single time series per participant and using random effects in mixed effect modelling My mixed effect model design involves assessing 42 different time series via a polynomial function. that is, fixed effects of time and time^2. I've incorporated a random slope of the individual time series. 
My hypothesis is effectively does each time series fit a quadratic curve when I allow the coefficients to vary. 
My question for here is, is there any issue with incorporating a random effect in this manner. Most documentation assumes multiple time series per random effect and I can't find much on if using it as I have is an issue? 
 A: This seems like a reasonable approach. The grouping factor for the random effects is the individual time series ID, and you have 42 of these. Each time series consists of repeated measures over time, and you are going to estimate fixed effects for an intercept, a linear term and a quadratic term. By specifying random intercepts, you allow each time series to have it's own intercept, and by fitting random slopes for the linear term and quadratic term you allow the shape and location of the curve to be different for each time series.
If the variances of the random effects are meaningfully above zero then this would be consistent with your hypothesis. Obviously they don't all have to be above zero and if one or more of them is close to zero then you may be justified in removing it/them. 
When you say "multiple time series per random effect" this implies that the time series' are nested within some other factor - such as subjects, which doesn't seem to be the case here, but if it was, then you would just specify the nested structure appropriately 
