Convolution sums of some functions on divisors
One of the main goals in this paper is to establish convolution sums of functions for the divisor sums σ̃s (n) = Σd/n (-1)d-1ds and σ̂ s = Σd/n(-l)(n/d)-1ds, for certain s, which were first defined by Glaisher. We first introduce three functions P(q), E(q), and Q(q) related to σ̃(n), σ̂(n), and σ̃3(n), respectively, and then we evaluate them in terms of two parameters x and z in Ramanujan's theory of elliptic functions. Using these formulas, we derive some identities from which we can deduce convolution sum identities. We discuss some formulae for determining r s(n) and rs(n), s = 4, 8, in terms of σ̃(n), σ̂(n), and σ̃3(n), where rs(n) denotes the number of representations of n as a sum of s squares and δs(n) denotes the number of representations of n as a sum of s triangular numbers. Finally, we find some partition congruences by using the notion of colored partitions. Copyright ©2007 Rocky Mountain Mathematics Consortium.
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