§13.5 Continued Fractions

Notes:: The two continued fractions in this section are Theorems 6.3 and 6.5 in Jones and Thron (1980).
Keywords:: Kummer functions, continued fractions
Permalink:: http://dlmf.nist.gov/13.5

If $a,b\in\Complex$ such that $a\neq-1,-2,-3,\dots$ , and $a-b\neq 0,1,2,\dots$ , then

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

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $z$ : complex variable and $u_{n}$ : continued fraction coefficients
Referenced by:: §13.31(ii)
Permalink:: http://dlmf.nist.gov/13.5.E1
Encodings:: TeX, pMML, png

where

13.5.2

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

$u_{{2n}}=\frac{a+n}{(b+2n-1)(b+2n)}$ .

Symbols:: $n$ : nonnegative integer and $u_{n}$ : continued fraction coefficients
Permalink:: http://dlmf.nist.gov/13.5.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

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

If $a,b\in\Complex$ such that $a\neq 0,-1,-2,\dots$ , and $b-a\neq 2,3,4,\dots$ , then

13.5.3 $\frac{\mathop{U\/}\nolimits\!\left(a,b,z\right)}{\mathop{U\/}\nolimits\!\left(a,b-1,z\right)}=1+\cfrac{v_{{1}}/z}{1+\cfrac{v_{{2}}/z}{1+\cdots}},$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : 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

where

13.5.4

$v_{{2n+1}}=a+n$ ,

$v_{{2n}}=a-b+n+1$ .

Symbols:: $n$ : nonnegative integer and $v_{n}$ : continued fraction coefficients
Permalink:: http://dlmf.nist.gov/13.5.E4
Encodings:: TeX, TeX, pMML, pMML, png, png

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

See also Cuyt et al. (2008, pp. 322–330).