The Lydian #2, like the Lydian and Ionian modes, have a major sound to them because of the major 3rd and major 7th. Because this mode has the augmented 2nd (same as the minor 3rd), this has a bluesy sound to it because the blues scale has a minor 3rd. The augmented second can also be thought of as a sharp 9, but this note isn't used very much in major chords.
|--------Octave------||----Extensions----|
1-#2-3-#4-5-6-7-8-#9-10-#11-12-13
Here are some of the chords we get by stacking 3rds from this mode:
1-3-5= Major Triad
1-3-5-7= Major 7th
1-3-5-7-#9= Major 7#9
1-3-5-7-#9-#11= Major 7(#9#11)
1-3-5-7-#9-#11-13= Major 13(#9#11)
Other possible chords:
1-3-b5=Major triad b5(b5 is enharmonic of #4)
1-3-b5-7= Major 7b5
1-3-5-7-#11= Major 7#11
1-3-5-6= Major 6
1-3-6-7= Major 6/7
1-3-6-#11= Major 6#11
Example 1 is just a one chord Amaj7 vamp. The "meat and bones" of this mode are in the #4, 3 and 7. The #2 will give a bluesy sound to it so try bending this note up a half step towards the natural 3rd. For example 2, stress the #4 note in the chord, so you can hang on this note in your solo as well.
Vamp #1:(Amaj7/Amaj7) (Use A Lydian #2.)
Vamp #2:(Emaj9/Emaj7b5/Emaj9/Emaj7) (Use E Lydian #2.)