The Crimson Binome
Hero
Since all Persuasion checks are based on Charisma, just like all Stealth checks are based on Dexterity, you don't have to spell it out. The rule books will usually spell it out every time, because that's the level of formality that they've agreed on, but it's not strictly necessary (or useful).Sorry, I'm still getting used to 5th edition parlance! I started with 4th edition.
So it would be Charisma (Persuasion) and Charisma (Performance) checks.
Note, also, that there is a variant rule about using some skills with different ability scores, like using Stealth with Charisma in order to hide in a crowd. As with most variant rules, these are rarely brought up for discussion, since there's no way to know what variant rules might be in effect for any given campaign. If someone is using a variant, then they will write out the whole thing as a Stealth (Charisma) check, but in the absence of clarification it's safe to assume that default abilities are in use.