Generalizations of the Bellavitis partition identities

Let p = p(a, b, c) be the number of partitions of a into b parts, no part exceeding c. Bellavitis and perhaps some earlier writers noted that p satisfies three very simple identities. Here p is generalized to a function of k + 1 variables in a natural way. One of the identities then generalizes; the proof of this (which depends on the P. Hall commutator collecting process) is given only for k = 3.