15 Hypergeometric Function Properties 15.11 Riemann’s Differential Equation 15.13 Zeros

§15.12 Asymptotic Approximations

ⓘ

Permalink:: http://dlmf.nist.gov/15.12
See also:: Annotations for Ch.15

§15.12(i) Large Variable
§15.12(ii) Large
§15.12(iii) Other Large Parameters

§15.12(i) Large Variable

ⓘ

Keywords:: asymptotic approximations, hypergeometric function, large variable
A&S Ref:: 15.7
Permalink:: http://dlmf.nist.gov/15.12.i
See also:: Annotations for §15.12 and Ch.15

For the asymptotic behavior of $\mathbf{F}\left(a,b;c;z\right)$ as $z\to\infty$ with , , fixed, combine (15.2.2) with (15.8.2) or (15.8.8).

§15.12(ii) Large

ⓘ

Keywords:: asymptotic approximations, hypergeometric function, large
A&S Ref:: 15.7
Notes:: See Wagner (1988). The region of validity given in Luke (1969a, p. 235) is incorrect.
Referenced by:: §14.15(i), §15.19(iv), §8.18(i), Erratum (V1.0.7) for References
Permalink:: http://dlmf.nist.gov/15.12.ii
Addition (effective with 1.0.7):: The reference to López and Pagola (2011) has been added at the end of this subsection.
See also:: Annotations for §15.12 and Ch.15

Let $\delta$ denote an arbitrary small positive constant. Also let be real or complex and fixed, and at least one of the following conditions be satisfied:

(a)

and/or $b\in\{0,-1,-2,\dots\}$ .
(b)

$\Re z<\tfrac{1}{2}$ and $|c+n|\geq\delta$ for all $n\in\{0,1,2,\dots\}$ .
(c)

$\Re z=\tfrac{1}{2}$ and $\left|\operatorname{ph}c\right|\leq\pi-\delta$ .
(d)

$\Re z>\tfrac{1}{2}$ and $\alpha_{-}-\tfrac{1}{2}\pi+\delta\leq\operatorname{ph}c\leq\alpha_{+}+\tfrac{1% }{2}\pi-\delta$ , where

15.12.1 $\alpha_{\pm}=\operatorname{arctan}\left(\frac{\operatorname{ph}z-\operatorname% {ph}\left(1-z\right)\mp\pi}{\ln|1-z^{-1}|}\right),$
ⓘ

Symbols:

$\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, : complex variable and $\alpha_{\pm}$

Permalink:

http://dlmf.nist.gov/15.12.E1

Encodings:

TeX, pMML, png

See also:

Annotations for §15.12(ii), §15.12 and Ch.15

with restricted so that $\pm\alpha_{\pm}\in[0,\tfrac{1}{2}\pi)$ .

Then for fixed $m\in\{0,1,2,\dots\}$ ,

15.12.2 $F\left(a,b;c;z\right)=\sum_{s=0}^{m-1}\frac{{\left(a\right)_{s}}{\left(b\right% )_{s}}}{{\left(c\right)_{s}}s!}z^{s}+O\left(c^{-m}\right),$ $|c|\to\infty$ .

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and : integer
Referenced by:: §15.19(iv)
Permalink:: http://dlmf.nist.gov/15.12.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §15.12(ii), §15.12 and Ch.15

Similar results for other sectors are given in Wagner (1988). For the more general case in which $a^{2}=o\left(c\right)$ and $b^{2}=o\left(c\right)$ see Wagner (1990).

For large and with see López and Pagola (2011).

§15.12(iii) Other Large Parameters

ⓘ

Keywords:: asymptotic approximations, hypergeometric function, large (or ) and , large and , large or , large , , and , large
Notes:: For (15.12.3) see Luke (1969a, §7.2) and Olver (1997b, p. 162). The sector of validity given in the first reference is incorrect. See the third footnote in the second reference.
Referenced by:: §15.19(iv), §2.8(iii), §2.8(iv), §2.8(vi), Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/15.12.iii
Addition (effective with 1.2.5):: Just below (15.12.4), the reference Bissi et al. (2025) was added which contains another expansion in powers of $\lambda$ with explicit coefficients, and expansions for the case that both and are large.
Additions and a Deletion (effective with 1.0.11):: In the last paragraph of this subsection, (i) a reference to Farid Khwaja and Olde Daalhuis (2014) was added, replacing the original reference to Olde Daalhuis (2010), and (ii) a reference to Temme (2015, Chapters 12 and 28) was also added.
See also:: Annotations for §15.12 and Ch.15

Again, throughout this subsection $\delta$ denotes an arbitrary small positive constant, and are real or complex and fixed.

