When Renu Kansal opened Bollywood West dance studio three years ago, the “slumdog” was not yet a millionaire, but Indian beats and melodies had already begun to seep into the American mainstream via Jay-Z’s “Beware of the Boys,” Britney Spears’ “Me Against the Music” and The Black Eyed Peas’ “Don’t Phunk with My Heart,” which featured samples of two 1970s Bollywood songs.