16 Generalized Hypergeometric Functions and Meijer

-Function Generalized Hypergeometric Functions 16.10 Expansions in Series of $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits$ Functions 16.12 Products

§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 $E_{{p,q}}(z)=(2\pi)^{{\ifrac{(p-q)}{2}}}\kappa^{{-\nu-(\ifrac{1}{2})}}e^{{% \kappa z^{{\ifrac{1}{\kappa}}}}}\sum_{{k=0}}^{\infty}c_{k}\left(\kappa z^{{% \ifrac{1}{\kappa}}}\right)^{{\nu-k}},$

Symbols:: : base of exponential function, : nonnegative integer, : nonnegative integer, : 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
Encodings:: TeX, pMML, png

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, : : factorial, : nonnegative integer, : nonnegative integer, : 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
Encodings:: TeX, pMML, png

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:: : nonnegative integer, : 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
Encodings:: TeX, TeX, pMML, pMML, png, png

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
Encodings:: TeX, TeX, pMML, pMML, png, png

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, : nonnegative integer, : 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
Encodings:: TeX, pMML, png

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), Other Changes
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 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

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, : nonnegative integer, : 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

compare (16.8.8).

¶ Case

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, : base of exponential function, $\sim$ : asymptotic equality, : nonnegative integer, : 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
Encodings:: TeX, pMML, png

Here the upper or lower signs are chosen according as 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 $a_{1}=1$ , explicit representations for the right-hand side of (16.11.7) in terms of generalized hypergeometric functions are given in Kim (1972).

¶ Case

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, : nonnegative integer, : 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

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

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, : : factorial, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and : large parameter
Permalink:: http://dlmf.nist.gov/16.11.E10
Encodings:: TeX, pMML, png

as $r\to+\infty$ . Here can have any integer value from 1 to . Also if , 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, : : factorial, : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and : 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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 16.10 Expansions in Series of $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits$ Functions 16.12 Products

§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 $p\leq q-2$