}}}\kappa^{{-\nu-(\ifrac{1}{2})}}e^{{\kappa z^{{\ifrac{1}{\kappa}}}}}\sum _{{k=0}}^{\infty}c_{k}\left(\kappa z^{{\ifrac{1}{\kappa}}}\right)^{{\nu-k}},$ $p<q+1$ ,

Symbols:: $e$ : base of exponential function, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $E_{{p,q}}(z)$ : formal infinite series, $\kappa$ and $\nu$
Referenced by:: §16.11(i), §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.11.E1
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16.11.2 $H_{{p,q}}(z)=\sum _{{m=1}}^{p}\sum _{{k=0}}^{\infty}\frac{(-1)^{k}}{k!}\mathop{\Gamma\/}\nolimits\!\left(a_{m}+k\right)\left({\textstyle\ifrac{\prod\limits _{{\substack{\ell=1\\ \ell\neq m}}}^{p}\mathop{\Gamma\/}\nolimits\!\left(a_{\ell}-a_{m}-k\right)}{\prod\limits _{{\ell=1}}^{q}\mathop{\Gamma\/}\nolimits\!\left(b_{\ell}-a_{m}-k\right)}}\right)z^{{-a_{m}-k}}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $!$ : $n!$ : factorial, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $H_{{p,q}}(z)$ : formal infinite series
Referenced by:: ¶ ‣ §16.11(ii), §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.11.E2
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In (16.11.1)

16.11.3

$\kappa=q-p+1,$

$\nu=a_{1}+\dots+a_{p}-b_{1}-\dots-b_{q}+\tfrac{1}{2}(q-p),$

Symbols:: $p$ : 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.E3
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and

16.11.4

$c_{0}=1,$

$c_{k}=-\frac{1}{k\kappa^{\kappa}}\sum _{{m=0}}^{{k-1}}c_{m}e_{{k,m}},$ $k\geq 1$ ,

Symbols:: $\kappa$ and $e_{{k,m}}$
Permalink:: http://dlmf.nist.gov/16.11.E4
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where

16.11.5 $e_{{k,m}}=\sum _{{j=1}}^{{q+1}}\left(1-\nu-\kappa b_{j}+m\right)_{{\kappa+k-m}}\left({\textstyle\ifrac{\prod\limits _{{\ell=1}}^{p}(a_{\ell}-b_{j})}{\prod\limits _{{\substack{\ell=1\\ \ell\neq j}}}^{{q+1}}(b_{\ell}-b_{j})}}\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $p$ : 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, $\kappa$ , $\nu$ and $e_{{k,m}}$
Permalink:: http://dlmf.nist.gov/16.11.E5
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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

Notes:: See Paris and Kaminski (2001, §2.3), Wright (1940a, p. 391), and Meijer (1946, p. 1172).
Keywords:: generalized hypergeometric functions
Referenced by:: ¶ ‣ §2.10(iii), Version 1.0.1 (June 27, 2011)
Permalink:: http://dlmf.nist.gov/16.11.ii
Clarification (effective with 1.0.1):: The choice of branches in (16.11.7)–(16.11.9) was not originally made clear; it was clarified that in the fractional powers (§4.2(iv)) of $z$ appearing in (16.11.1) and (16.11.2), the phases of $z\mathop{\exp\/}\nolimits\!\left(\mp\pi i\right)$ are $\mathop{\mathrm{ph}\/}\nolimits\!\left(z\right)\mp\pi$ , respectively.
Reported 2010-06-14 by Fredrik Johansson

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $H_{{p,q}}(z)$ : formal infinite series
Permalink:: http://dlmf.nist.gov/16.11.E6
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compare (16.8.8).

¶ Case $p=q$

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $e$ : base of exponential function, $\sim$ : asymptotic equality, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $E_{{p,q}}(z)$ : formal infinite series and $H_{{p,q}}(z)$ : formal infinite series
Referenced by:: ¶ ‣ §16.11(ii), §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.11.E7
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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.)

For the special case $a_{1}=1$ , $p=q=2$ explicit representations for the right-hand side of (16.11.7) in terms of generalized hypergeometric functions are given in Kim (1972).

¶ Case $p=q-1$

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\sim$ : asymptotic equality, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $E_{{p,q}}(z)$ : formal infinite series, $H_{{p,q}}(z)$ : formal infinite series and $e_{{k,m}}$
Permalink:: http://dlmf.nist.gov/16.11.E8
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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 $\left({\textstyle\ifrac{\prod\limits _{{\ell=1}}^{p}\mathop{\Gamma\/}\nolimits\!\left(a_{\ell}\right)}{\prod\limits _{{\ell=1}}^{q}\mathop{\Gamma\/}\nolimits\!\left(b_{\ell}\right)}}\right)\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};-z\right)\sim E_{{p,q}}(ze^{{-\pi i}})+E_{{p,q}}(ze^{{\pi i}}),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\sim$ : asymptotic equality, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $E_{{p,q}}(z)$ : formal infinite series and $e_{{k,m}}$
Referenced by:: §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.11.E9
Encodings:: TeX, pMML, png

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

§16.11(iii) Expansions for Large Parameters

Notes:: See Luke (1969a, Chapter 7).
Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/16.11.iii

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $p$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $r$ : large parameter
Permalink:: http://dlmf.nist.gov/16.11.E10
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as $r\to+\infty$ . Here $k$ can have any integer value from 1 to $p$ . Also if $p<q$ , then

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $r$ : large parameter
Permalink:: http://dlmf.nist.gov/16.11.E11
Encodings:: TeX, pMML, png

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).

§16.11 Asymptotic Expansions

Contents

§16.11(i) Formal Series

§16.11(ii) Expansions for Large Variable

¶ Case

¶ Case

¶ Case

¶ Case

§16.11(iii) Expansions for Large Parameters

¶ Case $p=q+1$

¶ Case $p=q$

¶ Case $p=q-1$

¶ Case $p\leq q-2$