We study identities for the distribution of the number of edges at time t back (i.e. measured backwards) in a coalescent tree whose subtrees have no mutations. This distribution is important in the infinitely-many-alleles model of mutation, where every mutation is unique. The model includes, as a special case, the number of edges in a coalescent tree at time t back when mutation is ignored. The identities take the form of expected values of functions of Zt=eiXt, where Xt is distributed as standard Brownian motion. Associated identities are also found for the distributions of the time to the most recent common ancestor, the time until loss of ancestral lines by coalescence or mutation, and the age of a mutation. Hypergeometric functions play an important role in the identities. The identities are of mathematical interest, as well as potentially being formulae to use for numerical integration or simulation to compute distributions that are usually expressed as alternating-sign series expansions, which are difficult to compute.
