§15.7 Continued Fractions

ⓘ

Keywords:: continued fractions, hypergeometric function
Notes:: See Andrews et al. (1999, pp. 94, 97–98, and Ex. 26 on p. 119), Lorentzen and Waadeland (1992, §6.1), and Berndt (1989, pp. 134–137). These references contain several restrictions on the parameters $a$ , $b$ , and $c$ . This is because they use the function $F\left(a,b;c;z\right)$ . No restrictions are needed for $\mathbf{F}\left(a,b;c;z\right)$ .
Referenced by:: §3.10(ii)
Permalink:: http://dlmf.nist.gov/15.7
See also:: Annotations for Ch.15

If $|\operatorname{ph}\left(1-z\right)|<\pi$ , then

15.7.1		$\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=t_{0% }-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},$
ⓘ Symbols: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $t_{n}$ : coefficient and $u_{n}$ : coefficient Permalink: http://dlmf.nist.gov/15.7.E1 Encodings: TeX, pMML, png See also: Annotations for §15.7 and Ch.15

where

15.7.2	$\displaystyle t_{n}$	$\displaystyle=c+n,$
	$\displaystyle u_{2n+1}$	$\displaystyle=(a+n)(c-b+n),$
	$\displaystyle u_{2n}$	$\displaystyle=(b+n)(c-a+n).$
ⓘ Symbols: $n$ : integer, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $t_{n}$ : coefficient and $u_{n}$ : coefficient Permalink: http://dlmf.nist.gov/15.7.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §15.7 and Ch.15

If $|z|<1$ , then

15.7.3		$\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=v_{0% }-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},$
ⓘ Symbols: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $v_{n}$ : coefficient and $w_{n}$ : coefficient Permalink: http://dlmf.nist.gov/15.7.E3 Encodings: TeX, pMML, png See also: Annotations for §15.7 and Ch.15

where

15.7.4	$\displaystyle v_{n}$	$\displaystyle=c+n+(b-a+n+1)z,$
15.7.4	$\displaystyle w_{n}$	$\displaystyle=(b+n)(c-a+n)z.$
ⓘ Symbols: $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $v_{n}$ : coefficient and $w_{n}$ : coefficient Permalink: http://dlmf.nist.gov/15.7.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §15.7 and Ch.15

If $\Re z<\tfrac{1}{2}$ , then

15.7.5		$\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a+1,b+1;c+1;z\right)}={x% _{0}+\cfrac{y_{1}}{x_{1}+\cfrac{y_{2}}{x_{2}+\cfrac{y_{3}}{x_{3}+\cdots}}}},$
ⓘ Symbols: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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, $x$ : real variable, $y$ : real variable, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.7.E5 Encodings: TeX, pMML, png See also: Annotations for §15.7 and Ch.15

where

15.7.6	$\displaystyle x_{n}$	$\displaystyle=c+n-(a+b+2n+1)z,$
15.7.6	$\displaystyle y_{n}$	$\displaystyle=(a+n)(b+n)z(1-z).$
ⓘ Symbols: $x$ : real variable, $n$ : integer, $y$ : real variable, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.7.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §15.7 and Ch.15

See also Cuyt et al. (2008, pp. 295–309).