Yeah, we have his birth certificate. Unlike some states Hawaii doesn't hand it out to just anyone, because we're not fond of identity theft.
Furthermore, even if he were born in Kenya, his mother's nationality would be taken into account and she could easily have arranged to have him get US citizenship. In most cases, the child of a US citizen can get US citizenship on that basis even if born elsewhere.
But that's all moot since he was born in Hawaii.