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13 Confluent Hypergeometric FunctionsKummer Functions

§13.11 Series

For z,

13.11.1 M(a,b,z)=Γ(a-12)e12z(14z)12-as=0(2a-1)s(2a-b)s(b)ss!(a-12+s)Ia-12+s(12z),
a+12,b0,-1,-2,.

(13.6.9) is a special case.

For additional expansions combine (13.14.4), (13.14.5), and §13.24. For other series expansions see Tricomi (1954, §1.8), Hansen (1975, §§66 and 87), Prudnikov et al. (1990, §6.6), and López and Temme (2010). See also §13.13.