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MacMahon in the paper Divisors of Numbers and their Continuations in the Theory of Partitions 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?

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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


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I don't know, but if you type $$\rm macmahon\ divisors\ continuations$$ into Google you'll find a number of papers by George Andrews, and one by Jim Huard, citing the MacMahon paper.

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Andrews might be a good person to ask directly; from reading some of his papers it looks like he's really interested in MacMahon's work. –  Qiaochu Yuan Oct 7 '10 at 22:27
I've already downloaded and skimmed over two of those, "stacked lattice boxes" and "partitions with designated summands" which don't seem to really help; they're mostly about the case where you simply count the number of such 'divisor partitions', instead of summing them. I could try email him though. –  Simon Rose Oct 7 '10 at 22:37

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