Burn it (page 31)
Fire fascinates you. Your spells and alchemical items that deal fire damage gain a conditional bonus to damage equal to half the spell’s level or the item’s level (minimum 1). You also gain a +1 conditional bonus to any persistent fire damage you deal, whether it’s from a spell or not.
Alchemist Fire (page 360)
Alchemist’s fire is combination of several volatile liquids that ignite when exposed to air, typically stored in a sealed flask. Alchemist’s fire deals 1d8 fire damage, 1 persistent fire damage, and 1 fire splash damage. The target can end this persistent damage by spending an Interact action or by becoming submerged in water or otherwise entering an area deprived of air. A creature adjacent to the target can also end the persistent damage by spending an Interact action.
I was talking with another member of my group and I mentioned that alchemists can eventually hit 15 persistent fire damage on alchemist fire (level 19 bomb from empower bombs + 1/2 level from burn it; 6+9).
They said that I was calculating it wrong and that burn it only gives the +1 persistent damage. To this I pointed out that the persistent damage is from alchemist fire and thus the 1/2 level should apply, we left it alone after that.
Is was my interpretation correct?