<p>This study presents an analytical investigation into the interaction dynamics between a q-Gaussian periodic chirped laser pulse and free electron in vacuum, aimed at optimizing energy gain. q-Gaussian profile is a generalized non-Gaussian beam structure. The q parameter is crucial in defining the laser envelope as it offers enhanced tunability in the transverse intensity distribution via q parameter. The introduction of a periodic frequency chirping term into the laser phase systematically modulates the temporal variation of instantaneous frequency. This modulation improves the overall energy exchange efficiency. Analytical formulations are established through the resolution of the relativistic equations of motion for an electron interacting with a radially polarized q-Gaussian periodic chirped laser field. The equations are normalized and solved for electron energy gain as functions of the laser and chirp parameters. The examination of key parameters, including chirp amplitude, chirp frequency, and q-parameter, on the energy transfer process is conducted in detail. The periodic chirping has been noted to introduce a sustained energy transfer in contrast to traditional Gaussian or linearly chirped pulses.</p>