Does the father live with you and the child? If so, he can claim the child, since he is related and meets the other rules for a Qualifying Child. He needs no paper work form you. The issue of the birth certificate, will only be a problem if the other father tries to claim the child and that is unlikely since you are the custodial parent. If the IRS challenges it (unlikely), then you'll have to go to court to establish paternity.
If the father does not live with you, see macuser22's answer.
