Colpitt oscillator transfer function
In order to derive the transfer function of a Colpitts oscillator, you’d need to use the small signal model of the transistor since it’s operating as an amplifier. Then use circuit analysis to get Vout in terms of Vin, then divide Vout equation by Vin and simplify. To help consider the link below where they derive equations for a coltpitts oscillator using nodal analysis. Please note that you’ll have to include your components to the the model that are not included in the circuit in the link. Also you’ll have to somewhat diverge from their analysis at some point as they are finding as somewhat different transfer function. 

Link: https://www.seas.upenn.edu/~ese319/Lecture_Notes/Lec_19_Colpitts_Osc_09.pdf