The Alameda County Coroner's Office tells TMZ ... Frankie's official cause of death was determined during her autopsy, which also revealed her manner of death as an accidental overdose.

As we reported ... she died on her birthday back in July after a several-decades-long battle with addiction. Frankie's brother, Sam, told us she overdosed at her Oakland home where she'd been partying for her birthday.

Sam had told us he checked in with her daily, trying to help her stay clean. Her battle was well documented ... and one Keyshia & co. shared publicly in recent years on TV, social media and other channels.

Frankie was regularly featured on Keyshia's BET reality show "Keyshia Cole: My New Life." The duo had many heart-to-heart moments ... and Keyshia was quite vocal about her mother's addiction struggles.

Keyshia was given up by Frankie and her biological father, Virgil Hunter, for adoption at a very young age. The singer was raised by family friends. Mother and daughter reunited when Keyshia was an adult ... and after she'd hit it big in music.