The Kummer functions KummerM(mu, nu, z) and KummerU(mu, nu, z) solve the differential equation

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

$1.297442541$

$\mathrm{evalf}\left(\mathrm{KummerU}\left(-\frac{1}{2}\,-\frac{1}{3}\,\frac{1}{7}\right)\right)$

$0.9025082951$

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

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

$\mathrm{series}\left(\mathrm{KummerU}\left(\frac{1}{2}\,\frac{1}{3}\,z\right)\,z\right)$

$\frac{3\mathrm{\Gamma }\left(\frac{5}{6}\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\pi }}-\frac{\sqrt{\mathrm{\pi }}\sqrt{3}{z}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{9\mathrm{\Gamma }\left(\frac{5}{6}\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)z}{2\mathrm{\pi }}-\frac{7\sqrt{\mathrm{\pi }}\sqrt{3}{z}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{10\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{81\mathrm{\Gamma }\left(\frac{5}{6}\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)z^2}{32\mathrm{\pi }}-\frac{91\sqrt{\mathrm{\pi }}\sqrt{3}{z}^{\raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{320\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{405\mathrm{\Gamma }\left(\frac{5}{6}\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)z^3}{448\mathrm{\pi }}-\frac{1729\sqrt{\mathrm{\pi }}\sqrt{3}{z}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{21120\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{243\mathrm{\Gamma }\left(\frac{5}{6}\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)z^4}{1024\mathrm{\pi }}-\frac{1235\sqrt{\mathrm{\pi }}\sqrt{3}{z}^{\raisebox{1ex}{$14$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{67584\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{6561\mathrm{\Gamma }\left(\frac{5}{6}\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)z^5}{133120\mathrm{\pi }}-\frac{7657\sqrt{\mathrm{\pi }}\sqrt{3}{z}^{\raisebox{1ex}{$17$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{2297856\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\mathrm{O}\left(z^6\right)$

Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover, 1972.