In plain English, you would need clear evidence, in a document or via credible witnesses, that your father acknowledged you as his child during your father's lifetime. If you can get the evidence, then the court can order a DNA test (assuming that your father's remains were not cremated).
The above represents the only way to prove you have any rights to your father's estate. The legal expenses related to the petition to the NY Surrogate's Court would be costly. You could expect to pay at least $5,000 to get the DNA order, and perhaps another $5,000 to actually exhume your father's remains, obtain the test information and deliver it to the court for judgment.
Consequently, unless the estate is large, there is little point in pursuing the matter. If the estate is large, and if you would like a link to a competent lawyer referral service, please let me know and I will be happy to provide.
And, if you need further clarification or assistance please let me know that, too.
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