in and .

Moreover, the long-term correlation studies between radio/optical and gamma-ray bands were already performed by @2014MNRAS.445..437M and @2014ApJ...797..137C using similar radio and optical datasets as those presented in this analysis. Therefore, we only attempt to study the long-term radio-optical cross-correlation behaviour of the source together with its optical polarization behaviour.

### Radio-Optical cross-correlation analysis {#3-4-1}

Figure \[fig\_longterm\] illustrates the long-term optical and radio data of S2 0109+22. The coverage is of 12 years in the optical band (R-band and open filters) and at 37GHz, and 10 years at 15GHz.

Following , we calculated the cross-correlation function between the optical and 15GHz LCs using the Discrete Correlation Function [DCF; @1988ApJ...333..646E] with local normalization [LCCF; @1999PASP..111.1347W]. We use temporal binning of 10 days and require that each LCCF bin has at least 10 elements. Following @2014MNRAS.445..437M, the significance of the correlation is estimated using simulated LC. In the simulations, we used a power spectral density index of -1.8 for the radio LC (Max-Moerbeck et al. in prep.), which is slightly smaller than the values between -1.4 and -1.7 reported in @2004MNRAS.348.1379C for the (8 to 37GHz) radio LCs. For the optical, we used a power spectral density index of -1.5 