  We propose a scenario in which massive stars form in a self-gravitating
gaseous disc around a supermassive black hole. We find that once the surface
density of the disc exceeds a critical value, the disc fragments into dense
clumps. The clumps accrete material from the remaining disc and merge into
larger clumps; the upper mass of a merged clump is a few tens to a few hundreds
of solar mass. This picture fits well with the observed young stellar discs
near the SgrA* black hole in the Galactic Center. In particular, we show how
the masses and spatial distribution of the young stars, and the total mass in
the Galactic Center discs can be explained. However, explaining the origin of
the several young stars closest to the black hole (the S-stars) is more
problematic: their orbits are compact, eccentric, and have random orientation.
We propose that the S-stars were born in a previous starburst(s), and then
migrated through their parent disc via type I or runaway migration. Their
orbits were then randomized by the Rauch-Tremaine resonant relaxation. We then
explore the consequences of the star-formation scenario for AGN discs, which
are continuously resupplied with gas. We argue that some compact remnants
generated by the starburst will get embedded in the disc. The disc-born
stellar-mass black holes will interact gravitationally with the massive
accretion disc and be dragged towards the central black hole. Merger of a
disc-born black hole with the central black hole will produce a burst of
gravitational waves. If the central black hole is accreting at a rate
comparable to the Eddington limit, the gas drag from the accretion disc will
not alter significantly the dynamics of the final year of merger, and the
gravitational waves should be observable by LISA.
