  Ramanujan gave a recurrence relation for the partition function in terms of
the sum of the divisor function $\sigma(n)$. In 1885, J.W. Glaisher considered
seven divisor sums closely related to the sum of the divisors function. We
develop a calculus to associate a generating function with each of these
divisor sums. This yields analogues of Ramanujan's recurrence relation for
several partition-theoretic functions as well as $r_k(n)$ and $t_k(n)$,
functions counting the number of ways of writing a number as a sum of squares
(respectively, triangular) numbers. As by-products of this association, we
obtain several convolutions, recurrences and congruences for divisor functions.
We give alternate proofs of two classical theorems, one due to Legendre and the
other -- Ramanujan's congruence $p(5n+4) \equiv 0 \pmod 5$.
