Digital Library of Mathematical Functions
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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.17 Continued Fractions

If κ,μ such that μ±(κ-12)-1,-2,-3,, then

13.17.1 zMκ,μ(z)Mκ-12,μ+12(z)=1+u1z1+u2z1+,

where

13.17.2 u2n+1 =-12+μ+κ+n(2μ+2n+1)(2μ+2n+2),
u2n =12+μ-κ+n(2μ+2n)(2μ+2n+1).

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

If κ,μ such that μ+12±(κ+1)-1,-2,-3,, then

13.17.3 Wκ,μ(z)zWκ-12,μ-12(z)=1+v1/z1+v2/z1+,

where

13.17.4 v2n+1 =12+μ-κ+n,
v2n =12-μ-κ+n.

This continued fraction converges to the meromorphic function of z on the left-hand side throughout the sector |phz|<π.

See also Cuyt et al. (2008, pp. 336–337).