I'm not sure of the exact birth date of Barack Obama Sr., although I have seen that the birth certificate for Jr. and some of the immigration documents for Sr. do not match. But I don't see how this makes Jr's birth certificate fraudulent as a record of his birth.
If Sr's age as recorded on the birth certificate is wrong, there could be various possible explanations, one of which is that he lied about his age to Stanley Ann. He seems to have each of two different dates on several of the immigration documents. Perhaps Sr. was a "slippery character", or perhaps he was one of those people from foreign countries who are not sure of their real birthdate. But this has nothing to do with whether Jr was born in a certain date and place.
Did you know that as late as the 1970s it was common throughout the nations of the British Commonwealth to REUSE birth reporting forms ~ just erase the old stuff and put in new stuff.
What do you imagine it was like in East Africa back in the Great Depression even BEFORE the Mau-Mau!?!?!?!
There were millions of people living in that part of the world who were just then grasping the concept of Solar Year and the dude was a Moslem and they were using a Lunar Calender and Lunar Year.