En route Controller described a TCAS RA event between an IFR and VFR aircraft within TRACON's airspace. The IFR was handed off and transferred to the Center; the reporter notes the conflicting traffic should have been issued by the TRACON.

Narrative

I was training on the RADAR controller position. I accepted a hand off on a Dash 8 from Approach when the aircraft was leaving 8;000 or 9;000 FT if I remember correctly. Inside of Approach's airspace was a VFR aircraft indicating 11;500 MSL of unknown type northbound and in direct conflict with the departing Dash 8 flight. The aircraft remained on Approach's frequency for a short time before checking on; which I assumed was because Approach was issuing the traffic. When the aircraft initially checked on he was climbing out of 10;000 for 12;000. I acknowledged the aircraft and issued a climb clearance to FL190. After the read back from the Dash 8 flight I issued the traffic of the VFR northbound at 11;500. The flight responded that they were responding to an RA to which I acknowledged. As soon as the conflict was clear; I advised the aircraft that the traffic had passed behind and was no factor. When asked; the flight stated that Approach never issued them any traffic in regards to the VFR aircraft. I believe this could have been avoided had approach issued the traffic to the flight before switching him to Center frequency. In hindsight I should have issued the traffic first before any sort of control instruction.