§13.11 Series

Notes:: See Slater (1960, §2.7.3: Eq. (2.7.14) has errors).
Keywords:: Kummer functions
Permalink:: http://dlmf.nist.gov/13.11

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, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $s$ : nonnegative integer and $z$ : 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 Hansen (1975, §§66 and 87), Prudnikov et al. (1990, §6.6), and Tricomi (1954, §1.8). See also §13.13.