I have been following this thread for a while and fail to understand the point of the question:
If the reference is to a person, it is likely to differentiate him/her from others, and is a point of reference for purposes of identification: e.g.. "The Bialystoker came to visit last night"
If it is in reference to an institution, it may refer to the NYC Landmark known as the Bialistoker Synagogue at 7 -11 Bialystoker Place (previously 7 Willett Street) on the Lower East Side, it was a very well known synagogue, is restored (1988), operative and there is information and photos on-line, including a Wikipedia article.
As a word, Bialystoker means of or from Bialystok, a city in North-East Poland .
In order to be of further assistance, context is important.