Tower controller experienced loss of separation event when failing to remember a touch and go operation on Runway 15 as air carrier was landing Runway 8.

Narrative

Aircraft Y was in the pattern; left downwind for Runway 15. I extended the downwind to allow spacing for a departure on Runway 15. Once I determined spacing was adequate from the departure; I instructed Aircraft Y to turn base; and cleared him for touch and go. Aircraft Y turned base; then final approximately 3 miles out. At that same time; Air Carrier X was on approximately 7 mile final to Runway Z; an intersecting runway. At the time Aircraft Y turned his base to final; I determined there would be adequate spacing for Aircraft Y to complete his touch and go and be through the runway intersection before Air Carrier X landed on Z; as per the required separation. At this point I became distracted with making several traffic calls to other aircraft in the airspace. Once I completed making traffic calls and frequency changes; I scanned my RACD (Remote ARTS Color Display); and did not see the tag for Aircraft Y; as it had already dropped into the arrival list since he was on short final. I went back to make another traffic call; and as I looked out the window; I saw Air Carrier X just touching down on Runway Z; just as Aircraft Y was starting his climb and passing through the runway intersection. As Aircraft Y went through the intersection; he was climbing at approximately 200 feet AGL; and Air Carrier X was approximately 2000 feet prior to the intersection on his landing roll. I flat out forgot about Aircraft Y; and became distracted with other duties. I didn't see him on my RACD because I checked too late; and I did not see him out the window until it was too late; as he was a small aircraft; and small aircraft can be difficult to see while at the approach end of the runway. Recommendation; the best thing I can do to prevent the recurrence of this event; is to make some sort of memory jogger for myself; perhaps on my pad; to remind me that I have an aircraft in the pattern.