As $\lambda\to\infty$ ,

15.12.3 $F\left({a,b\atop c+\lambda};z\right)\sim\frac{\Gamma\left(c+\lambda\right)}{% \Gamma\left(c-b+\lambda\right)}\sum_{s=0}^{\infty}q_{s}(z){\left(b\right)_{s}}% \lambda^{-s-b},$

where $q_{0}(z)=1$ and $q_{s}(z)$ , $s=1,2,\dots$ , are defined by the generating function

15.12.4 $\left(\frac{{\mathrm{e}}^{t}-1}{t}\right)^{b-1}{\mathrm{e}}^{t(1-c)}\left(1-z+% z{\mathrm{e}}^{-t}\right)^{-a}=\sum_{s=0}^{\infty}q_{s}(z)t^{s},$

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and $q_{s}(z)$ : coefficients
Referenced by:: §15.12(iii), Erratum (V1.2.5) for Chapter 15 Additions
Permalink:: http://dlmf.nist.gov/15.12.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

then (15.12.3) applies when $|\operatorname{ph}\lambda|\leq\pi-\delta$ . For another expansion in powers of $\lambda$ with explicit coefficients see Bissi et al. (2025, (C.4)). This reference also contains expansions for the case that both and are large. If $|\operatorname{ph}\left(1-z\right)|<\pi$ , then (15.12.3) applies when $|\operatorname{ph}\lambda|\leq\tfrac{1}{2}\pi-\delta$ . If $\Re z\leq\tfrac{1}{2}$ , then (15.12.3) applies when $|\operatorname{ph}\lambda|\leq\pi-\delta$ .

If $|\operatorname{ph}\left(z-1\right)|<\pi$ , then as $\lambda\to\infty$ with $|\operatorname{ph}\lambda|\leq\pi-\delta$ ,

15.12.5 $\mathbf{F}\left({a+\lambda,b-\lambda\atop c};\tfrac{1}{2}-\tfrac{1}{2}z\right)% =2^{(a+b-1)/2}\frac{(z+1)^{(c-a-b-1)/2}}{(z-1)^{c/2}}\sqrt{\zeta\sinh\zeta}% \left(\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b\right)^{1-c}\left(I_{c-1}\left((% \lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)(1+O(\lambda^{-2}))+\frac{I_{c% -2}\left((\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)}{2\lambda+a-b}\left% (\left(c-\tfrac{1}{2}\right)\left(c-\tfrac{3}{2}\right)\left(\frac{1}{\zeta}-% \coth\zeta\right)+\tfrac{1}{2}(2c-a-b-1)(a+b-1)\tanh\left(\tfrac{1}{2}\zeta% \right)+O(\lambda^{-2})\right)\right),$

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/15.12.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

where

15.12.6 $\zeta=\operatorname{arccosh}z.$

ⓘ

Symbols:: $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, : complex variable and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/15.12.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

For $I_{\nu}\left(z\right)$ see §10.25(ii). For this result and an extension to an asymptotic expansion with error bounds see Jones (2001).

See also Dunster (1999) where the asymptotics of Jacobi polynomials is described; compare (15.9.1).

If $|\operatorname{ph}z|<\pi$ , then as $\lambda\to\infty$ with $|\operatorname{ph}\lambda|\leq\pi-\delta$ ,

15.12.7 $F\left({a,b-\lambda\atop c+\lambda};-z\right)=2^{b-c+(1/2)}\left(\frac{z+1}{2% \sqrt{z}}\right)^{\lambda}\left(\lambda^{a/2}U\left(a-\tfrac{1}{2},-\alpha% \sqrt{\lambda}\right)\left((1+z)^{c-a-b}z^{1-c}\left(\frac{\alpha}{z-1}\right)% ^{1-a}+O(\lambda^{-1})\right)+\frac{\lambda^{\ifrac{(a-1)}{2}}}{\alpha}U\left(% a-\tfrac{3}{2},-\alpha\sqrt{\lambda}\right)\left((1+z)^{c-a-b}z^{1-c}\left(% \frac{\alpha}{z-1}\right)^{1-a}-2^{c-b-(\ifrac{1}{2})}\left(\frac{\alpha}{z-1}% \right)^{a}+O(\lambda^{-1})\right)\right),$

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and $\alpha$
Permalink:: http://dlmf.nist.gov/15.12.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

where

15.12.8 $\alpha=\left(-2\ln\left(1-\left(\frac{z-1}{z+1}\right)^{2}\right)\right)^{1/2},$

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, : complex variable and $\alpha$
Permalink:: http://dlmf.nist.gov/15.12.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

with the branch chosen to be continuous and $\Re\alpha>0$ when $\Re\left(\ifrac{(z-1)}{(z+1)}\right)>0$ . For $U\left(a,z\right)$ see §12.2, and for an extension to an asymptotic expansion see Olde Daalhuis (2003a).

