Hi Cheryl, do you need help researching Luka? There's a naturalization record for "Lewko Kokura" in Illinois that states he arrived on 3 July 1913 in Detroit. Is this your Luka?
Most Ukrainian records (birth/death/baptism/marriage) will be found in the metrical records for each oblast (in this case, Ternopil) in the State Archives of Ukraine. Due to the situation in Ukraine right now, I don't know how or if it's available for research. If possible, I recommend hiring a local Ukrainian genealogist who can research in the archives and translate their findings for you.
The LDS copied some Catholic records from this region, they're on file in some Family History Libraries run by LDS, but I don't know much about how to access them.
