# §13.24 Series

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

For expansions of arbitrary functions in series of $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ 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 $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)=\mathop{\Gamma\/}\nolimits% \!\left(\kappa+\mu\right)2^{2\kappa+2\mu}z^{\frac{1}{2}-\kappa}\*\sum_{s=0}^{% \infty}(-1)^{s}\frac{{\left(2\kappa+2\mu\right)_{s}}{\left(2\kappa\right)_{s}}% }{{\left(1+2\mu\right)_{s}}s!}\*\left(\kappa+\mu+s\right)\mathop{I_{\kappa+\mu% +s}\/}\nolimits\!\left(\tfrac{1}{2}z\right),$ $2\mu,\kappa+\mu\neq-1,-2,-3,\dots$,

and

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

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

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