Lie Theory and the Lauricella Functions FD

It is shown that the Lauricella functions F_D in n variables transform as basis vectors corresponding to irreducible representations of the Lie algebra $\text{sl(n+3,}\mathbb{C})$ . Group representation theory can then be applied to derive addition theorems, transformation formulas, and generating functions for the F_D. It is clear from this analysis that the use of $\mathrm{SL(m,}\mathbb{C})$ symmetry in atomic and elementary particle physics will lead inevitably to the remarkable functions F_D.