The fractional coalescent

A new approach to the coalescent is introduced. The fractional coalescent (f-coalescent) extends the Kingman9s n-coalescent allowing more variability in the patterns of the waiting times. These changes in the waiting times could be caused by population size changes or selection processes. The f-coalescent is derived by applying fractional calculus, in particular, the fractional Poisson process which itself is based on the expansion to the fractional waiting times of the conventional Poisson process. The f-coalescent introduces an additional parameter alpha; affecting the steepness of the waiting time function. With an alpha = 1, Kingman9s n-coalescent and the f-coalescent are equivalent. The f-coalescent has been implemented in the population genetic model inference software MIGRATE. With an alpha = 1, as expected, both, the n-coalescent and f-coalescent implementation deliver the same results. When either data is simulated under models that allow for alpha