# §13.24 Series

## §13.24(i) Expansions in Series of Whittaker Functions

For expansions of arbitrary functions in series of $M_{\kappa,\mu}\left(z\right)$ functions see Schäfke (1961b).

## §13.24(ii) Expansions in Series of Bessel Functions

For $z\in\mathbb{C}$, and again with the notation of §§10.2(ii) and 10.25(ii),

 13.24.1 $M_{\kappa,\mu}\left(z\right)=\Gamma\left(\kappa+\mu\right)2^{2\kappa+2\mu}z^{% \frac{1}{2}-\kappa}\*\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(2\kappa+2\mu% \right)_{s}}{\left(2\kappa\right)_{s}}}{{\left(1+2\mu\right)_{s}}s!}\*\left(% \kappa+\mu+s\right)I_{\kappa+\mu+s}\left(\tfrac{1}{2}z\right),$ $2\mu,\kappa+\mu\neq-1,-2,-3,\dots$,

and

 13.24.2 $\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(z\right)=2^{2\mu}z^{\mu% +\frac{1}{2}}\sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(2\sqrt{\kappa z}\right)^% {-2\mu-s}J_{2\mu+s}\left(2\sqrt{\kappa z}\right),$

where $p_{0}^{(\mu)}(z)=1$, $p_{1}^{(\mu)}(z)=\frac{1}{6}z^{2}$, and higher polynomials $p_{s}^{(\mu)}(z)$ are defined by

 13.24.3 $\exp\left(-\tfrac{1}{2}z\left(\coth t-\frac{1}{t}\right)\right)\left(\frac{t}{% \sinh t}\right)^{1-2\mu}=\sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(-\frac{t}{z}% \right)^{s}.$

(13.18.8) is a special case of (13.24.1).