I encountered some of these suckers in this solo piece I've been noodling around with, Variations on a March of Shostakovich by Arthur Frackenpohl. Here's what they look like:
The directions in the music say, in Courier New font: "half-valve type sound, slap palm on mouthpiece or snap fingers, etc". I have been working at trying to make a "half-valve type sound", mostly by wiggling my trigger while trying to play these notes. I've found that a F in the staff with a trigger half-pulled makes a really unusual sound, but I do have a hard time accenting it. I may settle for a Joe Alessi-style yell or maybe make some linguistic clicks.
My music dictionary tells me:
A muted or dampened notes that has rhythm but often no discernible pitch. It is often thought of as an implied note in a musical phrase and can be not performed or performed only faintly for effect. (...) The false note is indicated by a (parenthesis) around the notehead or with an "X" in place of the notehead, indicating that the note is to be played very quietly, as a ghost.
Trombonists have another use for the term "false note", although I first heard of them as "frog tones". This term refers to a sort of "ghost partial" between low E and pedal Bb played without the trigger. You can sort of make the notes half exist, with a thin sound and overtones that feel a little strange, and you can lip these notes to approximate the pitch. Some more advanced trombonists can produce false notes on and around a triple pedal Bb. Other brass instruments can do this too, as evidenced by the YouTube video below of a guy playing what sounds like the Meow Mix theme song on false notes on a tuba.