The Whittaker functions WhittakerM(mu, nu, z) and WhittakerW(mu, nu, z) solve the differential equation

They can be defined in terms of the hypergeometric and Kummer functions as follows:

$\mathrm{WhittakerM}\left(1\,2\,0.5\right)$

$0.1606687379$

$\mathrm{diff}\left(\mathrm{WhittakerW}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)\,z\right)$

$\left(\frac{1}{2}-\frac{\mathrm{\mu }}{z}\right)\mathrm{WhittakerW}\left(\mathrm{\mu }\,\mathrm{\nu }\,z\right)-\frac{\mathrm{WhittakerW}\left(\mathrm{\mu }+1\,\mathrm{\nu }\,z\right)}{z}$

$\mathrm{series}\left(\mathrm{WhittakerM}\left(2\,3\,x\right)\,x\right)$

$x^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-\frac{2x^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{7}+\frac{23x^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{448}+\mathrm{O}\left(x^{\raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$

$\mathrm{series}\left(\mathrm{WhittakerW}\left(-\frac{1}{2}\,-\frac{1}{3}\,x\right)\,x\right)$

$\frac{3\sqrt{3}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2{x}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{2\mathrm{\pi }}-\frac{\mathrm{\pi }\sqrt{3}{x}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2}+\frac{9\sqrt{3}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2{x}^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{4\mathrm{\pi }}-\frac{3\mathrm{\pi }\sqrt{3}{x}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{10{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2}+\frac{9\sqrt{3}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2{x}^{\raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{16\mathrm{\pi }}-\frac{3\mathrm{\pi }\sqrt{3}{x}^{\raisebox{1ex}{$17$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{40{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2}+\frac{27\sqrt{3}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2{x}^{\raisebox{1ex}{$19$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{224\mathrm{\pi }}-\frac{9\mathrm{\pi }\sqrt{3}{x}^{\raisebox{1ex}{$23$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{880{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2}+\frac{27\sqrt{3}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2{x}^{\raisebox{1ex}{$25$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{1792\mathrm{\pi }}-\frac{9\mathrm{\pi }\sqrt{3}{x}^{\raisebox{1ex}{$29$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{7040{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2}+\frac{81\sqrt{3}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2{x}^{\raisebox{1ex}{$31$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{46592\mathrm{\pi }}-\frac{27\mathrm{\pi }\sqrt{3}{x}^{\raisebox{1ex}{$35$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{239360{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2}+\mathrm{O}\left(x^{\raisebox{1ex}{$37$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\right)$

$\mathrm{simplify}\left(\mathrm{WhittakerW}\left(\mathrm{\mu }+\frac{7}{3}\,\mathrm{\nu }\,x\right)\right)$

$-\left(\mathrm{\mu }-\frac{1}{6}-\mathrm{\nu }\right)\left(\mathrm{\nu }+\mathrm{\mu }-\frac{1}{6}\right)\left(x-2\mathrm{\mu }-\frac{8}{3}\right)\mathrm{WhittakerW}\left(\mathrm{\mu }-\frac{2}{3}\,\mathrm{\nu }\,x\right)+\left(5{\mathrm{\mu }}^2+\left(-4x+\frac{25}{3}\right)\mathrm{\mu }+x^2-{\mathrm{\nu }}^2-\frac{10x}{3}+\frac{89}{36}\right)\mathrm{WhittakerW}\left(\mathrm{\mu }+\frac{1}{3}\,\mathrm{\nu }\,x\right)$

Abramowitz, M., and Stegun I. Handbook of Mathematical Functions. New York: Dover Publications.

Luke, Y. The Special Functions and Their Approximations. Vol 1. Academic Press, 1969.