MacMahon in the paper [Divisors of Numbers and their Continuations in the Theory of Partitions](http://plms.oxfordjournals.org/cgi/reprint/s2-19/1/75.pdf) defines several generalized notions of the sum-of-divisors function; for example, if we write $a\_{n,k}$ for the sum
$$
\sum s\_1 \cdots s\_k
$$
where this sum is taken over all ways of writing $n = s\_1m\_1 + \cdots s\_km\_k$ with $m\_1 < \cdots < m\_k$ (note the asymmetry in $s\_k, m\_k$), he then studies the generating functions
$$
A\_k(q) = \sum\_{n=1}^\infty a\_{n,k}q^n
$$
for fixed $k$, as well as a number of other variants. Note that for $k=1$, this is nothing but the generating function for the ordinary sum-of-divisors function.


These functions (Specifically, from his paper, the functions $A\_k$ and $C\_k$) have arisen in my research and I would like to know what literature there is on them. In particular, I would like to know if there are any well-known identities that hold between them. MacMahon himself lists the identity
$$
A\_2(q) = \tfrac{1}{8}\sum\_{n=1}^\infty\big(\sigma\_3(n) - (2n-1)\sigma\_1(n)\big)q^n
$$
as well as similar ones for $A\_3$ and $A\_4$, but the identities that I am looking for are more in line with an attempt to write these as quasi-modular forms, if possible. For example, it turns out that the following is true:
$$
A\_2(q) = \tfrac{1}{10}\Big(3A\_1(q)^2 + A\_1(q) - q\frac{d}{dq}A\_1(q)\Big)
$$
and I conjecture that you can always write $A\_k(q)$ (and similarly $C\_k(q)$) recursively in terms of previous such functions.


So my question is: What literature is there on these functions? Is the relation above a well known one?



What literature is known about MacMahon's generalized sum-of-divisors function?
For anyone who comes by this later, it turns out that the following relationship is true for the functions $A\_k(q)$:
$$
A\_k(q) = \frac{1}{(2k+1)2k}\Big(\big(6A\_1(q) + k(k-1)\big)A\_{k-1}(q) - 2q\frac{d}{dq}A\_{k-1}(q)\Big)
$$
After contacting George Andrews (as suggested), he and I wrote a joint paper proving this result (as well as a few other related ones), which can be found at


<http://arxiv.org/abs/1010.5769>


