When we suppose to use this special action button(right analog button)?

As far as I know, it's only used during one single event throughout the whole game. Depending on which storyline you go for, you might end up in a sort of tournament in a ring; you can choose to yield to the opponent in the fight by pressing the special action button ('F' on keyboards).

In other words, it's pretty much useless.

If you have to yield in the tournament arena, then I have no idea how you even made it that far.

As I've posted before, I believe it originally had much more use, but was hobbled in the final release.

Otherwise, I just can't see them dedicating the F key for just this one action.

You're right. In gameplay videos shown at E3 2010 you can see, that all context actions like jumping, opening doors or lighting torches were assigned to the F button. Then RED's invented their "one button for all" control scheme.