A Frank Sinatra recording of "New York, New York" sent the song into the stratosphere.
Copyright restrictions prevent reproducing the full lyrics here, but they do appear on other web sites.
New York City has no official song that's declared as such by the City Council. However, "New York, New York" was declared the official song by Mayor Ed Koch in 1985, so it's close to being official.
20 February 1985, New York Times, pg. B4:
Singing the Official Song
"I'm king of the hill, top of the heap." Mayor Koch sang merrily, obviously pleased with his choice of "New York, New York" as the city's official song.
He proclaimed it that yesterday in the City Hall Blue Room and then handed certification of appreciation to its composer and lyricist, John Kander and Fred Ebb. "We're the most exciting place on earth," Mr. Koch told them.
9 June 1998, New York Times, "A City In Search Of an Anthem" by Clyde Haberman, pg. B1:
There it was, the true New York anthem, ever present: a car alarm.