§13.5 Continued Fractions

If $a,b\in\mathbb{C}$ such that $a\neq-1,-2,-3,\dots$, and $a-b\neq 0,1,2,\dots$, then

 13.5.1 $\frac{M\left(a,b,z\right)}{M\left(a+1,b+1,z\right)}=1+\cfrac{u_{1}z}{1+\cfrac{% u_{2}z}{1+\cdots}},$

where

 13.5.2 $\displaystyle u_{2n+1}$ $\displaystyle=\frac{a-b-n}{(b+2n)(b+2n+1)}$, $\displaystyle u_{2n}$ $\displaystyle=\frac{a+n}{(b+2n-1)(b+2n)}$. ⓘ Defines: $u_{n}$: continued fraction coefficients (locally) Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/13.5.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §13.5 and Ch.13

This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $\mathbb{C}$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980).

If $a,b\in\mathbb{C}$ such that $a\neq 0,-1,-2,\dots$, and $b-a\neq 2,3,4,\dots$, then

 13.5.3 $\frac{U\left(a,b,z\right)}{U\left(a,b-1,z\right)}=1+\cfrac{v_{1}/z}{1+\cfrac{v% _{2}/z}{1+\cdots}},$ ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, $z$: complex variable and $v_{n}$: continued fraction coefficients Permalink: http://dlmf.nist.gov/13.5.E3 Encodings: TeX, pMML, png See also: Annotations for §13.5 and Ch.13

where

 13.5.4 $\displaystyle v_{2n+1}$ $\displaystyle=a+n$, $\displaystyle v_{2n}$ $\displaystyle=a-b+n+1$. ⓘ Defines: $v_{n}$: continued fraction coefficients (locally) Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/13.5.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §13.5 and Ch.13

This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector $|\operatorname{ph}{z}|<\pi$.