Wikipedia says "That'll Be the Day" was written by Buddy Holly and Jerry Allison. It was first recorded by Buddy Holly and the Three Tunes for Decca in 1956, but Decca did not release that recording. It was re-recorded on February 25, 1957 by Buddy Holly and his new band, The Crickets. And it gets even more convoluted and confusing. Read the Wikipedia article. It's an interesting insight into the music industry.