D to F in key of C: Secondary Dominant, or something else?

Question

In the Cee Lo Green song “Forget you”, the chord progression is C - D - F - C, the D has an F# which is out of key.

Within the limited music theory I learned so far, the closest thing I can relate this to is secondary dominant? Like C - D - G - C, the second D major chord is out of key and if you read it literally, it’s a D major chord, but really it functions like V/V, five of the five chord, which is G, which resolves to the next G chord, then back to C.

Is this C - D - F - C progression similar to C - D - G - C?

(UPDATE)

After reading answers and other resources I don't think the D is functioning as a secondary dominant in this particular case

Answer 1

IF C is the tonic, then you probably want to consider @AlbrechtHugli's point in comments.

If you omit the D7 you get something very, very familiar C - F - C. You could call that a kind plagal progression, basically it just shifts back and forth from tonic to subdominant.

Functionally that doesn't go anywhere. It just prolongs the tonic.

Interposing the non-functional D7 in that prolongation doesn't really change what is going on: an elaboration of a tonic.

The root movement by step from C to D creates a nice linear movement but that isn't the same as actual harmonic function.

Back to @AlbrechtHugli's point, some call this a Lydian progression because of the raised fourth degree. That particular wording Lydian progression suggests a kind of borrowing of chords from the Lydian mode. Borrowing like that is more about color (chromaticism beyond the diatonic) without those chromatic tones having actual function, the critical one being chromatic tones which become temporary leading tones.

If you want to label it with Roman numerals, you can use a capital letter for the major quality and add a seven: C: I II7 IV I

Answer 2

You're correct, it is a secondary dominant or V/V chord. It does function similarly to the Dm or ii chord, except that the non-diatonic F# tone provides a stronger leading tone to the G than the diatonic F natural in the Dm chord.

However, it should be pointed out that although that is the V/V chord, it does not resolve to the dominant, but instead to the subdominant. So I suppose the argument could be made that it serves more of a generic predominant borrowed chord function. And the plagal cadence from F to C differs from the ordinary G-C cadence also. At any rate, those are the differences between this progression and C-D-G-C.