Digital Library of Mathematical Functions
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13 Confluent Hypergeometric FunctionsKummer Functions

§13.5 Continued Fractions

If a,b such that a-1,-2,-3,, and a-b0,1,2,, then

13.5.1 M(a,b,z)M(a+1,b+1,z)=1+u1z1+u2z1+,

where

13.5.2 u2n+1 =a-b-n(b+2n)(b+2n+1),
u2n =a+n(b+2n-1)(b+2n).

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

If a,b such that a0,-1,-2,, and b-a2,3,4,, then

13.5.3 U(a,b,z)U(a,b-1,z)=1+v1/z1+v2/z1+,

where

13.5.4 v2n+1 =a+n,
v2n =a-b+n+1.

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. 322–330).