In the DR anyone is allowed to "give" his last name to another person through a name change process. In this case I guess you would be giving the child your last name.
The name change process would be as follows:
1) The person has to file a name change application to the executive branch through the Central Electoral Board. This application has to specify: a) Reason for the name change, b) Copy of the person?s birth certificate and any other document which may support the application.
2) If the executive branch considers the application has merits, it will authorize the applicant to: A) Insert an excerpt of his name change request in the Official Gazette of the Dominican Republic inviting anyone who may oppose to the name change to file their requests. This excerpt should also be published in a national circulation newspaper. B) Require a bailiff to post a legal announcement of the name change, with the excerpt, in the door of the Peace Court of the municipality of both the place of birth an actual residence of the applicant. This announcement should be posted for at least 60 consecutive days and would be certified by the bailiff himself at the bottom of the document.
After this time has passed and in order to have his name change decree dictated, the applicant would have to present the Attorney General of the Dominican Republic the following: a) An original copy of the announcement, with is date and duration certified by the bailiff and b) A copy of the Official Gazette in which the excerpt was published.
(These documents would be sent to the executive for their approval)
If approved, the executive would dictate a name change decree which would have to be published in the Official Gazette and in a national circulation newspaper.
To inform yourself about the recognition of a name change in NY state you would have to seek direct counsel there.
