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 $\widetildeσ_s(n)=\sum_{d|n}(-1)^{d-1}d^s$ and $\widehatσ_s(n)=\sum_{d|n}(-1)^{\frac{n}{d}-1}d^s$, for certain $s$, which were first defined by Glaisher. We first introduce three functions $\mathcal{P}(q)$, $\mathcal{E}(q)$, and $\mathcal{Q}(q)$ related to $\widetildeσ(n)$, $\widehatσ(n)$, and $\widetildeσ_3(n)$, 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 $δ_s(n)$, $s=4,$ $8$, in terms of $\widetildeσ(n)$, $\widehatσ(n)$, and $\widetildeσ_3(n)$, where $r_s(n)$ denotes the number of representations of $n$ as a sum of $s$ squares and $δ_s(n)$ 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.

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