§13.11 Series

For $z\in\mathbb{C}$,

 13.11.1 $M\left(a,b,z\right)=\Gamma\left(a-\tfrac{1}{2}\right)e^{\frac{1}{2}z}\left(% \tfrac{1}{4}z\right)^{\frac{1}{2}-a}\*\sum_{s=0}^{\infty}\frac{{\left(2a-1% \right)_{s}}{\left(2a-b\right)_{s}}}{{\left(b\right)_{s}}s!}\*\left(a-\tfrac{1% }{2}+s\right)\*I_{a-\frac{1}{2}+s}\left(\tfrac{1}{2}z\right),$ $a+\frac{1}{2},b\neq 0,-1,-2,\dots$,
 13.11.2 $M\left(a,b,z\right)=\Gamma\left(b-a-\tfrac{1}{2}\right){\mathrm{e}}^{\frac{1}{% 2}z}\left(\tfrac{1}{4}z\right)^{a-b+\frac{1}{2}}\sum_{s=0}^{\infty}(-1)^{s}% \frac{{\left(2b-2a-1\right)_{s}}{\left(b-2a\right)_{s}}(b-a-\frac{1}{2}+s)}{{% \left(b\right)_{s}}s!}I_{b-a-\frac{1}{2}+s}\left(\tfrac{1}{2}z\right),$ $b-a+\frac{1}{2},b\neq 0,-1,-2,\dots$.

(13.6.9), (13.6.11_1) and (13.6.11_2) are special cases.

 13.11.3 ${\mathbf{M}}\left(a,b,z\right)={\mathrm{e}}^{\frac{1}{2}z}\sum_{s=0}^{\infty}A% _{s}\left(b-2a\right)^{\frac{1}{2}(1-b-s)}\left(\tfrac{1}{2}z\right)^{\frac{1}% {2}(1-b+s)}J_{b-1+s}\left(\sqrt{2z(b-2a)}\right),$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer, $z$: complex variable and $A_{n}$: coefficients Source: Slater (1960, §3.8) A&S Ref: 13.3.7 Referenced by: §13.11, Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/13.11.E3 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. See also: Annotations for §13.11 and Ch.13

where

 13.11.4 $\displaystyle A_{0}$ $\displaystyle=1,$ $\displaystyle A_{1}$ $\displaystyle=0,$ $\displaystyle A_{2}$ $\displaystyle=\frac{1}{2}b,$ $\displaystyle(n+1)A_{n+1}$ $\displaystyle=(n+b-1)A_{n-1}+(2a-b)A_{n-2},$ $n=2,3,4,\dots$. ⓘ Defines: $A_{n}$: coefficients (locally) Symbols: $n$: nonnegative integer Referenced by: §13.11, Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/13.11.E4 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png Addition (effective with 1.1.0): This equation was added. See also: Annotations for §13.11 and Ch.13

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), López and Temme (2010a) and López and Pérez Sinusía (2014). See also §13.13.