§15.7 Continued Fractions

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 $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ . No restrictions are needed for $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ .
Keywords:: continued fractions, hypergeometric function
Referenced by:: ¶ ‣ §3.10(ii)
Permalink:: http://dlmf.nist.gov/15.7

If $|\mathop{\mathrm{ph}\/}\nolimits\!\left(1-z\right)|<\pi$ , then

15.7.1 $\frac{\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)}{\mathop{\mathbf{F}\/}\nolimits\!\left(a,b+1;c+1;z\right)}=t_{0}-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},$

Symbols:: $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : 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

where

15.7.2

$t_{n}=c+n,$

$u_{{2n+1}}=(a+n)(c-b+n),$

$u_{{2n}}=(b+n)(c-a+n).$

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

If $|z|<1$ , then

15.7.3 $\frac{\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)}{\mathop{\mathbf{F}\/}\nolimits\!\left(a,b+1;c+1;z\right)}=v_{0}-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},$

Symbols:: $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : 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

where

15.7.4

$v_{n}=c+n+(b-a+n+1)z,$

$w_{n}=(b+n)(c-a+n)z.$

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

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

15.7.5 $\frac{\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)}{\mathop{\mathbf{F}\/}\nolimits\!\left(a+1,b+1;c+1;z\right)}={x_{0}+\cfrac{y_{1}}{x_{1}+\cfrac{y_{2}}{x_{2}+\cfrac{y_{3}}{x_{3}+\cdots}}}},$

Symbols:: $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : 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

where

15.7.6

$x_{n}=c+n-(a+b+2n+1)z,$

$y_{n}=(a+n)(b+n)z(1-z).$

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 Cuyt et al. (2008, pp. 295–309).