In 1978, at one of his concerts, Frank Sinatra performed “Theme from New York, New York.” He did this as a reference to his own song “New York, New York,” which was at the time still far more famous than “Theme from New York, New York.” Without Sinatra having his own previous “New York” song, his choice to cover this recent movie theme would have been absurd. This new song was written to be performed by a flamboyant 30-year-old Liza Minelli, not a 60-year-old Frank Sinatra.
Nonetheless, people loved his version. A couple years later, he recorded it for an anthology album, and it went on to become a major New York anthem. A fair number of people know the Sinatra version without knowing Liza Minelli ever sang it. Even more people know it without knowing about Sinatra’s other “New York, New York” song, and still more people know it without having watched either New York, New York or On the Town.
Below is a video from 1982 of Sinatra doing a medley of the two different “New York, New York”s, but he wouldn’t make a habit of that, as the newer song would eventually far outstrip the older one in popularity.