You don't need any documentation. The only time you would need a birth certificate is if you were flying with them as a lap child. I just flew 3 southwest flights with my four year old and he was only asked what his name was by TSA. Didn't have to show anything.
Do you need a notarized letter to travel with a child alone? In most cases of a single parent traveling out of the country with their child, the answer is yes you do need a notarized consent to travel letter. You can download your own customizable travel consent letter here.
Children under 18 don't need an ID to fly domestically but do need a passport for international flights.
The letter of consent, preferably in English and notarized, from the other parent (if the child is traveling with one custodial parent) or signed by both parents (if the child is traveling with a guardian or alone) should say: "I acknowledge that my child is traveling outside the country with the name of the adult ...
TSA does not require children under 18 to provide identification when traveling within the United States. Contact the airline for questions regarding specific ID requirements for travelers under 18.
In the USA, though, unless there's a court order requiring permission from both parents, a child may travel (even to overseas) with either parent/guardian without showing proof of permission from the other parent.
Checklist for Flying with an Infant Birth Certificate. Passport (if flying internationally). Global Entry Card (if you use that service). Proper Clothes. Blanket. Breastmilk or Formula (plus accompanying bottles and s). Diapers, wipes, and changing pad. Hand Sanitizer (wipes or mini bottles)