A corporate jet descending for an ATC requested expedited descent from FL470 to FL290 experienced a TCAS TA at FL343. The reporter acknowledged the descent may have been at a slower rate than expected by ATC.

Narrative

I was PIC flying west in the vicinity of ZZZZZ intersection. I received descent instructions out of FL470 for what I remember was for FL290. The Center requested that I expedite below FL340. I increased my rate of vertical speed. The TCAS advisory (TA) sounded and I noted that I had about seven NM separation and 700 FT vertical on oncoming (one o'clock) traffic. At that data point I was at 34;300 FT and the traffic was at FL350. Simultaneously; Center told an aircraft to turn left to 075 degrees. I believe it was an air carrier flight; but I cannot be certain about the call sign. Whoever it was; they were probably cruising east at FL350. I take responsibility for not expediting as fast as the controller needed me too. I remember thinking that I need to hurry; but that I did not need to bounce anybody off the cabin ceiling. I should have used 1200 FPM down on the initial start of the descent (slowly increasing to 2400 FPM) approaching FL290. Early on in descent; I was expecting to hear Center tell me something like 'Cross 60 NM east of ZZZ below FL340.' At the relevant time; the SIC was trying to call ahead to our destination and to also obtain ATIS. I recall paying some attention to how that was going. However; in my opinion; the controller was conscientious about his traffic separation and relied on what he knows to be the 'norm' for his descents.