16 Generalized Hypergeometric Functions & Meijer G-Function Generalized Hypergeometric Functions 16.10 Expansions in Series of

{{}_{p}F_{q}}

§16.11 Asymptotic Expansions

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Referenced by:: Ch.16
Permalink:: http://dlmf.nist.gov/16.11
Addition (effective with 1.1.9):: Text referring to explicit representations for the coefficients $c_{k}$ given in Volkmer (2023) was inserted just below (16.11.5).
See also:: Annotations for Ch.16

§16.11(i) Formal Series

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Keywords:: asymptotic expansions, formal series, generalized hypergeometric functions
Notes:: See Paris and Kaminski (2001, §2.3).
Referenced by:: §2.10(iii), Erratum (V1.1.9) for Section 16.11(i)
Permalink:: http://dlmf.nist.gov/16.11.i
See also:: Annotations for §16.11 and Ch.16

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

16.11.1		$E_{p,q}(z)=(2\pi)^{\ifrac{(p-q)}{2}}\kappa^{-\nu-(\ifrac{1}{2})}{\mathrm{e}}^{% \kappa z^{\ifrac{1}{\kappa}}}\sum_{k=0}^{\infty}c_{k}\left(\kappa z^{\ifrac{1}% {\kappa}}\right)^{\nu-k},$
$p<q 1$ ,
ⓘ 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	$\displaystyle\kappa$	$\displaystyle=q-p 1,$
16.11.3	$\displaystyle\nu$	$\displaystyle=a_{1} \dots a_{p}-b_{1}-\dots-b_{q} \tfrac{1}{2}(q-p),$
ⓘ 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 $\nu$ Referenced by: §16.11, Erratum (V1.1.9) for Section 16.11(i) 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 $b_{q 1}=1$ . Explicit representations for the coefficients $c_{k}$ are given in Volkmer (2023).

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

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Keywords:: asymptotic expansions, generalized hypergeometric functions, large variable
Notes:: See Paris and Kaminski (2001, §2.3), Wright (1940a, p. 391), and Meijer (1946, p. 1172).
Referenced by:: §2.10(iii), Erratum (V1.0.1) for Clarifications, Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/16.11.ii
Changes and Addition (effective with 1.0.11):: The paragraph below (16.11.7), which read originally

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).

was rewritten and a new reference to Volkmer and Wood (2014) was added.
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 $z\exp\left(\mp\pi i\right)$ are $\operatorname{ph}\left(z\right)\mp\pi$ , respectively.
Suggested 2010-06-14 by Fredrik Johansson
See also:: Annotations for §16.11 and Ch.16

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

Case $p=q 1$

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See also:: Annotations for §16.11(ii), §16.11 and Ch.16

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

16.11.6		$\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q 1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q 1}F_{q}}% \left({a_{1},\dots,a_{q 1}\atop b_{1},\dots,b_{q}};z\right)=H_{q 1,q}(-z),$
$\|\operatorname{ph}\left(-z\right)\|\leq\pi$ ;
ⓘ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

compare (16.8.8).

Case $p=q$

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See also:: Annotations for §16.11(ii), §16.11 and Ch.16

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

16.11.7		$\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q}F_{q}}% \left({a_{1},\dots,a_{q}\atop b_{1},\dots,b_{q}};z\right)\sim H_{q,q}(z{% \mathrm{e}}^{\mp\pi\mathrm{i}}) E_{q,q}(z).$
ⓘ 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, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $q$ : 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) Permalink: http://dlmf.nist.gov/16.11.E7 Encodings: TeX, pMML, png See also: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

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 $p=q-1$

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See also:: Annotations for §16.11(ii), §16.11 and Ch.16

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

16.11.8		$\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q-1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q-1}F_{q}}% \left({a_{1},\dots,a_{q-1}\atop b_{1},\dots,b_{q}};-z\right)\sim H_{q-1,q}(z) % E_{q-1,q}(ze^{-\pi\mathrm{i}}) E_{q-1,q}(ze^{\pi\mathrm{i}}),$
ⓘ 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, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $q$ : 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 Encodings: TeX, pMML, png See also: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

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

Case $p\leq q-2$

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See also:: Annotations for §16.11(ii), §16.11 and Ch.16

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

16.11.9		$\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{p}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{p}F_{q}}% \left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};-z\right)\sim E_{p,q}(ze^{-% \pi\mathrm{i}}) E_{p,q}(ze^{\pi\mathrm{i}}),$
ⓘ 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, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $p$ : 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 See also: Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

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

§16.11(iii) Expansions for Large Parameters

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Keywords:: asymptotic expansions, generalized hypergeometric functions, large parameters
Notes:: See Luke (1969a, Chapter 7).
Permalink:: http://dlmf.nist.gov/16.11.iii
See also:: Annotations for §16.11 and Ch.16

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

16.11.10		${{}_{p 1}F_{p}}\left({a_{1} r,\dots,a_{k-1} r,a_{k},\dots,a_{p 1}\atop b_{1} r% ,\dots,b_{k} r,b_{k 1},\dots,b_{p}};z\right)=\sum_{n=0}^{m-1}\frac{{\left(a_{1% } r\right)_{n}}\cdots{\left(a_{k-1} r\right)_{n}}{\left(a_{k}\right)_{n}}% \cdots{\left(a_{p 1}\right)_{n}}}{{\left(b_{1} r\right)_{n}}\cdots{\left(b_{k}% r\right)_{n}}{\left(b_{k 1}\right)_{n}}\cdots{\left(b_{p}\right)_{n}}}\frac{z% ^{n}}{n!} O\left(\frac{1}{r^{m}}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $z$ : 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 Encodings: TeX, pMML, png See also: Annotations for §16.11(iii), §16.11 and Ch.16

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

16.11.11		${{}_{p}F_{q}}\left({a_{1} r,\dots,a_{p} r\atop b_{1} r,\dots,b_{q} r};z\right)% =\sum_{n=0}^{m-1}\frac{{\left(a_{1} r\right)_{n}}\cdots{\left(a_{p} r\right)_{% n}}}{{\left(b_{1} r\right)_{n}}\cdots{\left(b_{q} r\right)_{n}}}\frac{z^{n}}{n% !} O\left(\frac{1}{r^{(q-p)m}}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $r$ : large parameter Permalink: http://dlmf.nist.gov/16.11.E11 Encodings: TeX, pMML, png See also: Annotations for §16.11(iii), §16.11 and Ch.16

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).