Glaisher's Divisors and Infinite Products

Ramanujan gave a recurrence relation for the partition function in terms of the sum of the divisors function sigma (n). In 1885, Glaisher considered seven divisor sums closely related to the sum of the divisors function. We develop a calculus to associate a partition generating function with each of these divisor sums. This yields analogues of Ramanujan's recurrence relation for several partition-theoretic functions as well as rk(n) and tk(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) equivalent to 0 (mod 5).