The convert( expr, StandardFunctions ) command converts functions that are represented in terms of MeijerG or hypergeom functions into the standard special and elementary functions found in such texts as Handbook of Mathematical Functions by Abramowitz and Stegun, and Integrals and Series, Volume 3: More Special Functions by A.P. Prudnikov, Yu. Brychkov, and O. Marichev. Examples of these standard functions include Bessel and Legendre functions, elliptic integrals, etc.

$\mathrm{convert}\left(\mathrm{hypergeom}\left(\left[1\,2\right]\,\left[\frac{3}{2}\right]\,z\right)\,\mathrm{StandardFunctions}\right)$

$\frac{1}{-2z+2}+\frac{\arcsin \left(\sqrt{z}\right)}{2{\left(1-z\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{z}}$

$\mathrm{convert}\left(\mathrm{MeijerG}\left(\left[\left[\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[0\right]\right]\,\frac{z^2}{4}\right)\,\mathrm{StandardFunctions}\right)$

$\mathrm{BesselJ}\left(0\,z\right)$

$\mathrm{assume}\left(n\,\mathrm{nonnegint}\right)$

$\mathrm{convert}\left(\mathrm{hypergeom}\left(\left[-n\,-n\right]\,\left[\frac{1}{2}-2n\right]\,z\right)\,\mathrm{StandardFunctions}\right)$

$\frac{{\left(\mathrm{-1}\right)}^{\mathrm{n~}}{\left(2\mathrm{n~}\right)!}^2{\left(4z\right)}^{\mathrm{n~}}\mathrm{LegendreP}\left(2\mathrm{n~}\,\sqrt{\frac{z-1}{z}}\right)}{\left(4\mathrm{n~}\right)!}$

Abramowitz, M. and Stegun, I. A. eds. Handbook of Mathematical Functions. New York: Dover Publications, 1974.

Prudnikov, A.P.; Brychkov, Yu.; and Marichev, O. Integrals and Series. Gordon and Breach Science Publishers, 1990. Vol. 3: More Special Functions.

Roach, K. "Hypergeometric Function Representations." Proceedings of ISSAC '96, ACM, New York, pp. 301-308. 1996.

Roach, K. "Meijer G Function Representations." Proceedings of ISSAC '97, ACM, New York, pp. 205-211. 1997.