The region of space very close to a black hole can be a very messy environment. Mathematically it is a very simple region dominated by the black holes' outer boundary called the event horizon. Matter is trying to flow into the black hole at the equator, and through friction, is being heated to thousands and perhaps even millions of degrees. Magnetic fields dragged in by the flow get amplified and concentrated. They eventually pop out of the gas disk like solar prominences and flares, releasing bursts of x-ray, and perhaps even gamma ray, energy. Clumps of clouds and asteroids orbit the black hole in seconds and are shredded by gravity, to produce flickering 'Quasi-Periodic' bursts of x-ray light as they slide into the horizon zone and are finally lost from our universe. Does fusion happen just outside a black hole? Not very easily. To produce thermonuclear reactions you need temperatures in excess of about one million degrees to cause protons to collide with deuterium and produce tritium. Deuterium fusion is the lowest energy fusion reaction we know about. To get to one million degrees, protons have to collide with deuterium nuclei at speeds of about 290,000 miles per hour. Near stellar-mass black holes, gravitational tidal forces are substantially higher and it might be possible for some of the gas to reach these kinds of conditions, at least in a limited volume of space near the horizon, perhaps even in 'solar' flares that pop out of the magnetized accreting matter. A signature of this would be a black hole emitting bursts of gamma rays.
