Glaisher's divisors and infinite products

TL;DR

This paper explores divisor sums related to Glaisher's work, developing generating functions that lead to new recurrence relations, identities, and proofs for classical theorems in partition theory and divisor functions.

Contribution

It introduces a calculus linking divisor sums to generating functions, producing analogues of Ramanujan's recurrence relations and new proofs of classical theorems.

Findings

Derived analogues of Ramanujan's recurrence relations for partition functions.

Established new convolutions, recurrences, and congruences for divisor functions.

Provided alternative proofs for Legendre's and Ramanujan's classical theorems.

Abstract

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 $p(5n+4) \equiv 0 \pmod 5$.