§15.12 Asymptotic Approximations

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

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

• (a)

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

• (b)

  and $|c+n|\ge \delta$ for all $n\in \{0,1,2,\mathrm{\dots }\}$.

• (c)

  $\mathrm{\Re }z=\frac{1}{2}$ and $\left|\mathrm{ph}c\right|\le \pi -\delta$.

• (d)

  $\mathrm{\Re }z>\frac{1}{2}$ and ${\alpha }_{-}-\frac{1}{2}\pi +\delta \le \mathrm{ph}c\le {\alpha }_{+}+\frac{1}{2}\pi -\delta$, where

  15.12.1		$${\alpha }_{\pm }=\arctan \left(\frac{\mathrm{ph}z-\mathrm{ph}\left(1-z\right)\mp \pi }{\ln |1-z^{-1}|}\right),$$
  ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: 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 $z$ restricted so that $\pm {\alpha }_{\pm }\in [0,\frac{1}{2}\pi )$.

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

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 $b$ and $c$ with $c>b+1$ see López and Pagola (2011).

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

As $\lambda \to \mathrm{\infty }$,

where $q_0(z)=1$ and $q_s(z)$, $s=1,2,\mathrm{\dots }$, are defined by the generating function

then (15.12.3) applies when $|\mathrm{ph}\lambda |\le \pi -\delta$. 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 $b$ and $c$ are large. If , then (15.12.3) applies when $|\mathrm{ph}\lambda |\le \frac{1}{2}\pi -\delta$. If $\mathrm{\Re }z\le \frac{1}{2}$, then (15.12.3) applies when $|\mathrm{ph}\lambda |\le \pi -\delta$.

If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \pi -\delta$,

where

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 , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \pi -\delta$,

where

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

If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \frac{1}{2}\pi -\delta$,

where

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

where

For $\mathrm{Ai}\left(z\right)$ 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 $F(a+e_1\lambda ,b+e_2\lambda ;c+e_3\lambda ;z)$ can be obtained with $e_j=\pm 1$ or $0$, $j=1,2,3$. 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).