13 Confluent Hypergeometric Functions Kummer Functions 13.10 Integrals 13.12 Products

§13.11 Series

Notes:: See Slater (1960, §2.7.3: Eq. (2.7.14) has errors).
Keywords:: Kummer functions
Referenced by:: Other Changes
Permalink:: http://dlmf.nist.gov/13.11
Addition (effective with 1.0.5):: The reference to López and Temme (2010) has been added at the end of this subsection.
Reported 2012-07-30

For $z\in\Complex$ ,

13.11.1 $\mathop{M\/}\nolimits\!\left(a,b,z\right)=\mathop{\Gamma\/}\nolimits\!\left(a-% \tfrac{1}{2}\right)e^{{\frac{1}{2}z}}\left(\tfrac{1}{4}z\right)^{{\frac{1}{2}-% a}}\*\sum_{{s=0}}^{{\infty}}\frac{\left(2a-1\right)_{{s}}\left(2a-b\right)_{{s% }}}{\left(b\right)_{{s}}s!}\*\left(a-\tfrac{1}{2}+s\right)\*\mathop{I_{{a-% \frac{1}{2}+s}}\/}\nolimits\!\left(\tfrac{1}{2}z\right),$ $a+\frac{1}{2},b\neq 0,-1,-2,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : base of exponential function, : : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : nonnegative integer and : complex variable
Referenced by:: §8.7
Permalink:: http://dlmf.nist.gov/13.11.E1
Encodings:: TeX, pMML, png

(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.