If $|\operatorname{ph}z|<\pi$ , then as $\lambda\to\infty$ with $|\operatorname{ph}\lambda|\leq\tfrac{1}{2}\pi-\delta$ ,

15.12.9 $(z+1)^{3\lambda/2}(2\lambda)^{c-1}\mathbf{F}\left({a+\lambda,b+2\lambda\atop c% };-z\right)={\lambda^{-1/3}\left({\mathrm{e}}^{\pi\mathrm{i}(a-c+\lambda+(1/3)% )}\operatorname{Ai}\left({\mathrm{e}}^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{% \ifrac{2}{3}}\beta^{2}\right)+{\mathrm{e}}^{\pi\mathrm{i}(c-a-\lambda-(1/3))}% \operatorname{Ai}\left({\mathrm{e}}^{\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac% {2}{3}}\beta^{2}\right)\right)\left(a_{0}(\zeta)+O(\lambda^{-1})\right)}+% \lambda^{-2/3}\left({\mathrm{e}}^{\pi\mathrm{i}(a-c+\lambda+(2/3))}% \operatorname{Ai}'\left({\mathrm{e}}^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{% \ifrac{2}{3}}\beta^{2}\right)+{\mathrm{e}}^{\pi\mathrm{i}(c-a-\lambda-(2/3))}% \operatorname{Ai}'\left({\mathrm{e}}^{\ifrac{2\pi\mathrm{i}}{3}}\lambda^{% \ifrac{2}{3}}\beta^{2}\right)\right)\left(a_{1}(\zeta)+O(\lambda^{-1})\right),$

where

15.12.10 $\zeta=\operatorname{arccosh}\left(\tfrac{1}{4}z-1\right),$

ⓘ

Symbols:: $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, : complex variable and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/15.12.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

15.12.11 $\beta=\left(-\frac{3}{2}\zeta+\frac{9}{4}\ln\left(\frac{2+{\mathrm{e}}^{\zeta}% }{2+{\mathrm{e}}^{-\zeta}}\right)\right)^{1/3},$

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $\ln\NVar{z}$ : principal branch of logarithm function, $\zeta$ : change of variable and $\beta$
Permalink:: http://dlmf.nist.gov/15.12.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

with the branch chosen to be continuous and $\beta>0$ when $\zeta>0$ . Also,

15.12.12

$a_{0}(\zeta)=\tfrac{1}{2}G_{0}(\beta)+\tfrac{1}{2}G_{0}(-\beta),$

$a_{1}(\zeta)=\left(\tfrac{1}{2}G_{0}(\beta)-\tfrac{1}{2}G_{0}(-\beta)\right)/\beta,$

ⓘ

Symbols:: $\zeta$ : change of variable, $\beta$ , $a_{j}(\zeta)$ : function and $G_{0}(\pm\zeta)$ : function
Permalink:: http://dlmf.nist.gov/15.12.E12
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

where

15.12.13 $G_{0}(\pm\beta)=\left(2+{\mathrm{e}}^{\pm\zeta}\right)^{c-b-(\ifrac{1}{2})}% \left(1+{\mathrm{e}}^{\pm\zeta}\right)^{a-c+(\ifrac{1}{2})}\left(z-1-{\mathrm{% e}}^{\pm\zeta}\right)^{-a+(\ifrac{1}{2})}\sqrt{\frac{\beta}{{\mathrm{e}}^{% \zeta}-{\mathrm{e}}^{-\zeta}}}.$

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, $\zeta$ : change of variable, $\beta$ and $G_{0}(\pm\zeta)$ : function
Permalink:: http://dlmf.nist.gov/15.12.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

For $\operatorname{Ai}\left(z\right)$ see §9.2, and for further information and an extension to an asymptotic expansion see Olde Daalhuis (2003b). (Two errors in this reference are corrected in (15.12.9).)

By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for $F\left(a+e_{1}\lambda,b+e_{2}\lambda;c+e_{3}\lambda;z\right)$ can be obtained with $e_{j}=\pm 1$ or 0, . For more details see Farid Khwaja and Olde Daalhuis (2014). For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

15.11 Riemann’s Differential Equation 15.13 Zeros

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§15.12 Asymptotic Approximations

Contents

§15.12(i) Large Variable

§15.12(ii) Large

§15.12(iii) Other Large Parameters