@nigul has implemented periodic one https://github.com/stan-dev/math/pull/677
It would be good to check that what you and he have been doing has similar structure.
For some of my experiments I would like to have dot-product (e.g. p. 80 in GPML book). It corresponds to integrated linear model with Gaussian prior on coefficients, but sometimes it’s handy to include that part in the GP covariance instead of modeling the linear model part explicitly.
Matern would be good choice in general. It would probably be enough to have Matern with nu=1/2 (same as exponential), nu=3/2, and nu=5/2, as then we don’t need to compute modified Bessel function (see e.g. p. 84-85 in GPML book). Of course there could be cases where nu could be continuous and unknown parameter, but I don’t think that kind of applications where nu would be well specified and GP would be fast would be common.
The set of exp_quad, periodic, dot-product and Matern would be that good, that I would then start thinking how to speed-up GPs (fast basis function approximations for low dimensional cases D<=2, inducing point approximations for moderate dimensional cases D>=2, approximative integration over latent values).