Does nucleosynthesis happen near a black hole or during its formation? During its lifetime, a star can synthesize elements to iron because of the pressure and temperature. A neutron star is composed of iron nuclei 56 and up to krypton 118.

According to that, a black hole (which concentrates matter to an increasingly tiny space during infall) must also have an unimaginable pressure/temperature during its formation and as additional matter falls into it.

Can it therefore synthesize new elements?
After a google search the answer is yes, in theory. There is nucleosynthesis associated with the Black Hole accretion disks, see here, here and here. These sources look at the nucleosynthesis in the hotter parts of the accretion disks surrounding black holes, and how this is effected by the neutrinos produced.

It is also proposed that its possible for very small mass (10e-9 Mass of sun) black holes to be devoured by a neutron star (see here) and produce many of the heavier elements. This method starts with primordial black holes which can be of the required mass compacted into a region the size of a single atom. This tiny black hole could crash into a neutron star and initiate the r-process for nucleosynthesis.