Pillow Talk is the 1959 movie starring Doris Day, Rock Hudson and Tony Randall. This was the first of three films with these three acting together. “Lover Come Back” and “Send Me No Flowers” followed.
 
Funny thing, but Rock Hudson supposedly turned this film down three times, thinking it was too risque. 
 
Doris Day and Rock Hudson are Jan Morrow and Brad Allen, unknown to each other except by a party-line phone line. Back in the day, phone lines were not always private, and customers had to share lines with complete strangers — often called a party-line. This allowed intentional or unintentional eavesdropping. If you picked up your phone to make a call and you could hear an on-going conversation, then you had to wait to place your call. 
 
Doris Day overhears Rock Hudson playing the piano and singing the same song over and over, each time to a different female. Jan becomes irritated that her phone line is always tied up with Brad Allen’s wooing of his females. — “You are my inspiration, Eileen.. “ or Yvette.. or any number of other females.

Brad Allen does not realize that Jan Morrow is quite the femme fatale until he sees her. To make his move, he pretends to be Rex from Texas.

Thelma Ritter always adds a comedic touch to these films. Tony Randall gets knocked it one scene, and in the filming the actor taking the swing at Tony actually hit him and knocked him out — they kept that in the film.

Tony Randall, Doris Day and Rock Hudson:
The party line — Jan Morrow overhears Brad Allen singing his one famous song:

This is the Roly Poly scene, Rock Hudson as Brad Allen, pretending to be Rex: