Possibly from central Poland (around Warsaw, according to family lore). His great-grandson's Ancestry DNA indicates matches other test-takers with the Lodz region.
He had immigrated to the US before 1900, as he is found in the NY census that year. I cannot find a record of a person with this name immigrating to NY, marrying in NY, or being born in Poland. Sadly, there have been no DNA matches with this last name either. There are lots of DNA matches on Ancestry, but none with this name anywhere in their tree.
In 1900, he is already married and living with his in-laws. In 1905, there is a census record of him living with his known wife and children, plus a brother named Stein Dembuski [sic] (maybe Stan? Stanislaw?) in his home, listed as "Russia-Pole". Could possibly be a misunderstanding of "Rusyn"?
Death likely in Olean NY, as he lived his whole life in America there.
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D > Dembowski > Frank Dembowski
Categories: New York, Immigrants from Poland | Poland, Emigrant Unknown Birth Location | Olean, New York | Saint Bonaventure Cemetery, Allegany, New York | 1900 US Census, Cattaraugus County, New York | 1905 New York State Census, Cattaraugus County | 1910 US Census, Cattaraugus County, New York | 1915 New York State Census, Cattaraugus County | 1920 US Census, Cattaraugus County, New York | 1925 New York State Census, Cattaraugus County | 1930 US Census, Cattaraugus County, New York | 1940 US Census, Cattaraugus County, New York